Smooth and Nonsmooth High Dimensional Chaos and the Melnidov-Type Methods

SMOOTH AND NONSMOOTH HIGH DIMENSIONAL CHAOS AND THE MELNIKOV-TYPE METHODS WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENC...

Author: Jan Awrejcewicz | Mariusz M. Holicke

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SMOOTH AND NONSMOOTH HIGH DIMENSIONAL CHAOS AND THE MELNIKOV-TYPE METHODS

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A.

MONOGRAPHS AND TREATISES

Volume 41:

Synchronization in Coupled Chaotic Circuits and Systems C. W. Wu

Volume 42:

Chaotic Synchronization: Applications to Living Systems E. Mosekilde, Y. Maistrenko & D. Postnov

Volume 43:

Universality and Emergent Computation in Cellular Neural Networks R. Dogaru

Volume 44:

Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems Z. T. Zhusubaliyev & E. Mosekilde

Volume 45:

Bifurcation and Chaos in Nonsmooth Mechanical Systems J. Awrejcewicz & C.-H. Lamarque

Volume 46:

Synchronization of Mechanical Systems H. Nijmeijer & A. Rodriguez-Angeles

Volume 47:

Chaos, Bifurcations and Fractals Around Us W. Szemplińska-Stupnicka

Volume 48:

Bio-Inspired Emergent Control of Locomotion Systems M. Frasca, P. Arena & L. Fortuna

Volume 49:

Nonlinear and Parametric Phenomena V. Damgov

Volume 50:

Cellular Neural Networks, Multi-Scroll Chaos and Synchronization M. E. Yalcin, J. A. K. Suykens & J. P. L. Vandewalle

Volume 51:

Symmetry and Complexity K. Mainzer

Volume 52:

Applied Nonlinear Time Series Analysis M. Small

Volume 53:

Bifurcation Theory and Applications T. Ma & S. Wang

Volume 54:

Dynamics of Crowd-Minds A. Adamatzky

Volume 55:

Control of Homoclinic Chaos by Weak Periodic Perturbations R. Chacón

Volume 56:

Strange Nonchaotic Attractors U. Feudel, S. Kuznetsov & A. Pikovsky

Volume 57:

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science L. O. Chua

Volume 58:

New Methods for Chaotic Dynamics N. A. Magnitskii & S. V. Sidorov

Volume 59:

Equations of Phase-Locked Loops J. Kudrewicz & S. Wasowicz

Lakshmi - Smooth & Nonsmooth.pmd

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NONLINEAR SCIENCE WORLD SCIENTIFIC SERIES ON

Series A

Vol. 60

Series Editor: Leon O. Chua

SMOOTH AND NONSMOOTH HIGH DIMENSIONAL CHAOS AND THE MELNIKOV-TYPE METHODS Jan Awrejcewicz Mariusz M Holicke Technical University of Łód´z, Poland

World Scientific NEW JERSEY

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SMOOTH AND NONSMOOTH HIGH DIMENSIONAL CHAOS AND THE MELNIKOV-TYPE METHODS Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-270-909-7 ISBN-10 981-270-909-6

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Preface

This monograph is devoted to the development and application of Melnikovtype methods applied to high dimensional dynamical systems governed by ordinary differential equations. There is no doubt that the analytical approaches are more powerful than numerical ones, but the key problem regarding a study of nonlinear dynamical systems concerns a narrow band of examples taken from both real live as well as engineering systems dynamics which can be studied directly in a purely analytical way. This observation is clearly exhibited by an attempt to predict chaotic behaviour of high dimensional dynamical systems. Although the classical Melnikov’s technique have found various applications to predict homoclinic intersections, it is devoted only to the analysis of three dimensional systems (in the case of mechanics, they represent one-degree-of-freedom non-autonomous systems). This monograph contributes to nonlinear dynamical systems development mainly via two features. First, we show and illustrate a way to extend the classical Melnikov’s approach to study high dimensional dynamical systems. Second, our considerations include simple models of dry friction and hence, both stick-slip and slip-slip chaotic orbits occurrence can be analytically predicted. It should be noted that this part of research is very rarely reported in the existing bibliography even regarding one-degree-of-freedom non-autonomous dynamics. In Chapter 1 a role of the Melnikov-type methods in applied sciences (applied physics, mathematics and engineering) is described. The state-ofart of the original Melnikov and the so-called Melnikov-type approaches in various branches of sciences with emphasis on their advantages and drawbacks are illustrated and discussed. In addition, special attention is paid to the Melnikov (Melnikov-type) methods devoted to the analysis of both lumped and continuous mechanical, civil engineering and electrical systems.

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vi Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods

The critical review of the existing results in the field of classical Melnikov method application and beyond allows to discover a gap and limitation in the so-far applied techniques to predict homo- and heteroclinic bifurcation yielding chaotic orbits. First, it has been strongly emphasized that the Melnikov’s method belongs to the perturbational (asymptotic) techniques , and hence it is opened for further development and modification within the field of applied mathematics. Second, the critical overview of the research devoted to either application or extension of the Melnikov technique indicates a lack of modification in the method towards a study of higher order dynamical systems, i.e. systems with a few degrees-of-freedom in the language of mechanics. The two mentioned important observations motivate the authors to focus on the problems of developing the existing classical Melnikov approach to analyse multibody mechanical systems. Another important feature of the book is a hybrid and unified approach which draws on both smooth and nonsmooth (friction) dynamical systems. In Chapter 2 the classical Melnikov approach regarding the analysis of smooth systems is revisited. This chapter plays an introductory role, and is slightly modified with respect to the original Melnikov approaches in order to fit suitably with the rest of the book material. Splitting of the homoclinic orbit that occurred in a studied three dimensional dynamical system is illustrated and rigorously discussed yielding the Melnikov function used as a criterion of chaotic threshold occurrence. In Chapter 3, a Froude pendulum harmonically driven and with dry friction is studied. Since the studied system possesses one-degree-of-freedom, the classical Melnikov approach is applied to derive Melnikov function governing chaotic dynamics occurrence. However, in contrary to the standard application of the Melnikov method, our studied systems exhibit (for some parameters) two new homoclinic orbits giving additional analytical thresholds of stable and unstable homoclinic manifold intersections. In addition, we show how to deal with Melnikov integrals to include a role of Coulomb type friction on the system dynamics, i.e. to derive analytical formulas for the Melnikov function including both smooth and non-smooth dynamic chaos. The obtained analytical thresholds of smooth/nonsmooth potential chaotic dynamics have been verified numerically to show a good match between the theoretical analysis and the introduced mathematical model governing the dynamics of the planar Froude pendulum as non-autonomous dynamics with dry friction.

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vii

In Chapter 4 a more precise definition of the domains of the stick-slip and slip-slip chaotic dynamics of a one degree-of-freedom very weakly forced oscillator using a novel approach based on the analysis of wandering trajectories is formulated. A comparison with analytical prediction obtained using Melnikov’s technique has demonstrated a good agreement with the results presented. Note that all the standard numerical methods, in particular the direct computations of Lyapunov exponents, are time consuming. The presented approach is effective, convenient to use and requires much less computational time in comparison with other approaches. Chapter 5 is devoted to the extension of the classical Melnikov technique to the multibody mechanical systems. First, modeling of lumped mechanical systems are applied in a way suitable for further application of the Melnikov-type extension. Namely, the so-called Melnikov-Gruendler method is introduced, and the asymptotical behaviour of the fundamental matrix solutions and hence of the associated Melnikov functions are studied. The second step includes the analysis of two-degrees-of-freedom mechanical systems, and the linearization procedure along a homoclinic orbit. It is shown, among others, that a lack of coupling of variational equations computed along a homoclinic orbit as well as their symmetry allow for an efficient derivation of the associated Melnikov functions. Finally, we show that the Melnikov-Gruendler approach is reduced to the Melnikov original method for one-degree-of-freedom systems, and hence leading to the study of high dimensional systems via the extension of Melnikov’s original approach. Chapter 6 is mainly focused on analytical prediction of chaos occurrence in a self-excited (due to Coulomb type friction) and harmonically driven spherical pendulum. First, dimensionless two differential equations are derived and then transformed to a suitable form for further application of the earlier introduced Melnikov-Gruendler approach. The obtained analytical criterions for chaos occurrence (homoclinic intersection of stable and unstable manifolds) have been verified via standard numerical tools, i.e. bifurcation diagrams and phase space projections, showing surprisingly accurate prediction. A double self-excited Duffing-type oscillator with dry friction modeled as the third order polynomial with the relative velocity and being harmonically excited is studied in Chapter 7. The studied system is formulated in a general form, since it includes internal, aerodynamic and hydrodynamic frictions, and hence the further obtained results can be directly used by

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viii Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods

engineers/researchers dealing with the particular examples of 2-DOF oscillators. The system is transformed to the non-dimensional form and then it is further studied with the help of the Melnikov-Gruendler approach. Finally, various criterions of smooth/nonsmooth chaos are derived analytically and are then successfully verified numerically. In addition, the critical surface in three-dimensional parameter space for homoclinic intersections is constructed and the predicted occurrence of chaos is also verified numerically. In Chapter 8 a triple self-excited (friction) Duffing-type oscillator is studied. It consists of three masses m lying on a rigid belt moving at constant velocity v. Dry friction (approximated by a modified function composed of a “sign” and a polynomial) occurs between masses and the belt. External masses are linked to a basis through elastic Duffing-type elements and viscous dampers with relatively small damping forces. Middle mass is coupled to external masses by nonlinear springs with arbitrary characteristics satisfying the condition k0 (−z) = −k0 (z). One of the external masses is harmonically driven with frequency Ω and relatively small amplitude Γ. Since the studied system has three-degrees-of-freedom, the MelnikovGruendler technique is applied to predict homoclinic bifurcations. The various analytical criterions for homoclinic chaos occurrence are derived. Finally, one of the co-authors (J. Awrejcewicz) acknowledges the financial support by the Polish Ministry of Science and Higher Education for years 2005-2008 (grant No 4 T07A 031 28). Mr D. Sendkowski and Mr M. Ka´zmierczak are thanked for their time and consideration paid to the preparation of this book. Jan Awrejcewicz and Mariusz M. Holicke

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Contents

Preface

v

1.

A Role of the Melnikov-Type Methods in Applied Sciences

1

1.1 1.2

1 3

2.

3.

4.

5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Application of the Melnikov-type methods . . . . . . . . .

Classical Melnikov Approach

11

2.1 2.2 2.3

11 13 20

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric interpretation . . . . . . . . . . . . . . . . . . . Melnikov’s function . . . . . . . . . . . . . . . . . . . . . .

Homoclinic Chaos Criterion in a Rotated Froude Pendulum with Dry Friction

25

3.1 3.2 3.3

25 53 54

Mathematical Model . . . . . . . . . . . . . . . . . . . . . Homoclinic Chaos Criterion . . . . . . . . . . . . . . . . . Numerical Simulations . . . . . . . . . . . . . . . . . . . .

Smooth and Nonsmooth Dynamics of a QuasiAutonomous Oscillator with Coulomb and Viscous Frictions

57

4.1 4.2 4.3

57 59 62

Stick-Slip Oscillator with Periodic Excitation . . . . . . . Analysis of the Wandering Trajectories . . . . . . . . . . . Comparison of Analytical and Numerical Results . . . . .

Application of the Melnikov-Gruendler Method to Mechanical Systems ix

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x Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods

5.1 5.2 5.3

6.

7.

65 68 78

A Self-Excited Spherical Pendulum

79

6.1 6.2

79 99

Analytical Prediction of Chaos . . . . . . . . . . . . . . . Numerical Results . . . . . . . . . . . . . . . . . . . . . .

A Double Self-excited Duffing-type Oscillator 7.1 7.2 7.3

8.

Mechanical Systems with Finite Number of Degrees-ofFreedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-DOFs Mechanical Systems . . . . . . . . . . . . . . . . Reduction of the Melnikov-Gruendler Method for 1-DOF Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Analytical Prediction of Chaos . . . . . . . . . . . . . . . 103 Numerical Simulations . . . . . . . . . . . . . . . . . . . . 180 Additional Numerical Example . . . . . . . . . . . . . . . 188

A Triple Self-Excited Duffing-type Oscillator 8.1 8.2

103

193

Physical and Mathematical Models . . . . . . . . . . . . . 193 Analytical Prediction of Homoclinic Intersections . . . . . 194

Bibliography

285

Index

291

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Chapter 1


In this chapter an important role of the classical Melnikov method and its extension to study engineering systems is briefly described. Advantages and disadvantages of the Melnikov-type approaches are discussed. A state of the art development and application of analytical homo- and heteroclinic intersections is presented and various applications of the mentioned techniques to analyze and control dynamical systems are reviewed.

1.1

Introduction

The celebrated for many years classical Melnikov method is used in many cases to predict the occurrence of chaotic orbits in non-autonomous smooth one-degree-of-freedom nonlinear systems. It is applied construct the Melnikov function, and hence to predict either regular or chaotic behaviour of a studied simple dynamical system. On the other hand, it belongs to one of the widely studied perturbative approaches, and hence it has many advantages associated with application of these methods described in numerous monographs, including the frequently cited reference [58]. This group of methods contains the contribution of Igor V. Melnikov [53], who proposed the method, further referred to as MM. In its classical form it concerns two-dimensional dynamical systems, perturbed by relatively small periodic external excitations. It is possible to determine a set of the system parameters with perturbation, assuming a knowledge of a homoclinic bifurcation of a system without perturbation, which has a singular saddle point along with a homoclinic orbit governed by a parametric type equation. The MM is one of the so-called small (perturbation) parameter methods. The above statement follows from the fact that the original system is perturbed by a relatively small excitation. This implies that the Melnikov 1

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2 Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods

function determines a measure of distance between stable and unstable manifolds in the Poincaré map using a linear term in the Taylor expansion. A question arises in this situation if such an approach is adequate to the considered problem. This problem was studied in [62]. The authors, analyzing the Duffing oscillator as a discretization of the system of masses distributed along spatial coordinates continuously, included in the analysis the terms of an order higher than the first one. In the conclusion to this work one can find a statement that taking into account only relatively small perturbations is sufficient for a qualitative description in the parameter space of the criteria of chaos occurrence, and taking into consideration higher order terms only small quantitative improvements are implied. Although the Melnikov method is merely approximative, it is one of a few methods allowing analytical prediction of chaos occurrence. Moreover, it can be applied to a relatively large class of dynamical systems. It is needless to emphasize that both the classical Melnikov method and its generalization in the form of the Melnikov-Gruendler method (MGM) (see [33-35]) found their application in the analysis of dynamical systems. The main advantages of both methods cover: (i) possibility of obtaining analytical results; (ii) possibility of applying the method in dynamical systems characterized by arbitrary but integrable characteristics (including discontinuities which occur in a finite number of points like e.g. friction characteristics); (iii) high efficiency of the verification of numerically generated results; (iv) possibility of examination of strongly nonlinear systems. Both mentioned methods (MM and MGM) are not ideal, since they exhibit the following drawbacks: (i) they are applicable to systems characterized by a specific phaseportrait, namely homoclinic orbits of a critical saddle point; (ii) they are not exact but approximative methods which use a small parameter; (iii) non-perturbed system should be integrable; (iv) they enable prediction of values of the parameters associated only with the so-called homoclinic and chaos ; (v) they are associated with rather complicated algebraic computations.

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1.2

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3

Application of the Melnikov-type methods

Despite the aforementioned imperfections, on the current level of nonlinear dynamical systems development, the Melnikov method has been accepted by the scientific community and it is a useful tool to perform advanced investigations in nonlinear dynamics, including deterministic chaos. In order to illustrate the importance of the MM and its impact on applied sciences, we briefly describe a direct use of this method together with its slight modifications to solve many important problems for both applied mathematicians and engineers. Bulsara et al. [18] applied the MM to a system with stochastic excitation of probability distribution of finite mean value and variation. A formula, defining the Melnikov function, was derived for a considered system in a form of probability distribution (random variable) and its mean value. The obtained analytical results were compared with numerical simulations, performed for a system of the Duffing oscillator, mathematical pendulum and an abstract system, while high compliance of predictions and numerical experiments was achieved. Formulas derived in reference [9] enable determination of the Melnikov function for oscillators (in particular mechanical systems with stick-slip behavior are addressed) in a simple form. The proposed approach simplifies the calculations needed to define conditions for homoclinic bifurcation exhibited by oscillating dynamical systems. Taki [65] applied the Melnikov method to a bistable optical system which is characterized by two homoclinic orbits connected with the same saddle point, while one of them is contained in the other. A criterion of the homoclinic chaos for one of the orbits was numerically determined on the basis of the formula describing the Melnikov function. A criterion corresponding to the second orbit was not considered, since it predicted chaos occurrence for relatively large values of the parameter of the considered dynamical system which should be relatively small on the assumption. The obtained results were compared with the numerical simulations. High compliance was achieved between the simulations and predictions of chaotic motions obtained by the Melnikov method. Holmes and Marsden [41] applied the method to a periodically driven buckled beam. By applying the Galerkin method they obtained an equivalent mathematical model in a form of the Duffing oscillator. Mielke and Holmes [54] used the MM to examine a problem of buckling of a strongly curved rod. In the work, a special attention was paid to chaos

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which can occur in systems with at least two heteroclinic orbits and where the Melnikov function has no single roots. This happens when stable and unstable differentiable manifolds of the same saddle point (which belongs to different homoclinic orbits when there is no perturbation) approach each other and form a chaotic attractor represented by infinite, countable number of intersections of stable and unstable manifolds but they will not intersect simultaneously with other manifolds of the second equilibrium point. In this case, the Melnikov function evaluated along the heteroclinic orbit, being a measure of distance between non-intersecting manifolds, will not have roots. Thus, it was proved that in some dynamical systems the chaos could occur although the Melnikov method showed regular motions. Moon and Li [57] considered a dependence of the fractal structure of a basin of attraction on the occurrence of homoclinic bifurcation taking as an example the Duffing oscillator. The threshold parameters predicted by MM imply the presence of a chaotic attractor. Next, basins of attractions for several selected values of the parameters were reported. It was shown, among the others, that the structure of the basin of attraction boundary was related to homoclinic bifurcation and chaos. Moon [55], and Moon and Holmes [56] examined the system of an elastic and harmonically excited pendulum of uniformly distributed mass in the form of a steel flat bar of small thickness which was located in a specific magnetic field. By applying the Galerkin method to a system governed by partial differential equations a discrete mathematical model of the examined system in a form of the Duffing equation was derived. Results obtained using the classical MM were compared with laboratory investigations. A quantitative divergence was observed between predictions, which followed from the Melnikov method and experimental investigations at qualitative compliance. Generally, a larger amplitude of external excitation turned out to be necessary to make deterministic chaos arise in the examined system than it was predicted by the method. Although the approximation obtained by Holmes [40] was better than this given by the Melnikov method, the mentioned results should be approached critically, since instead of using the Lyapunov exponents the Fourier transform was applied. It seems that methodological imperfections could have been a reason of incorrect identification of weak chaos as regular motions for smaller values of the external excitation. Guckenheimer and Holmes [35] applied the Melnikov method to determine limiting parameters of a harmonically driven and damped Duffing oscillator at which the homoclinic bifurcation occurred. Conditions of oc-

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currence of subharmonic vibrations in the parameters space by example of the mentioned oscillator were detected. Thus, the Melnikov method was applied to a system with heteroclinic orbits in the phase plane. Koch and Leven [44] applied the Melnikov method to examine a parametrically driven mathematical pendulum. The Melnikov function and criterion of homoclinic and subharmonic bifurcations were determined for the considered system. The numerical investigations confirmed the occurrence of tangency of stable and unstable orbits for critical parameters. The Melnikov method was used as a detector of global homoclinic structures in [17]. As a result of the performed investigations the averaged frequency spectra were obtained analytically (the Fourier integral transformation applied) for several arbitrarily selected values of the parameters. Salam [63] used the Melnikov method to study dissipative systems by example of a strongly damped mathematical pendulum at relatively small time-periodic external excitation and constant driving torque. He showed numerically that the system without excitation had a heteroclinic orbit and gave intersection of stable and unstable differentiable manifolds for relatively small excitation. Predictions based on the Melnikov method were numerically confirmed for several selected parameters of the given system. Some self-excited systems with dry friction are included in a group of discontinuous systems and have been studied extensively for a long time. The Melnikov method was applied to non-smooth systems by example of a self-excited, relatively weakly driven (quasi-autonomous) Duffing oscillator of one-degree-of-freedom with polynomial-type friction [7] for the first time derived in [9]. The analytically defined Melnikov function for one degree-offreedom oscillators, simultaneously described both types of motions: stick and slip. Taking into account only one criterion of chaos was a disadvantage of this method, though two such criteria could have been expected. In reference [7], due to the application of the Melnikov method, one proved analytically that chaotic attractors could occur in autonomous systems with almost zero initial excitation and with dry friction. Hence, with a help of the Melnikov method one verified the phenomenon of deterministic chaos, known from numerical and laboratory investigations [11, 29, 61], in selfexcited systems with dry friction without external excitation. The results, obtained by means of the Melnikov method, were numerically confirmed in reference [3]. On the other hand, on the basis of references [1] and [7], an interesting application of the Melnikov method to a rotary cylinder-bush system was

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presented in [12]. One considered a physical-mathematical model governing the influence of heat emitted during the dry friction process on the friction force magnitude. However, only a numerical analysis of the Melnikov function was performed due to its rather complex form. Litak et al. [51] examined a classical Froude pendulum numerically. The numerical investigations showed the occurrence of deterministic chaos, despite numerous simplifications of restitutive characteristics and friction in the form of Taylor expansions up to the third order. Authors of [39] applied the Melnikov method to analyze the Froude pendulum with dry friction, however analytical calculations were not verified by numerical investigations. Systems with nonlinearities of dry friction-type are typical of mechanical engineering, but a similar behavior can be found in majority of electric circuits containing diodes, transistors, logical systems etc., with elements of piecewise differentiable or discontinuous (jump variable) characteristics. Endo and Chua [28] applied the Melnikov method to an electronic system, governed by equations equivalent to the equations of motion of a mathematical pendulum and attempted to apply the described method of homoclinic chaos prediction to a system characterized by linear restitutive characteristics (commonly known as the Chua system [2, 10]). Two different criteria of chaos for each of the examined systems, i.e. one for each of the homoclinic orbits, were derived. Chaotic dynamics of a rotated pendulum using the Melnikov method was predicted in reference [38]. It was shown, among the others, that the one degree-of-freedom system could exhibit either two or four homoclinic orbits and one or two Melnikov criteria could be applied, respectively. A large rotating nonlinear multibody system (drag line) with energy dissipation exhibiting chaotic instabilities was studied in reference [52]. The sufficient analytical criterion for critical parameters set responsible for occurrence of chaotic motion was derived using the Melnikov method. Asymmetric spacecraft dynamics perturbed by small aerodynamic drag torque with periodic in time moments of inertia was investigated in reference [42]. Transientchaotic behaviour of the system was predicted by means of the Melnikov method. Chaotic critical thresholds of a nonlinear elastic beam large deflections were formulated with a help of the Melnikov function method in reference [36]. Existence, stability and bifurcation theorems for subharmonics of the planar Hamiltonian systems were reported in reference [68] using the modified Melnikov theory.

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7

Both Galerkin and Melnikov approaches were used to study chaotic motion of an elastic cylindrical shell in reference [37]. In reference [43] global bifurcation behavior of 2-DOF nonlinear oscillator was analyzed using the Melnikov method. A Smale horseshoe type of chaos was illustrated, among the others. The noise-induced chaos predicted by the Melnikov method (MM) was investigated in the softening Duffing oscillator in reference [30]. In reference [27] MM was extended also to study an inverted pendulum externally driven and impacting rigid walls. The Melnikov function up to the n-th order was computed and a critical set of parameters for the persistence of homoclinic impact cycles were estimated. Yagasaki [69] studied periodic and homoclinic motions in periodically forced and weakly coupled oscillators with perturbation of two independent planar Hamiltonian systems. The relationship between the subharmonic and homoclinic Melnikov theories was illustrated, and the modified homoclinic MM was directly applied to study two types of periodic orbits. Chen [22] extended MM to perturbed planar non-Hamiltonian integrable systems with slowly-varying angle variables, giving the condition of transversely homoclinic intersection. Lamarque and Bastien [45] studied a forced pendulum with viscous damping and Coulomb friction. Lyapunov exponents were computed and a Melnikov relation was obtained as a limit of regularised Coulomb friction. Chaotic attitude motion and its control of a magnetic rigid spacecraft in an arbitrary circular orbit round the Earth with a help of MM was investigated by Chen and Liu [23]. The feedback control and its local linearization were used to control chaotic attitude motions to the given either fixed point or periodic motion. Yagasaki [70] analyzed a homoclinic behavior in resonance cases of nonconservative forced oscillators. He applied MM to get a simple condition under which separatrix splittings with exponentially small upper bounds might appear. He also used MM to study codimension-two BogdanovTakens bifurcation for subharmonics in periodic perturbations of planar Hamiltonian systems [71]. Criteria of the Bogdanov-Takens bifurcations, approximate relations for saddle-node, Hopf and homoclinic bifurcation sets and the Bogdanov-Takens bifurcation points were derived. MM regarding generalized Hamiltonian systems was used to explain the control rules of directing chaotic motion towards low-periodic motion in the Lorenz equations in reference [19]. Numerical simulation verified the effectiveness of the applied Melnikov’s technique.

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Casasayas et al. [21] showed that the conditions for the persistence under perturbation of the invariant manifolds also ensured the convergence of the Melnikov integral in the case when the unperturbed system had a parabolic orbit with a homoclinic loop. MM was applied to analyze the intermittency transition between order and in the chaotic Duffing-type system [66]. Weak signals were detected and estimated. The application and extension of Melnikov’s idea by considering Poincaré sections non-orthogonal to the flux and by applying both the so called “one-half” and “full” Melnikov functions are proposed by Lenci and Rega [46–49]. In the mentioned papers it is shown (among the others) that classical Melnikov’s technique is practically inaccurate for both small and large excitation frequencies with respect to degenerated homo/heteroclinic bifurcations and in the case of generic periodic excitations. A sufficient condition for controlling chaos using a weak resonant excitation is given in reference [20]. In addition, it is shown how the MM illustrates a vital role of the initial phase difference in suppressing or inducing either chaotic or quasi-periodic dynamics. Some new methods to suppress chaos via slightly modified Melnikov function are proposed by Leung and Zengrong [50]. The dynamics of both stochastic and resonant layers for rotated pendulum equations with a use of MM were studied in reference [24]. A simultaneous occurrence of chaotic and subharmonic dynamics was proven. MM is applied to derive criteria for chaos of a cylindrical shell single model approximation in reference [25]. A discussion of one and double modes approximation is supplemented. Zhu and Liu [72] derived the random Melnikov process, and they proposed a mean square criterion to determine the threshold amplitude of the bounded noise for the onset of chaos in coupled pendulums and harmonic oscillator under bounded noise excitation. Control and Melnikov chaos of an harmonically excited particle from a catastrophic single well potential are studied in reference [59]. Both energy and Melnikov methods are used to obtain critical external forcing amplitudes for catastrophe and chaos. Dynamics of a nonlinear elastic shallow shell of large deflection subjected to constant boundary loading and harmonic lateral excitation is studied in [26]. Three different dynamic equations of the shell were derived and the associated Melnikov functions were formulated. Critical condition for chaos occurrence were provided, and then regular and chaotic vibrations of the shell were studied using history plots, phase diagrams and Poincaré maps.

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9

The Melnikov method, despite its numerous applications in the classical form, has an essential disadvantage: the method can be applied to mechanical systems of one degree-of-freedom with excitation. This disadvantage does not occur in the Melnikov-Gruendler method, as it will be shown in this book. Guckenheimer and Holmes [35] were the first who based on the KAM theorem, determined a set of parameters of relatively weak perturbed Hamiltonian systems of two-degrees-of-freedom, at which homoclinic bifurcation occurred. Another possibility is the approach based on the assumption that if a mechanical system of a finite number of degrees-of-freedom is an integrable system, then it is described by the Routh equation [16]. If the number of first integrals allows the Routh equation to satisfy the assumptions of the Melnikov method, then it is possible to apply the classical Melnikov method to a system of a larger number of degrees-of-freedom. In the case when the number of first integrals is unknown, one needs to apply the MelnikovGruendler method [34], which is an extension of the Melnikov method to non-Hamiltonian dynamical systems of arbitrary, finite number of degreesof-freedom. For some types of systems (symmetry of variational equations on a homoclinic orbit, no coupling, etc.) the formulas determining chaos criteria can be simplified. In contrast to the classical Melnikov method, the Melnikov-Gruendler method is not widely known. It was applied in [34] to consider small vibrations of a spherical pendulum subjected to magnetic field with a relatively small and then large viscous damping (a non-Hamiltonian system). Moreover, the Melnikov-Gruendler method was used in both physical interpretation and abstract dynamical systems [32, 33], in order to detect deterministic chaos and analyze structural changes of homoclinic orbits. Reference [14] played a key role in the attempt to apply the MelnikovGruendler method to self-excited systems with polynomial-type friction of two degrees-of-freedom. However, the obtained results did not reflect complexity of the multi-dimensional problem entirely, by making use of only 8% of the method capabilities. For this reason, one can say that the problem of homoclinic chaos occurrence in two degrees-of-freedom systems with dry friction has not been examined by means of the Melnikov-Gruendler method so far. Mechanical systems with friction of three degrees-of-freedom need investigation with a help of the Melnikov-Gruendler method. Other and these types of problems are addressed in this monograph.

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Chapter 2


In this chapter we present the classical Melnikov’s approach in order to introduce a reader to the book material. Phase portraits, homoclinic orbits and their splitting are illustrated and discussed, and finally the Melnikov function is derived.

2.1

Introduction

The classical Melnikov approach will be presented using the following second order differential equation y¨ + S (y) = εF (y, y, ˙ t) , (2.1) where 0 < ε ≤ 1 is the formally introduced small (perturbation) parameter to exhibit sufficiently small perturbation εF (y, y, ˙ t). We assume that function S (y) is continuous together with its derivative and function F (y, y, ˙ t) can be integrated (it can be discontinuous only in some countable set of points, i.e. it is segmental continuous and differentiable) and periodic with respect to time F (y, y, ˙ t) = F (y, y, ˙ t + T ). Equation (2.1) can be transformed to its first order differential equations cast in the form v˙ = −S (y) + εF (y, v, t) , (2.2) y˙ = v, where y˙ = v is the system velocity. Owing to the introduced periodicity of function F (y, y, ˙ t) with respect to time, system (2.2) can be transformed to the autonomous form expressed by three first order differential equations of the form v˙ = −S (y) + εF (y, v, η) , (2.3)

y˙ = v, η˙ = ω, 11

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where the introduced new coordinate η is of cyclic type and ω = 2π/T . Integrating the third equation of (2.3) one gets η = ωt + η0 ,

(2.4)

where η0 is the constant of integration. In vector notation system (2.3) is given in the form q˙ = g (q) + εh (q, η) , η˙ = ω,

(2.5)

where q = [y, v]T , g (q) = [−S (y) , v]T and h (q, η) = [F (q, η) , 0]T . In a general case, we cannot integrate this equation, and hence either approximate analytical or numerical methods are applied. Melnikov’s method belongs to one of the analytical techniques frequently applied to detect intersection of both stable and unstable manifolds associated with a saddle. Analytical approximate methods allow us to define the solutions of system (2.1) assuming full solution knowledge of an associated system. Such associated system is usually obtained from (2.1) after taking ε = 0. Proceeding in the mentioned manner one gets the following autonomous system y¨ + S (y) = 0

(2.6)

being equivalent to the following two first order differential equations v˙ = −S (y) , y˙ = v.

(2.7)

From (2.3) one obtains v˙ = −S (y) , y˙ = v,

(2.8)

η˙ = ω. Equation (2.6) can be solved analytically. Two sides of (2.6) are first multiplied by dy to get dy dy˙ + S (y) dy = 0. dt Bearing in mind that y˙ = dy/dt = v one finds vdv + S (y) dy = 0. In the above a differential equation with separated variables is obtained, and this integration gives v2 = C − V (y) , 2

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R where C is the integration constant and V (y) = S (y) dy. Solving the last equation with respect to v allows us to get an equation governing a family of the phase trajectories in the form p v = ± 2 (C − V (y)). (2.9) Applying formula v = dy/dt one gets again the equation with the separated values as follows dy . dt = ± p 2 (C − V (y)) Integration of both sides of this equation gives Z dy t − t0 = ± p , 2 (C − V (y)) where t0 is the integration constant (initial time value). Solution to equation (2.6) can be defined regarding to an inverse function in the following form y (t − t0 ) = yF (t − to , C) ,

(2.10)

where the inverse function y

−1

=±

Z

dy p . 2 (C − V (y))

Differentiation with respect to time gives v (t − t0 ) = vF (t − to , C) ,

(2.11)

where vF = dyF /dt. Formulas (2.10) and (2.11) define solution to system (2.7) in the following parametric form ( y = yF (t − t0 , C) q (t, t0 ) = . (2.12) v = vF (t − t0 , C) 2.2

Geometric interpretation

Observe that an essential influence on the qualitative behaviour of equations (2.6) and (2.1) have the system equilibrium positions. In those positions the system stays for an arbitrarily long time. Coordinates of the mentioned equilibrium positions are defined by the real roots to the algebraic equation S (y0 ) = 0.

(2.13)

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Fig. 2.1

Geometric interpretation of algebraic equation S (y0 ) = 0.

Equation (2.13) may have zero, one or more real solutions. Every real solution of (2.13) corresponds to an equilibrium position of equation (2.6). In the case when equation (2.13) has not real solutions, then the studied system does not exhibit equilibrium positions. Figure (2.1) shows the geometric interpretation of the system static configuration in variables (y, S (y)). It exhibits three types of equilibria: single roots y0s (red) and y0c (green), as well as a multiple root y0d (black). The difference between roots y0s and y0c is as follows. The slope of function S (y) at the point corresponding to y0s is negative, and positive in the case of y0c , which is equivalent to satisfying the conditions: S 0 (y) < 0 for root y0s , and S 0 (y) >0 y=y0s

y=y0c

for root y0c . If one deals with the multiple root y0d , then the following re lation is satisfied S 0 (y) = 0. As it has been already mentioned, each y=y0d

solution to algebraic equation (2.13) corresponds to its equilibrium position of equation (2.6). In this way one may define three types of equilibrium positions. Namely, each of real root type corresponds to the equilibrium position called saddle, whereas the root y0c corresponds to the so called center. In the case of multiple roots y0d one deals with the so called degenerate case and a degree of degeneracy is defined by the number of root y0d multiplicities. Since the mentioned degenerate cases do not play an important role during applications of the Melnikov method, we study only the non-degenerated cases. In the case of interpretation by the system of two differential equations (2.7) each of real solution of algebraic equation (2.13) as well as the defined equilibrium position corresponds to the critical point (y0 , 0) of the phase plane (y, v). If the equilibrium position is of a certain

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15

type, then the same type definition is applied to the corresponding critical point in the phase space. It is clear that depending on the equilibrium type we have various and qualitatively different phase neighborhoods. For instance in Figure (2.2) phase portraits of two critical points are reported: (a) center and (b) saddle. Note that equilibrium called the center is sur-

Fig. 2.2

Phase portraits of two critical points: (a) center, (b) saddle.

rounded by closed phase trajectories in the shape of concentric ellipses. The equilibrium position is located in the common center of the mentioned conical curves. A characteristic feature of the equilibrium position of the type of center is that none of the surrounding phase trajectories belongs to the discussed equilibrium position. Furthermore, a relatively small deviation from the center defines the proportional system movement on the corresponding ellipse. In the case of saddle type equilibrium the system behaves in a qualitatively different manner. The equilibrium is surrounded by the open phase trajectories having hyperbolic shapes excluding four trajectories being the half-straight lines. The mentioned trajectories are called separatrices, because they separate two qualitatively different motions. Two of them represent a phase point movement with time going to infinity and they create an unstable differential manifold Wu , whereas two other ones create stable difference manifold Ws . Even relatively small system deviation from the saddle equilibrium yields an exponential divergence of the distance between phase point and the saddle point. If a given system has two critical points lying relatively close to each other and one of them is the center, whereas the other one is the saddle, then after a small perturbation the phase trajectories representing the phase point movement in time

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have the forms similar to those shown in Figure (2.2) (for simplicity we take the saddle as the coordinates origin). However, for large phase point deviations the corresponding phase curves become deformed. Figure (2.3) shows two critical points (saddle and center) and the surrounding phase trajectories. Phase trajectories associated with the center have the shapes

Fig. 2.3

Phase portrait of trajectories surrounding the saddle and center.

of ellipses but when they approach the critical point of saddle type their shapes become similar to those of hyperbolas. Similar deformations take place also for stable Ws and unstable Wu manifolds associated with the saddle. The deformations increase with a decrease of the distance between them. It may happen that the distance approaches zero, and hence the T only one phase trajectory Wh = Ws Wu occurs, which begins and ends in the saddle point surrounding the center. Such trajectory is called the homoclinic trajectory. One of the simplest physical objects exhibiting the homoclinic trajectory is a pendulum. The pendulum has the saddle type equilibrium being in its vertical position. Another equilibrium is associated with its down position and is the center. Small initial pendulum deviation generates oscillations with constant amplitude. On the other hand, even an infinitely small deviation from the vertical equilibrium state yields an exponential increase of the distance between initial deviation and the saddle. However, if we put to the system the full energy equal to the potential vertical pendulum position energy, then the pendulum will be moving on the homoclinic orbit. First it starts to increase its velocity until achieving the down equilibrium, and then it begins to decrease its velocity down to zero

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for an infinite time value, and the pendulum stops in its vertical position. A parametric equation of the homoclinic orbit defined by formulas (2.10) and (2.11) for a certain constant C = CH is chosen in such a way that the trajectory belongs to the saddle point. Applying formula (2.11) and taking into account the assumption that the saddle lies at the coordinates origin one gets CH = V (0) . Substituting the last equation into (2.12) one gets the homoclinic trajectory in the following parametric form ( y (t − t0 ) = yF (t − t0 , V (0)) qH (t − t0 ) = . (2.14) v (t − t0 ) = vF (t − t0 , V (0)) The above formula shows that there are infinitely many ways of the homoclinic orbit parameterization depending on the choice of t0 . Owing to our assumption we have lim qH (t − t0 ) = 0,

t→±∞

(2.15)

which means that the saddle situated at the coordinates origin is achieved by the homoclinic trajectories in an asymptotic way. We determine components of the tangent vector to the phase trajectory in the given point of coordinates (y, v). Dividing the sides of equations (2.7) one gets S (y) dv =− . dy v

(2.16)

The above formula defines the tangent direction to the phase trajectory. There are infinitely many vectors of such a direction but of different length. One of them is as follows f|| (y, v) = [−S (y) , v] . The tangent vector is defined by the formula s|| (y, v) =

f|| (y, v) , |f|| (y, v) |

p where |f|| (y, v) | = v 2 + S 2 (y) is the length of vector f|| (y, v). On the other hand, the vector perpendicular to the phase trajectory in the phase point (y, v) and of the length equal to vector (2.16) is defined by the formula f⊥ (y, v) = [−v, −S (y)] .

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Hence, a versor perpendicular to the phase trajectory at point (y, v) is defined as follows n⊥ (y, v) =

f⊥ (y, v) , |f⊥ (y, v) |

(2.17)

where |f|| (y, v) | = |f⊥ (y, v) |. Observe that in view of three ODEs interpretation with cyclic coordinate η defined by (2.8), a critical point of the studied dynamic system corresponds to the straight line governed by the equation (y, v, η) = (y0 , 0, η (t)) ,

(2.18)

and all trajectories corresponding to the homoclinic orbits, irrespective of the choice of t0 , lie on the cylinder surface with the directrix defined by formula (2.14). Figure (2.4) presents the set of critical points saddle-center and the homoclinic orbit in space (y, v, η). In addition, part of the homoclinic trajectory for given t0 and η1 ≤ η (t) ≤ η2 is shown. Note that

Fig. 2.4 Two critical points (saddle and center) exposition and phase trajectory in space (y, v, η).

considering the studied system as the autonomous one in space (y, v, η) has

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some advantages. For instance, the phase point movement along the homoclinic orbit within the given parameterization (t0 = const) is associated not only with a change of the system on the homoclinic trajectory, but also the position on the coordinate η (t) is changed. On the other hand, fixation of time instant (t = const), and hence also of variable η (t) = η0 = const and substitution of different values of constant t0 yields (in plane η (t) = η0 ) a graph of the homoclinic orbit. For t0 equal to the time instant corresponding to the intersection of trajectory with plane η (t) = η0 the phase point lies in plane η (t) = η0 at point qH (0). Furthermore, if one takes constants η1 and η2 such that η1 = η2 modT then owing to cyclic property of the variable η one may identify plane η = η1 with plane η = η2 . It means that if we distinguish a certain plane in space (y, v, η) defined by formula Ση0 {(y, v, η) |η = η0 }, it is also identified with an infinite sequence of planes Ση0 +nT , where n is the natural number. It means that on plane Ση0 one may follow a motion of the investigated system. Namely, marking the subsequent system positions in time instant corresponding to plane Ση0 intersections one gets the drawing of the system movement represented by a discrete mapping of plane Ση0 into itself being called either a Poincaré section or a Poincaré map. In the introduced mapping straight line defined by formula (2.18) governing the position of critical point in the coordinates (y, v, η) is exhibited in the Poincaré map by a fixed point situated in the origin of system (y, v), whereas the chosen (t0 = const) homoclinic trajectory is a set of points partitioned in a non-uniform manner on the curve (2.9) for C = CH = V (0). Having the introduced background we will study the investigated system (2.3) for ε 6= 0. For this purpose the Poincaré map on plane Ση0 is used, where η0 ∈ (0, T ). The occurrence of a relatively small magnitude perturbation εF (y, y, ˙ η) being periodic with respect to η causes a shift of the saddle of the Poincaré section from the origin of coordinates associated with plane Ση0 to a certain neighborhood position (y0ε ) = (O (ε) , 0) assuming that for an arbitrary η0 choice the saddle point will not be degenerated. Perturbation εF (y, y, ˙ η) causes a collapse of the s homoclinic orbit into stable Wε and unstable Wεu differential manifolds. The so far described situation is illustrated in Figure (2.5). Let us define a distance between two differential manifolds in plane Ση0 . As a natural reference we take the homoclinic orbit qH marked in blue, in which all points of the Poincaré map are situated irrespective of its parameterization. Let us take the parameterization regarding time instant, when the homoclinic orbit intersects the plane Ση0 in t = 0, i.e. we consider orbit qH (−t0 ). Next, in the parameterization t0 we follow two points lying on stable Wεs

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Fig. 2.5

Splitting of the homoclinic orbit qH in plane Ση0 .

and unstable Wεu differential manifolds qεs,u (t, t0 ) appearing in plane Ση0 in time instant t = 0. The distance between these points defines the sought distance between stable Wεs and unstable Wεu differential manifolds defined as follows dˆ(t0 ) = q u (0, t0 ) − q s (0, t0 ) . (2.19) ε

ε

A more accurate definition of dˆ(t0 ) can be found numerically.

2.3

Melnikov’s function

In order to get the approximate analytical result we separate from the trajectory qεs,u (t, t0 ) the homoclinic trajectory qH (t − t0 ), which yields the assumption of the relatively small shift of the saddle one gets |qεs (t, t0 ) − qH (t − t0 ) | = O (ε) , 0 ≤ t < ∞, |qεu (t, t0 ) − qH (t − t0 ) | = O (ε) , −∞ < t ≤ 0. Hence, one gets up to the second order terms qεs (t, t0 ) = qH (t − t0 ) + εq1s (t, t0 ) + O ε2 , 0 ≤ t < ∞, qεu (t, t0 ) = qH (t − t0 ) + εq1u (t, t0 ) + O ε2 , −∞ < t ≤ 0. where ( y1s,u (t, t0 ) s,u q1 (t, t0 ) = v1s,u (t, t0 )

(2.20)

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are certain functions. Let us investigate the properties of function q1s,u (t, t0 ). Differentiating (2.20) with respect to ε and keeping first order terms only one gets ∂ s,u q (t, t0 ) = q1s,u (t, t0 ) . ∂ε ε Since qεs,u (t, t0 ) is a solution to system (2.5), hence

(2.21)

q˙εs,u (t, t0 ) = g (qεs,u (t, t0 )) + εh (qεs,u (t, t0 ) , η (t)) . Differentiation with respect to ε gives ∂ d s,u ∂ qε (t, t0 ) = g (qεs,u (t, t0 )) + h (qεs,u (t, t0 ) , η (t)) . ∂ε dt ∂ε Differentiation of the composite function gives d ∂ s,u dg (q) ∂ s,u qε (t, t0 ) = q (t, t0 ) + h (qεs,u (t, t0 ) , η (t)) . s,u dt ∂ε dq q=qε (t,t0 ) ∂ε ε From formula (2.21) we get dg (q) q˙1s,u (t, t0 ) = q s,u (t, t0 ) + h (qεs,u (t, t0 ) , η (t)) . dq q=qεs,u (t,t0 ) 1

In the limiting case we have dg (q) q1s,u (t, t0 ) + lim h (qεs,u (t, t0 ) , η (t)) . lim q˙1s,u (t, t0 ) = lim s,u ε→0 ε→0 dq ε→0 q=qε (t,t0 ) Since limε→0 qεs,u (t, t0 ) = qH (t − t0 ), one may write dg (q) q˙1s,u (t, t0 ) = q s,u (t, t0 ) + h (qH (t − t0 ) , η (t)) . dq q=qH (t−t0 ) 1

(2.22)

The above formula allows us to conclude that functions q1s,u (t, t0 ) are solutions to the variation equations along the homoclinic orbit associated with system (2.5). Applying (2.20) for t = 0 in formula (2.19) one gets dˆ(t0 ) ≈ ε (q1u (0, t0 ) − q1s (0, t0 )) . It follows from the above formula that functions q1s,u (t, t0 ) determine (for given t0 ) in time instant t = 0 two points whose distance is the approximation of distance dˆ(0, t0 ) between stable and unstable manifolds. However, there is no guarantee that the points lie in plane Ση0 . In this situation, as the distance measure between the studied manifolds we take the projection of distance dˆ(t0 ) onto normal to the homoclinic trajectory belonging to plane Ση0 . For this purpose the versor normal to the homoclinic trajectory qH (−t0 ) is built and vectors defined by the origin of coordinates in plane

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Ση0 and points q1s,u (0, t0 ) are constructed. The sought projection d (t0 ) is defined by a dot product defined by the formula d (t0 ) = εn (qH (−t0 )) (q1u (0, t0 ) − q1s (0, t0 )) . From (2.17) one gets f⊥ (qH (−t0 )) (q u (0, t0 ) − q1s (0, t0 )) . |f⊥ (qH (−t0 )) | 1 Let us introduce the supplementary relations d (t0 ) = ε

df

∆s,u (t) = f⊥ (qH (t − t0 )) q1s,u (t, t0 ) .

(2.23)

(2.24)

Substitution of (2.24) to (2.23) yields d (t0 ) = ε

∆u (0) − ∆s (0) . |f⊥ (qH (−t0 )) |

(2.25)

Differentiation of (2.24) with respect to time gives ˙ s,u (t) = d (f⊥ (qH (t − t0 ))) q s,u (t, t0 ) + f⊥ (qH (−t0 )) q˙s,u (t, t0 ) . ∆ 1 1 dt Differentiation of the composite function yields df⊥ (q) s,u ˙ ∆ (t) = q˙H (t − t0 ) q1s,u (t, t0 ) dq q=qH (t−t0 ) + f⊥ (qH (−t0 )) q˙1s,u (t, t0 ) .

Since the homoclinic orbit is a solution to system (2.5) for ε = 0 and taking into account formula (2.22) one gets df⊥ (q) s,u ˙ ∆ (t) = g (qH (t − t0 )) q1s,u (t, t0 ) dq q=qH (t−t0 ) dg (q) s,u + f⊥ (qH (−t0 )) q (t, t ) + h (q (t − t ) , η (t)) , 0 H 0 dq q=qH (t−t0 ) 1 or after some transformations df⊥ (q) dg (q) s,u ˙ ∆ (t) = g (qH (t − t0 )) + f⊥ (q) q1s,u (t, t0 ) dq dq q=qH (t−t0 ) + f⊥ (qH (t − t0 )) h (qH (t − t0 ) , η (t)) .

Observe that g (y, v) = [−S (y) , v]T , f⊥ (y, v) = [−v, −S (y)] and hence ˙ s,u (t) = ∆ T ! dS (y) dS (y) T −1, − [−S (y) , v] + [−v, −S (y)] − ,1 x = xH (t − t0 ) dy dy y = yH (t − t0 )

×

q1s,u

(t, t0 ) + f⊥ (qH (t − t0 )) h (qH (t − t0 ) , η (t))

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23

and ˙ s,u (t) = ∆ dS (y) dS (y) S (y) − v +v − S (y) x = x (t − t ) q1s,u (t, t0 ) H 0 dy dy y = yH (t − t0 )

+ f⊥ (qH (t − t0 )) h (qH (t − t0 ) , η (t)) . Carrying out reduction of the similar terms one gets ˙ s,u (t) = f⊥ (qH (t − t0 )) h (qH (t − t0 ) , η (t)) . ∆ ˙ s,u (t) without The above transformations allowed us to define magnitudes ∆ s,u ˙ s (t) and an explicit definition of function q1 (t, t0 ). Upon integration of ∆ u ˙ ∆ (t) in the limits from 0 to ∞ and from −∞ to 0 one gets ∞ ∆s (t) = ∆s (∞) − ∆s (0) 0 Z ∞ = f⊥ (qH (t − t0 )) h (qH (t − t0 ) , η (t)) dt, 0

0 ∆ (t) u

−∞

= ∆u (0) − ∆s (−∞) Z 0 = f⊥ (qH (t − t0 )) h (qH (t − t0 ) , η (t)) dt.

(2.26)

−∞

Applying (2.20) for t → ±∞, one gets |qεs,u (t, t0 ) − qH (t − t0 ) | → 0 and qεs,u (t, t0 ) → 0. Taking into account the above consideration in (2.24) one gets ∆s (∞) = ∆u (−∞) = 0. The considerations presented so far are taken into account in the sum of equations (2.26) to yield Z ∞ ∆u (0) − ∆s (0) = f⊥ (qH (t − t0 )) h (qH (t − t0 ) , ωt + η0 ) dt, −∞

where also formula (2.4) has been applied. Finally, formula (2.25) takes the form M (t0 ) d (t0 ) = ε , |f⊥ (qH (−t0 )) | where the expression Z ∞ M (t0 ) = f⊥ (qH (t − t0 )) h (qH (t − t0 ) , ωt + η0 ) dt −∞

is called the Melnikov function and q 2 (−t ) + S 2 (y (−t )). |f⊥ (qH (−t0 )) | = vH 0 H 0

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A more convenient form of the Melnikov function for application is obtained by changing variables t → t + t0 , namely Z ∞ M (θ0 ) = f⊥ (qH (t)) h (qH (t − t0 ) , ωt + θ0 ) dt, −∞

where θ0 = ωt0 + η0 . Recalling the definitions of f⊥ (y, v) = [−v, −S (y)] and h (q, t) = [F (q, t) , 0]T one gets a formula Z ∞ −S (yH (t)) F (qH (t) , ωt + θ0 ) dt, M (θ0 ) = vH (t) 0 −∞ which is further used in the analysis of systems of many degrees-of-freedom. Expanding the determinant one gets Z ∞ M (θ0 ) = − vH (t) F (qH (t) , ωt + θ0 ) dt. −∞

Therefore, we have obtained finally classical formulas for the Melnikov function regarding intersection of stable and unstable manifolds of the onedegree-of-freedom smooth dynamical (mechanical) systems.

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Chapter 3

Homoclinic Chaos Criterion in a Rotated Froude Pendulum with Dry Friction In this chapter a rotated Froude pendulum with dry friction is studied. We introduce a reader to computation point by point of the chaos criterion on a basis of homoclinic intersections via the classical Melnikov’s technique. Two distinct homoclinic loops configurations depending on the rotation frequency are analysed yielding two various criteria for stick-slip and slip chaotic orbits occurrence.

3.1

Mathematical Model

Consider a Froude pendulum (see Figure 3.1) with mass m and length l, where dry friction occurs between a pivot and a shaft. It is assumed that

Fig. 3.1

Scheme of a rotating Froude pendulum.

the pivot has radius r and rotates at angular velocity $ = const> 0. The 25

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suspension point of this pendulum rotates at angular velocity ω = const > 0. Movement of the analyzed system is monitored by the angle coordinate ϕ. Our investigated system is harmonically driven along coordinate ϕ by external excitation with amplitude Γ and frequency Ω, and is also damped in a viscous type manner characterized by coefficient D. In addition, following reference [60] the Coulomb friction characterized by coefficient µ is added. Note that during relative clockwise rotation of the mass m around the pendulum axis, a normal reaction is shifted of friction angle ρ (see Figure 3.2). A similar system has been studied in reference [38]. Since it has one degree-of-freedom, Melnikov’s technique is applied to define the boundaries between regular and chaotic motions.

Fig. 3.2

Computational scheme of the analyzed system.

The equation of system dynamics is given in the following form 1 ml2 ϕ¨ + εDϕ˙ + εT r sgn (ϕ˙ − $) − ml2 ω 2 sin 2ϕ + mgl sin ϕ = εΓ cos ω1 t, 2 (3.1) where ε is the small parameter, and friction force is described by the equation µ g T = ml p ϕ˙ 2 + ω 2 sin2 ϕ + cos ϕ , l 1 + µ2 (1 − λ2 )

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27

where: µ = tan ρ is the friction coefficient and λ = r/l < 1. Denoting by .p µλ = µ 1 + µ2 (1 − λ2 ) the modified frictional coefficient, the friction force is g T = mlµλ ϕ˙ 2 + ω 2 sin2 ϕ + cos ϕ . l Observe that µλ < µ, and the governing equation takes the form ml2 ϕ¨ + εDϕ˙ + εmlrµλ ϕ˙ 2 + ω 2 sin2 ϕ + gl cos ϕ sgn (ϕ˙ − $) − 12 ml2 ω 2 sin 2ϕ + mgl sin ϕ = εΓ cos ω1 t, and after some rearrangements one gets ϕ¨ + εδ ϕ˙ + ελµλ ϕ˙ 2 + ω 2 sin2 ϕ + ω02 cos ϕ sgn (ϕ˙ − $) − 12 ω 2 sin 2ϕ + ω02 sin ϕ = εγ cos ω1 t,

where the following notation is applied: ω0 = Finally, we obtain

p

g/l, λ = r/l, γ = Γ ml2 .

ϕ¨ − 12 ω 2 sin 2ϕ + ω02 sin ϕ = = ε γ cos ω1 t − δ ϕ˙ − λµλ ϕ˙ 2 + ω 2 sin2 ϕ + ω02 cos ϕ sgn (ϕ˙ − $) . (3.2) Introducing a new variable defined by ψ = ϕ + π, the following equation is obtained ψ¨ − 12 ω 2 sin 2ψ − ω02 sin ψ = = ε γ cos ω1 t − δ ψ˙ − λµλ ψ˙ 2 + ω 2 sin2 ψ − ω02 cos ψ sgn ψ˙ − $ . (3.3) For ε = 0 one gets an autonomous system described by the equation ϕ¨ − 12 ω 2 sin 2ϕ + ω02 sin ϕ = 0.

(3.4)

ψ¨ − 21 ω 2 sin 2ψ − ω02 sin ψ = 0.

(3.5)

Equation (3.3) gives

Equilibria of our system satisfy the following equation −ω 2 sin ϕ0 cos ϕ0 + ω02 sin ϕ0 = 0, which after a simple rearrangement gives −ω 2 cos ϕ0 + ω02 sin ϕ0 = 0 and cos ϕ0 =

ω02 > 0 ∨ sin ϕ0 = 0. ω2

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Assuming −π ≤ ϕ0 ≤ π one may notice that for an arbitrary value of ω the two following equilibria exist: ϕ01 = 0 and ϕ02 = ±π. For |ω| > ω0 > 0 two additional equilibria appear in the form 2 ω0 ϕ03,4 = ± arccos , ω2 where |ϕ03,4 | < π/2. The equilibria associated with coordinate ψ are defined by the formulas: 2 ψ01 = π, ψ02 = 0 and for |ω| > ω0 > 0 by ω ψ03,4 = ± arccos ω02 + π. Owing to linearization of equation (3.4) one gets in the vicinity of equilibrium ϕ¨ + ω02 cos ϕ0 − ω 2 cos 2ϕ0 (ϕ − ϕ0 ) = 0. For ϕ01 = 0 we get ϕ¨ + ω02 − ω 2 (ϕ − ϕ01 ) = 0. Observe that for |ω| < ω0 the equilibrium is a center, whereas for |ω| > ω0 it is a saddle. For ϕ02 = ±π one gets ϕ¨ − ω 2 (ϕ − ϕ0 ) = 0, which defines the saddle. For remaining critical points ϕ03,4 one obtains ω 2 − ω02 ω 2 + ω02 ϕ¨ + (ϕ − ϕ0 ) = 0. ω2 The mentioned critical points are centers and they exist only for |ω| > ω0 . Phase trajectory is defined by equation (3.4), which is multiplied by ϕ˙ to give dϕ˙ ϕ˙ − 12 ω 2 ϕ˙ sin 2ϕ + ω02 ϕ˙ sin ϕ = 0, dt and after integration one gets ϕ˙ 2 1 C + ω 2 cos 2ϕ − ω02 cos ϕ = , 2 4 2 where C is the integration constant. The last equation defines a phase trajectory of the form q ϕ˙ = ± C + 2ω02 cos ϕ − 12 ω 2 cos 2ϕ. Figure 3.3 shows phase trajectories of the analyzed system for ω0 = 2, ω = 1 and for C ∈ {−2; 3; 8, 5; 15; 25} (up), as well as for ω0 = 1, ω = 2 ω = 1 and C ∈ {−1, 5; 0; 2; 4; 6, 5; 9} (down). The up phase portrait is associated with pendulum movement for |ω| < ω0 , i.e. for a relatively low angular velocity. In this case the analyzed system behaves similarly like

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29

Fig. 3.3 Phase portrait of the system governed by equation (3.4) for ω0 = 2, ω = 1 (up) and for ω0 = 1, ω = 2 (down).

a mathematical pendulum. Its lower equilibrium position is a center, and the system exhibits periodic oscillations around this equilibrium. Upper equilibrium position of the pendulum is a saddle, which is associated with two homoclinic trajectories. The latter ones play a role of a curve separating different motion of the pendulum. Inside this trajectory, the pendulum oscillates periodically, whereas outside this curve the pendulum rotates. For |ω| > ω0 (see Figure 3.3 down) a more complicated picture of the phase portrait is visible. Owing to the investigations carried out, inside the homoclinic orbit two additional equilibria (centers) appear, whereas the equilibrium being previously a center is now a saddle. Additionally, two new homoclinic orbits are born. They have internal area of the homoclinic orbit divided into two oscillation domains of the newly born equilibria characterized by a relatively small period of oscillations, and into one area of phase space associated with periodic oscillations around three critical points. Stable and unstable differential manifolds associated with the sad-

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dle ϕ01 = 0 and for |ω| > ω0 are defined by the equation q ϕ˙ s1 = ± 2ω02 (cos ϕ − 1) + 12 ω 2 (1 − cos 2ϕ), and hence one gets ±dϕ

dt = p

2ω02 (cos ϕ − 1) + ω 2 (1 − cos2 ϕ)

.

Integrating the last equation and computing an inverse function one gets   2 2 4 ω − ω0  . (3.6) p ϕs1 (t) = ± arccos 1 − 2 2 2ω − ω0 1 − cosh 2 ω 2 − ω02 t Differentiation of (3.6) with respect to time yields p ω 2 − ω02 t 4ω0 ω 2 − ω02 sinh p ϕ˙ s1 (t) = ∓ 2ω 2 − ω02 1 − cosh 2 ω 2 − ω02 t p 2ω0 ω 2 − ω02 sinh ω 2 − ω02 t p . =∓ ω 2 + ω02 sinh2 ω 2 − ω02 t

(3.7)

The last two equations for |ω| > ω0 define two homoclinic orbits q01,2 = (ϕs1 , ϕ˙ s1 ) associated with the critical point ϕ01 = 0. Stable and unstable differential manifolds associated with the saddle ϕ02 = ±π are defined by the following equation q ϕ˙ s2 = ± 2ω02 (1 + cos ϕ) + 12 ω 2 (1 − cos 2ϕ). After some rearrangements one gets ±dϕ

dt = p

2ω02

(1 + cos ϕ) + ω 2 (1 − cos2 ϕ)

.

Integration of the last equation and computation of the inverse function yields   2 2 4 ω + ω0 p − 1 , ϕs2 (t) = ± arccos  2 2 2ω + ω0 1 + cosh 2 ω 2 + ω02 t where sign “+” is taken for t ≥ 0, whereas sign “-” holds for t < 0. Differentiation with respect to time gives p 4ω02 ω 2 + ω02 cosh ω 2 + ω02 t p , ϕ˙ s2 (t) = ± 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t

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31

and hence 

 4 ω 2 + ω02 p − 1 sgn t ϕs2 (t) = arccos  2 2 2 2 2ω + ω0 1 + cosh 2 ω + ω0 t and consequently p 4ω02 ω 2 + ω02 cosh ω 2 + ω02 t p . ϕ˙ s2 (t) = ± 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t The last two equations define two successive homoclinic orbits (ϕs2 , ϕ˙ s2 ) associated with the critical point ϕ02 = ±π. Along the coordinate ψ the homoclinic orbits are defined by the formulas ψs2 (t) = π + ϕs2 (t) = π  + sgn t · arccos 

2

2ω 2

+

ω02

ω02



(3.8) 4 ω + p − 1 , 2 2 1 + cosh 2 ω + ω0 t

and p 4ω02 ω 2 + ω02 cosh ω 2 + ω02 t p . (3.9) ψ˙ s2 (t) = ϕ˙ s2 (t) = ± 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t Equations (3.8) and (3.9) define two homoclinic orbits q03,4 = ψs2 , ψ˙ s2 associated with the critical point ψ02 = 0. Melnikov function associated with homoclinic orbits (ϕs1 , ϕ˙ s1 ) occurring for |ω| > ω0 is described by the following formula Z∞ Ms1 (t0 ) = ϕ˙ s1 (t) (γ cos ω1 (t + t0 ) − δ ϕ˙ s1 (t) −∞

2 −λµλ ϕ˙ s1 (t) + ω 2 sin2 ϕs1 (t) + ω02 cos ϕs1 (t) sgn (ϕ˙ s1 (t) − $) dt. (3.10) Substituting (3.7) for “+” into (3.10) one gets (|ω| > ω0 ) Ms1+ (t0 ) = I6 (t0 ) − I7 − I8 − I9 − I10 ,

(3.11)

where: 2

I6 (t0 ) = 4γω0 ω 2 − ω0

Z∞

−∞

ω 2 − ω02 t cos ω1 (t + t0 ) p dt, 2ω 2 − ω02 1 − cosh 2 ω 2 − ω02 t sinh

p

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2 2

I7 = 16δω02 ω 2 − ω0

Z∞

−∞

I8 = 64λµλ ω03 ω 2 − ω02 ×

Z∞

−∞

p sinh2 ω 2 − ω02 t p 2 dt, 2ω 2 − ω02 1 − cosh 2 ω 2 − ω02 t

3

p sinh3 ω 2 − ω02 t p 3 sgn (ϕ˙ s1 (t) − $) dt, 2ω 2 − ω02 1 − cosh 2 ω 2 − ω02 t

2 I9 = 64λµλ ω 2 ω03 ω 2 − ω02 p p Z∞ sinh ω 2 − ω02 t cosh2 ω 2 − ω02 t × p 3 sgn (ϕ˙ s1 (t) − $) dt, 2 − ω 2 1 − cosh 2 ω 2 − ω 2 t 2ω −∞ 0 0 I10 = 8λµλ ω03 ω 2 − ω02 p p Z∞ sinh ω 2 − ω02 t ω02 1 + cosh2 ω 2 − ω02 t − ω 2 × p 2 2ω 2 − ω02 1 − cosh 2 ω 2 − ω02 t −∞ × sgn (ϕ˙ s1 (t) − $) dt. Formula I6 is I6 (t0 ) = I61 (t0 ) − I62 (t0 ) , where I61 (t0 ) = 4γω0 ω 2 − ω02 cos ω1 t0 p Z∞ sinh ω 2 − ω02 t cos ω1 t p dt, × 2ω 2 − ω02 1 − cosh 2 ω 2 − ω02 t −∞ I62 (t0 ) = 4γω0 ω 2 − ω02 sin ω1 t0 p Z∞ ω 2 − ω02 t sin ω1 t sinh p dt. × 2 − ω 2 1 − cosh 2 ω 2 − ω 2 t 2ω 0 0 −∞ In order to get I61 one may notice that the under integral expression is an odd function, and hence I61 (t0 ) = 0.

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33

Therefore, it follows that I6 (t0 ) = −I62 (t0 ) = −4γω0 ω 2 − ω02 A62 sin ω1 t0 ,

(3.12)

where constant A62 is A62 =

Z∞ −∞

p ω 2 − ω02 t sin ω1 t p dt. 2ω 2 − ω02 1 − cosh 2 ω 2 − ω02 t sinh

Since the formula of I7 gives p sinh 2 ω 2 − ω02 t p I7 = δω02 ω 2 + ω02 sinh2 ω 2 − ω02 t √ p ω 2 −ω02 2 2 2 atanh tanh ω − ω0 t !∞ ω p − , ω ω 2 − ω02 −∞ hence I7 = 8δω02

1 1 − p atanh ω02 ω ω 2 − ω02

p

ω 2 − ω02 ω

!!

.

(3.13)

Recall the following identity sgn (ϕ˙ s1 (t) − $) =

−1, $ > ϕ˙ s1 (t) . 1, $ < ϕ˙ s1 (t)

(3.14)

A change of sign appears at time instant when the bush angular velocity ϕ˙ s1 (t) is equal to the pivot velocity $. Applying (3.7) with reference to sign “+” gives p 2ω0 ω 2 − ω02 sinh ω 2 − ω02 t p . $ = ϕ˙ s1 (t) = (3.15) ω 2 + ω02 sinh2 ω 2 − ω02 t Formula (3.15) for ω0 = 1, ω = 3, $ = 1 is illustrated in Figure 3.4. After some rearrangements one gets $ω02 x2 − 2ω0 ω 2 − ω02 x + $ω 2 = 0, (3.16) where the following formula has been applied q 2 2 x = sinh ω − ω0 t . Equation (3.16) has the following discrimi 2 2 2 2 2 4 nant ∆ = 4ω0 ω − 2ω0 − $ ω + ω0 . Its sign governs a number of

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Fig. 3.4

Drawing of formula (3.15) for ω0 = 1, ω = 3, $ = 1.

roots of this equation. For ∆ < 0; |$| > ω2 − ω02 ω real roots do not exist. Assuming ∆ ≥ 0; |$| ≤ ω 2 − ω02 ω , equation (3.16) has the two following roots x1,2 =

ω 2 − ω02 ±

p

(ω 2 − 2ω02 − $2 ) ω 2 + ω04 . $ω0

(3.17)

Therefore, one may write sinh

q

ω2

−

ω02 t1,2

=

ω 2 − ω02 ±

p

(ω 2 − 2ω02 − $2 ) ω 2 + ω04 , $ω0

(3.18)

and also asinh x1 t1 = p ω 2 − ω02 1 =p asinh 2 ω − ω02 asinh x2 t2 = p ω 2 − ω02 1 =p asinh ω 2 − ω02

ω 2 − ω02 −

ω 2 − ω02 +

p

(ω 2 − 2ω02 − $2 ) ω 2 + ω04 $ω0

p

(ω 2 − 2ω02 − $2 ) ω 2 + ω04 $ω0

!

, (3.19)

!

.

To conclude, formula (3.14) in the considered case for ∆ < 0 at each time instant has the value of -1, whereas for ∆ ≥ 0 in time interval t ∈ (t1 , t2 ) it has the value of 1, and for t ∈ (−∞, t1 )∪(t2 , ∞) it is equal to -1. Proceeding

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35

√

ω 2 −ω02 ω

in computation of integral I8 , the following original p function is computed 3 Z sinh ω 2 − ω02 t 3 2 2 3 ¯ I8 (t) = 64λµλ ω0 ω − ω0 p 3 dt 2ω 2 − ω02 1 − cosh 2 ω 2 − ω02 t q ! 2 2 ω − 4ω ω 0 0 atanh p cosh ω 2 − ω02 t = −λµλ ω 2 − ω02 p ω 2 − ω02 ω 2 − ω02 p ω0 ω02 ω 2 − 4ω02 cosh 3 ω 2 + ω02 t − p 2 4 ω 2 + ω02 sinh2 ω 2 + ω02 t p ω0 4ω 4 + 9ω 2 ω02 − 4ω04 cosh ω 2 + ω02 t + . p 2 4 ω 2 + ω02 sinh2 ω 2 + ω02 t For ∆ < 0 one has ∞ I8 = −I¯8 (t) −∞ = 0, whereas for ∆ ≥ 0 we have t2 ∞ t1 I8 = −I¯8 (t) −∞ + I¯8 (t) t1 − I¯8 (t) t2 = −2 I¯8 (t1 ) − I¯8 (t2 ) . Finally, taking (3.19) into account, one gets I8 = 2λµλ ω 2 − ω02 " ! !! p p ω 2 − 4ω02 ω0 x21 + 1 ω0 x22 + 1 p atanh p − atanh p ω 2 − ω02 ω 2 − ω02 ω 2 − ω02 p ω0 x21 + 1 ω02 ω 2 − 4ω02 1 + 4x21 − 4ω 4 − 9ω 2 ω02 + 4ω04 − 2 4 (ω 2 + ω02 x21 ) p # ω0 x22 + 1 ω02 ω 2 − 4ω02 1 + 4x22 − 4ω 4 − 9ω 2 ω02 + 4ω04 + , 2 4 (ω 2 + ω02 x22 ) where x1,2 are defined by formula (3.17). Hence, the following formulas are obtained I8 = 2λµλ ω 2 − ω02 × √ 2 2  ω −ω0   0 for $ ≥  ω   √ √    ω 2 − 4ω02  ω0 x22 +1 ω0 x21 +1    √ √  p atanh − atanh   2 −ω 2 2 −ω 2  2  2 ω ω  0 0   pω − ω0  2 2 2 2 4 2 2 4  2 × ω x + 1 ω ω − 4ω 1 + 4x − 4ω − 9ω ω + 4ω 0 0 0 1 0 0 1   −  2    4 (ω2 + ω02 x21 )  p     2 2 2 2 4 2 2 4 2   ω0 x2 + 1 ω0 ω − 4ω0 1 + 4x2 − 4ω − 9ω ω0 + 4ω0        +  2 2 2 2 4 (ω + ω0 x2 ) (3.20) for $
ω0 ) q 8δω 2 0 2 4ω0 ω 2 − ω0 |γA62 | > p ω 2 − ω02 √ 2 2 !!  p ω −ω  2 2 ω − ω0 1 1 0 for $ ≥ √ ω 0 p × − atanh +  B for $ < ω2 −ω02 ω02 ω ω ω 2 − ω02 s1 ω Bs1 = 2λµλ

(3.26)

Substituting (3.7) for sign “-” into (3.10) one may write (|ω| > ω0 ) Ms1− (t0 ) = −I6 (t0 ) − I7 + I11 + I12 + I13 ,

(3.27)

where I6 and I7 are defined by formulas (3.12) and (3.13). In addition, one gets I11 = 64λµλ ω03 ω 2 − ω02

3 2

p Z∞ sinh3 ω 2 − ω02 t × p 3 sgn (−ϕ˙ s1 (t) − $) dt, 2ω 2 − ω02 1 − cosh 2 ω 2 − ω02 t −∞ 3 I12 = 64λµλ ω 2 ω03 ω 2 − ω02 2 p p Z∞ sinh ω 2 − ω02 t cosh2 ω 2 − ω02 t × p 3 sgn (−ϕ˙ s1 (t) − $) dt, 2 − ω 2 1 − cosh 2 ω 2 − ω 2 t 2ω −∞ 0 0

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I13 = 8λµλ ω03

39

q

ω 2 − ω02 p p Z∞ sinh ω 2 − ω02 t −ω 2 + ω02 1 + cosh2 ω 2 − ω02 t × p 2 2ω 2 − ω02 1 − cosh 2 ω 2 − ω02 t −∞ × sgn (−ϕ˙ s1 (t) − $) dt.

Note that sgn (−ϕ˙ s1 (t) − $) =

−1, $ > −ϕ˙ s1 (t) , 1, $ < −ϕ˙ s1 (t)

(3.28)

and the sign changes when the bush angular velocity ϕ˙ s1 (t) is equal to the pilot angular velocity $. According to (3.7) and for “-” one gets p p 2ω0 ω 2 − ω02 sinh ω 2 − ω02 t p . $ = ϕ˙ s1 (t) = − (3.29) ω 2 + ω02 sinh2 ω 2 − ω02 t After some rearrangements we get q $ω02 x2 + 2ω0 x ω 2 − ω02 + $ω 2 = 0,

(3.30)

where the following formula has been used q 2 2 x = sinh ω − ω0 t . Discriminant of equation (3.30) is ∆ = 4ω02 1 − $2 ω 2 − ω02 , and its sign defines the. number of roots of this equation. For ∆ < p 0; $ > ω 2 − ω02 ω no real roots exist. Assuming ∆ ≥ 0; . p $ ≤ ω 2 − ω02 ω , equation (3.30) has two roots defined by the formula r r ω 2 − ω02 ω 2 − ω02 x1,2 = − ± − ω2 , (3.31) $2 $2 and since r r q 2 − ω2 ω ω 2 − ω02 0 2 sinh ω 2 − ω0 t3,4 = − ± − ω2 (3.32) 2 $ $2 one gets ! r r −1 ω 2 − ω02 ω 2 − ω02 2 t3 = −t2 = p asinh + −ω , $2 $2 ω 2 − ω02 ! (3.33) r r −1 ω 2 − ω02 ω 2 − ω02 t4 = −t1 = p asinh − − ω2 . $2 $2 ω 2 − ω02

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To conclude, for ∆ < 0 formula (3.28) has its value -1 for any time, for ∆ ≥ 0 in time interval t ∈ (t3 , t4 ) its value is 1, whereas for t ∈ (−∞, t3 )∪(t4 , ∞) its value is -1. In order to compute integral I11 for ∆ < 0 one has ∞ I11 = −I¯11 (t) = 0, −∞

where I¯11 (t) = I¯8 (t) is defined by (5.20). In contrary, at ∆ ≥ 0 one gets t3 t4 ∞ I11 = −I¯11 (t) −∞ + I¯11 (t) t − I¯11 (t) t = −2 I¯11 (t3 ) − I¯11 (t4 ) 3 4 −2 I¯8 (−t2 ) − I¯8 (−t1 ) . Observe that I¯8 (−t) = I¯8 (t), and hence I11 = −I8 . According to (3.19) for this case one gets I11 = −2λµλ ω 2 − ω02 " ! !! p p ω 2 − 4ω02 ω0 x21 + 1 ω0 x22 + 1 × p atanh p − atanh p ω 2 − ω02 ω 2 − ω02 ω 2 − ω02 p ω0 x21 + 1 ω02 ω 2 − 4ω02 1 + 4x21 − 4ω 4 − 9ω 2 ω02 + 4ω04 − 2 4 (ω 2 + ω02 x21 ) p # ω0 x22 + 1 ω02 ω 2 − 4ω02 1 + 4x22 − 4ω 4 − 9ω 2 ω02 + 4ω04 + , 2 4 (ω 2 + ω02 x22 ) where x1,2 are defined by formula (3.17). Finally, one gets I11= −2λµλ ω 2 − ω02 0 for $ ≥ $ ˆ    √ √  2 2  2 +1 ω − ω   ω0 x22 +1 ω x 0 0   1   p atanh √ 2 2 − atanh √ 2 2    2  2 ω −ω ω −ω  0 0  ω − ω  0  p 2 2 2 2 4 2 2 4  2 ω0 x1 + 1 ω0 ω − ω0 1 + 4x1 − 4ω − 9ω ω0 + 4ω0  × −   2 2 2 2   4 (ω  p  + ω0 x21)    2 2 2 4 2 2 4  2   ω x + 1 ω ω − ω 1 + 4x − 4ω − 9ω ω + 4ω  0 0 0 2 0 0  2      +  2 2 2 4 (ω 2 + ω0 x1 ) (3.34) √ for$ < $ ˆ

July 19, 2007

where $ ˆ :=

ω 2 −ω02 . ω

Computing I12 for ∆ < 0 one gets ∞ ¯ I12 = −I12 (t) = 0, −∞

where I¯12 (t) = I¯9 (t) is defined by formula (3.21). However, for ∆ ≥ 0 one obtains t3 t4 ∞ I12 = −I¯12 (t) −∞ + I¯12 (t) t3 − I¯12 (t) t4 = −2 I¯12 (t3 ) − I¯12 (t4 )

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41

−2 I¯9 (−t2 ) − I¯9 (−t1 ) .

Since I¯9 (−t) = I¯9 (t) then I12 = −I9 . According to (3.19), the following formula is found I12 = −2λµλ ω 2 ! !! " p p ω0 x21 + 1 ω0 x22 + 1 1 × p atanh p − atanh p ω 2 − ω02 ω 2 − ω02 ω 2 − ω02 # p p −ω 2 + ω02 2 + x21 ω0 x21 + 1 −ω 2 + ω02 2 + x22 ω0 x22 + 1 − + , 2 2 (ω 2 + ω02 x21 ) (ω 2 + ω02 x22 ) where x1,2 is defined by (3.17). Finally, one gets I12= −2λµλ ω 2  ˆ  √ 0 √  for $ ≥ $   1  ω0 x21 +1 ω0 x22 +1    p  atanh √ 2 2 − atanh √ 2 2    ω −ω0 ω −ω0   ω 2 − ω02    p  2 2 2 2  −ω + ω0 2 + x2 ω0 x2 + 1 × +  for $ < $ ˆ 2    (ω 2 + ω02 x22 ) p         −ω 2 + ω02 2 + x21 ω0 x21 + 1      −   2 2 2 2 (ω + ω0 x1 ) (3.35) √ where $ ˆ :=

ω 2 −ω02 . ω

Computing integral I13 for ∆ < 0 one gets ∞ ¯ I13 = −I13 (t) = 0, −∞

where I¯13 (t) = I¯10 (t) is defined by formula (3.22). However, for ∆ ≥ 0 one gets t3 t4 ∞ I13 = −I¯13 (t) −∞ + I¯13 (t) t3 − I¯13 (t) t4 = −2 I¯13 (t3 ) − I¯13 (t4 ) = −2 I¯10 (−t2 ) − I¯10 (−t1 ) . Observe that I¯10 (−t) = I¯10 (t) and hence I13 = −I10 . According to (3.19), for this case we have I13 =

−4λµλ ω03

! p p x21 + 1 x22 + 1 − 2 , ω 2 + ω02 x21 ω + ω02 x22

where x1,2 are defined by (3.17). To conclude, one gets  √ 2 2 ω −ω0   0 for $ ≥ ω p p 3 √ I13 = −4λµλ ω0 . x21 + 1 x22 + 1 ω 2 −ω02   2 − 2 for $ < 2 2 2 2 ω ω + ω0 x1 ω + ω0 x2

(3.36)

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Substituting (3.12), (3.13) and (3.34), (3.36) into (3.27) we obtain q Ms1− (t0 ) = 4γω0 ω 2 − ω02 A62 sin ω1 t0 !! p ω 2 − ω02 8δω02 1 1 +p − p atanh ω ω 2 − ω02 ω02 ω ω 2 − ω02 (3.37)  √ 2 2 ω −ω0 0 for $ ≥ √ 2ω 2 . − ω −ω0 B for $ < s1

ω

where Bs1 is defined by formula (3.25). Criterion of chaos occurrence for the orbit q02 = (ϕs1 , −ϕ˙ s1 ) is defined by (|ω| > ω0 ) q 8δω 2 0 2 2 4ω0 ω − ω0 |γA62 | > p ω 2 − ω02 √ 2 2 !!  p ω −ω  ω 2 − ω02 1 1 0 for $ ≥ √ ω 0 p × − atanh − .  B for $ < ω2 −ω02 ω02 ω ω ω 2 − ω02 s1 ω (3.38) Using (3.11) and (3.27) one may match both Melnikov’s functions into the formula Ms1 (t0 ) = ±I6 (t0 ) − I7 ∓ I8 ∓ I9 ∓ I10 .

(3.39)

Applying (3.24) and (3.37) both chaos criteria are given by the following formula q 8δω 2 0 2 2 4ω0 ω − ω0 |γA62 | > p ω 2 − ω02 √ 2 2 !!  p ω −ω  2 2 ω − ω0 1 1 0 for $ ≥ √ ω 0 p × − atanh ± .  B for $ < ω2 −ω02 ω02 ω ω ω 2 − ω02 s1

ω

(3.40)

Melnikov’s function for the orbit q03,4 = ψs2 , ψ˙ s2 is defined by Ms2 (t0 ) =

Z∞

ψ˙ s2 (t) γ cos ω1 (t + t0 ) − δ ψ˙ s2 (t)

−∞

2 −λµλ ψ˙ s2 (t) + ω 2 sin2 ψs2 (t) − ω02 cos ψs2 (t) sgn ψ˙ s2 (t) − $ dt. Applying of (3.8) and (3.9) for sign “+” one gets Ms2+ (t0 ) = I1 (t0 ) − I2 − I3 − I4 + I5 ,

(3.41)

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where 2

I1 (t0 ) = 4γω0 ω 2 + ω0

Z∞

−∞

I2 = 16δω02 ω 2 + ω02

2

Z∞

−∞

I3 = 64λµλ ω03 ω 2 + ω02

3

p cosh2 ω 2 + ω02 t p 2 dt, 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t Z∞

−∞

p ω 2 + ω02 t cos ω1 (t + t0 ) p dt, 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t cosh

p cosh3 ω 2 + ω02 t p 3 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t

× sgn ψ˙ s2 (t) − $ dt, 2 I4 = −32λµλ ω 2 ω03 ω 2 + ω02 p p Z∞ 1 − cosh 2 ω 2 + ω 2 t cosh ω 2 + ω02 t 0 × p 3 2 + ω 2 1 + cosh 2 ω 2 + ω 2 t 2ω −∞ 0 0 × sgn ψ˙ s2 (t) − $ dt, I5 = −4λµλ ω0 ω 2 + ω02 p p Z∞ 2ω 2 + ω 2 3 − cosh 2 ω 2 + ω 2 t cosh ω 2 + ω02 t 0 0 × p 2 2 + ω 2 1 + cosh 2 ω 2 + ω 2 t 2ω −∞ 0 0 × sgn ψ˙ s2 (t) − $ dt. On the other hand, I1 (t0 ) is given in the form I1 (t0 ) = I11 (t0 ) + I12 (t0 ) , where: I11 (t0 ) = −4γω0 ω 2 + ω02 sin ω1 t0 p Z∞ cosh ω 2 + ω02 t sin ω1 t p dt, × 2 + ω 2 1 + cosh 2 ω 2 + ω 2 t 2ω 0 0 −∞

43

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I12 (t0 ) = 4γω0 ω 2 + ω02 cos ω1 t0 p Z∞ cosh ω 2 + ω02 t cos ω1 t p dt. × 2 + ω 2 1 + cosh 2 ω 2 + ω 2 t 2ω 0 0 −∞ Before computing I11 (t0 ), one may observe that the under integral function is odd and hence I11 (t0 ) = 0, which gives I1 (t0 ) = γA12 cos ω1 t0 ,

(3.42)

where constant A12 is defined by p Z∞ ω 2 + ω02 t cos ω1 t cosh p dt. A12 = 8ω0 ω 2 + ω02 2 + ω 2 1 + cosh 2 ω 2 + ω 2 t 2ω 0 0 0 Since q ! ω02 ω 2 I2 = 2δ atanh p tanh ω 2 + ω0 t ω ω 2 + ω02 p ! p ∞ ω ω 2 + ω02 sinh 2 ω 2 + ω02 t p + 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t −∞ hence I2 = 4δ

ω02 atanh ω

ω

!

! q 2 2 + ω + ω0 .

(3.43) ω 2 + ω02 In order to compute other terms of Melnikov function, some relations useful for further considerations are introduced. At the beginning, the following relation is introduced −1, $ > ψ˙ (t) s2 ˙ sgn ψs2 (t) − $ = . (3.44) 1, $ < ψ˙ s2 (t) Sign change occurs when angular velocity of the bush ψ˙ is equal to the p

pivot velocity $. Using (3.9) and sign “+” one gets p ω 2 + ω02 t 4ω0 ω 2 + ω02 cosh p $ = ψ˙ s2 (t) = 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t p 2ω0 ω 2 + ω02 cosh ω 2 + ω02 t p . = ω 2 + ω02 cosh2 ω 2 + ω02 t

(3.45)

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45

Fig. 3.5 Graph of relation (3.45) for ω0 = 3, ω = 1, $ = 2, 5 (left) and for ω0 = 1, ω = 3, $ = 2, 5 (right).

Figure 3.5 illustrates relation (3.45) for ω0 = 3, ω = 1, $ = 2, 5 (left) and for ω0 = 1, ω = 3, $ = 2, 5 (right). Owing to the application of hyperbolic trigonometry transformations, the following equation is obtained $ω02 x2 − 2xω0 ω 2 + ω02 + $ω 2 = 0,

(3.46)

where the following formula has been applied q x = cosh ω 2 + ω02 t ≥ 1. 2 Equation (3.46) has the determinant ∆ = 4ω02 ω 2 + ω02 − $2 ω 2 , and its sign governs a number ofroots of the mentioned equation. 2 2 For ∆ < 0 |$| > ω + ω ω no real roots exist. Assuming ∆ ≥ 0 0 2 2 |$| ≤ ω + ω0 ω , equation (3.46) provides the following two solutions ω 2 + ω02 x1 = + $ω0

s

ω 2 + ω02 $ω0

2

ω 2 + ω02 x2 = − $ω0

s

ω 2 + ω02 $ω0

2

−

ω ω0

2

−

ω ω0

2

, (3.47) ,

which are physically realized when x1,2 ≥ 1. The latter condition is satisfied for x1 , when $ > 0; it holds for x2 , when $ > 2ω0 . Owing to the mentioned constraints one gets s q 2 2 2 2 2 ω + ω ω + ω02 ω 0 2 2 cosh ω + ω0 t1,2 = ± − , (3.48) $ω0 $ω0 ω0

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and

acosh x1 t1 = − p ω 2 + ω02



2 2  ω + ω0 + acosh = −p $ω0 ω 2 + ω02

1

acosh x1 t2 = p ω 2 + ω02



2 2  ω + ω0 + acosh =p $ω0 ω 2 + ω02

1

s

s

ω 2 + ω02 $ω0

ω 2 + ω02 $ω0

2

2

−

ω ω0

ω ω0

−

−

2

2



,

(3.49)



,

and also

acosh x2 t3 = − p ω 2 + ω02



2 2  ω + ω0 − = −p acosh $ω0 ω 2 + ω02

1

acosh x2 t4 = p ω 2 + ω02



2 2  ω + ω0 − =p acosh $ω0 ω 2 + ω02

1

s

s

ω 2 + ω02 $ω0

ω 2 + ω02 $ω0

2

2

−

ω ω0

ω ω0

2

2



,

(3.50)



.

Observe that formula (3.44) for ∆ < 0 for any time instant has the value of -1, whereas for ∆ ≥ 0 two cases are possible: for $ < 2ω0 in interval t ∈ (t1 , t2 ) the mentioned formula has the value 1. For t ∈ (−∞, t1 )∪(t2 , ∞) it is equal to -1, whereas for $ ≥ 2ω0 formula (3.44) is equal to -1 for t ∈ (−∞, t1 )∪(t3 , t4 )∪(t2 , ∞), and has the value of 1 for t ∈ (t1 , t3 )∪(t4 , t2 ).

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47

In order to compute integral I3 the following original function is defined p 3 Z 2 + ω2 t cosh ω 3 0 I¯3 (t) = 64λµλ ω03 ω 2 + ω02 p 3 dt 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t " q ! ω0 2 2 2 2 = λµλ ω + 4ω0 arctan p sinh ω + ω0 t ω 2 + ω02 p p ω0 ω 2 + ω02 ω02 ω 2 + 4ω02 sinh 3 ω 2 + ω02 t + p 2 4 ω 2 + ω02 cosh2 ω 2 + ω02 t p # 4ω02 + 9ω 2 ω02 − 4ω 2 sinh ω 2 + ω02 t + . p 2 4 ω 2 + ω02 cosh2 ω 2 + ω02 t (3.51) For ∆ < 0 one gets ∞ I3 = −I¯3 (t) −∞ = −2 lim I¯3 (t) = −πλµλ ω 2 + 4ω02 , t→∞

whereas for ∆ ≥ 0 the two following cases should be considered. First, for $ < 2ω0 one gets t1 t2 ∞ I3 = −I¯3 (t) −∞ + I¯3 (t) t1 − I¯3 (t) t2 = −2 lim I¯3 (t) − 4I¯3 (t1 ) , t→∞

where the following relation has been applied t2 = −t1 . According to (3.49) for the considered case one gets " ! √ ω x − 1 0 1 I3 = −πλµλ ω 2 + 4ω02 + λµλ 4 ω 2 + 4ω02 arctan p ω 2 + ω02 p # √ ω0 ω 2 + ω02 x1 − 1 ω02 ω 2 + 4ω02 1 − 4x21 + 4ω02 + 9ω 2 ω02 − 4ω 2 + , 2 (ω 2 + ω02 x21 ) where x1,2 are defined by (3.47). In the second case, i.e. for $ ≥ 2ω0 one obtains t1 t3 t4 t2 ∞ I3 = − I¯3 (t) + I¯3 (t) − I¯3 (t) + I¯3 (t) − I¯3 (t) −∞

t1

t3

t4

−2 lim I¯3 (t) − 4 I¯3 (t1 ) − I¯3 (t3 ) . t→∞

t2

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Applying (3.49) and (3.50) we get I3 = −πλµλ ω 2 + 4ω02 " ! !! √ √ ω0 x1 − 1 ω0 x2 − 1 2 2 + λµλ 4 ω + 4ω0 arctan p − arctan p ω 2 + ω02 ω 2 + ω02 √ q x1 − 1 ω02 ω 2 + 4ω02 1 − 4x21 + 4ω02 + 9ω 2 ω02 − 4ω 2 2 2 + ω0 ω + ω0 2 (ω 2 + ω02 x21 ) !# √ x2 − 1 ω02 ω 2 + 4ω02 1 − 4x22 + 4ω02 + 9ω 2 ω02 − 4ω 2 − , 2 (ω 2 + ω02 x22 )

for $ < $k

where x1,2 are defined by formula (3.47). Finally, we get I3 = −πλµλ ω 2 + 4ω02 + λµλ   for $ ≥ $k 0 √      x −1 ω  2 2 1 0   √ 4 ω + 4ω arctan   0   ω 2 +ω02    √  √   2 2 2 2 2 2 2 2 2 2   ω0 ω +ω0 x1 −1(ω0 (ω +4ω0 )(1−4x1 )+(4ω0 +9ω ω0 −4ω ))     + 2   2 2 2   ω +ω0 x1 ) (         0 for $ < 2ω 0    ×                   √     −     4 ω 2 + 4ω 2 arctan ω√0 x2 −1         0 2 +ω 2    ω     0  √  √    2 2 2 2 2 2 2 2 2 +ω 2 x −1 ω    ω ω ω +4ω 1−4x + 4ω +9ω ω −4ω ( ( )( ) ( ))  0 2 0 0 2 0 0 + 0        2 (ω2 +ω02 x22 ) where $k = ω 2 + ω02 ω. In order to compute integral I4 we proceed in a similar way to the previous one to get 2 I¯4 (t) = −32λµλ ω 2 ω03 ω 2 + ω02 p p Z 1 − cosh 2 ω 2 + ω 2 t cosh ω 2 + ω02 t 0 × p 3 dt 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t " ! q q ω0 2 2 + ω2 t = λµλ ω arctan p sinh ω − ω ω 2 + ω02 0 0 ω 2 + ω02 p p  2ω 2 − ω02 −3 + cosh 2 ω 2 + ω02 t sinh ω 2 + ω02 t  × . p 2 2 ω 2 + ω02 cosh2 ω 2 + ω02 t for $ ≥ 2ω0

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(3.52)

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49

Note that I¯4 (−t) = −I¯4 (t). For ∆ < 0 one obtains ∞ I4 = − I¯4 (t) = −2 lim I¯4 (t) = −πλµλ ω 2 , −∞

t→∞

whereas for ∆ ≥ 0 two different cases should be studied. First, for $ < 2ω0 we have t1 t2 ∞ I4 = − I¯4 (t) + I¯4 (t) − I¯4 (t) = −2 lim I¯4 (t) − 4I¯4 (t1 ) , −∞

t1

t2

t→∞

where t2 = −t1 . According to (3.49) one finally obtains ( " ! √ ω0 2 I4 = −λµλ ω π − 4 arctan p x1 − 1 ω 2 + ω02 p #) √ ω0 ω 2 + ω02 x1 − 1 ω 2 + ω02 2 − x21 − , 2 (ω 2 + ω02 x21 )

where x1,2 is defined by (3.47). In the second case, i.e. for $ ≥ 2ω0 one obtains t1 t3 t4 t2 ∞ I4 = − I¯4 (t) + I¯4 (t) − I¯4 (t) + I¯4 (t) − I¯4 (t) = −∞

t1

t3

t4

t2

−2 lim I¯4 (t) − 4 I¯4 (t1 ) − I¯4 (t3 ) . t→∞

Applying (3.49) and (3.50) we have ! ! ( " √ √ ω0 x1 − 1 ω0 x2 − 1 2 − arctan p I4 = −λµλ ω π − 4 arctan p ω 2 + ω02 ω 2 + ω02 q − ω0 ω 2 + ω02 √ !#) √ x1 − 1 ω 2 + ω02 2 − x21 x2 − 1 ω 2 + ω02 2 − x22 × − , 2 2 (ω 2 + ω02 x21 ) (ω 2 + ω02 x22 ) where x1,2 are defined by formula (3.47). To conclude, the following relation is obtained

          

for $ < $k

I4 = −πλµλ ω 2 + 4λµλ ω 2  for $ ≥ $k  √ 0 √  √  2 +ω 2 x −1 ω 2 +ω 2 2−x2  ω ω ( 1  0( 1 )) 0 arctan ω√0 x1 −12 − 0 2   ω 2 +ω02 x21 ) ω 2 +ω0 (       0 for $ < 2ω0   ×         −    √  √ 2 2√      ω ω +ω0 x2 −1(ω 2 +ω02 (2−x22 ))    arctan ω√0 2x2 −12 − 0 2 ω +ω0 (ω2 +ω02 x21 ) for $ ≥ 2ω0

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         

(3.53)

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where $k = ω 2 + ω02 ω. In order to compute I5 we first define I¯5 (t) = −4λµλ ω0 ω 2 + ω02 p p Z 2ω 2 + ω 2 3 − cosh 2 ω 2 + ω 2 t 2 + ω2 t cosh ω 0 0 0 dt × p 2 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t p p 2λµλ ω0 ω 2 + ω02 sinh ω 2 + ω02 t . p =− (3.54) ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t For ∆ < 0 we have ∞ I5 = − I¯5 (t) −∞ = −2 lim I¯5 (t) = 0, t→∞

(3.55)

whereas for ∆ ≥ 0 the two following cases are considered. First, for $ < 2ω0 one obtains t1 t2 ∞ I5 = − I¯5 (t) + I¯5 (t) − I¯5 (t) = −4I¯5 (t1 ) . −∞

t1

t2

Application of (3.49) and (3.50) yields p 8λµλ ω0 (ω 2 + ω02 ) (x1 − 1) I5 = . ω 2 + 2x1 ω02 In the second case, i.e. for $ ≥ 2ω0 one gets t1 t3 t4 t2 ∞ I5 = − I¯5 (t) −∞ + I¯5 (t) t − I¯5 (t) t + I¯5 (t) t − I¯5 (t) t 1 3 4 2 = −4 I¯5 (t1 ) − I¯5 (t3 ) . According to (3.49) and (3.50) we get √ √ q x1 − 1 x2 − 1 2 2 I5 = 32λµλ ω0 ω + ω0 − 2 , ω 2 + 2x1 ω02 ω + 2x2 ω02 where x1,2 are defined by formula (3.47). As a final result we obtain   0 for $ ≥ ωk     q  I5 = 32λµλ ω0 ω 2 + ω02 ( ) √   0 for $ < 2ω x − 1 0 1  √   x2 −1 ω 2 + 2x1 ω 2 − for $ ≥ 2ω0 0 ω 2 +2x2 ω02 (3.56) Substituting (3.42), (3.43) and (3.54), (3.56) into (3.41) gives p ω2 Ms2+ (t0 ) = γA12 cos ω1 t0 − 4δ ω0 atanh √ 2ω 2 + ω 2 + ω02 ω +ω0 +2πλµλ ω 2 + 2ω02  0 for $ ≥ ω 2 + ω02 ω  , −λµλ 0 for $ < 2ω0 for $ < ω 2 + ω02 ω  B (x1 ) − B (x2 ) for $ ≥ 2ω0 for $ < ωk

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51

where the following notation has been applied 2

B (x) = 4 ω +

4ω02

+ω

2

arctan

! √ ω0 x − 1 p ω 2 + ω02

√ q x−1 − 32ω0 ω 2 + ω02 2 ω + 2x1 ω02 p √ ω0 ω 2 + ω02 x − 1 ω02 ω 2 + 4ω02 1 − 4x2 + 2 (ω 2 + ω02 x2 ) 4ω02 + 9ω 2 ω02 − 4ω 2 − ω 2 ω 2 + ω02 2 − x2 + . 2 (ω 2 + ω02 x2 ) Homoclinic chaos criterion for the orbit q03 = ψs2 , ψ˙ s2 is defined by the formula ! q ! ω ω02 2 2 2 2 atanh p + ω + ω0 |γA12 | > 2πλµλ ω + 2ω0 − 4δ ω ω 2 + ω02  0 for $ ≥ ω 2 + ω02 ω  . −λµλ 0 for $ < 2ω0  B (x1 ) − for $ < ω 2 + ω02 ω B (x2 ) for $ ≥ 2ω0 (3.57) Matching (3.8) and (3.9) for sign “-” we get Ms2− (t0 ) = −I1 (t0 ) − I2 + I30 + I40 − I50 ,

(3.58)

where I1 (t0 ) and I2 are defined by (3.42) and (3.43), respectively: p 3 Z∞ 2 + ω2 t cosh ω 3 0 I30 = 64λµλ ω03 ω 2 + ω02 p 3 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t −∞ × sgn −ψ˙ s2 (t) − $ dt, 2 I40 = −32λµλ ω 2 ω03 ω 2 + ω02 p p Z∞ 1 − cosh 2 ω 2 + ω 2 t cosh ω 2 + ω02 t 0 × p 3 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t −∞ × sgn −ψ˙ s2 (t) − $ dt,

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I50 = −4λµλ ω0 ω 2 + ω02 p p Z∞ 2ω 2 + ω 2 3 − cosh 2 ω 2 + ω 2 t 2 + ω2 t cosh ω 0 0 0 × p 2 2ω 2 + ω02 1 + cosh 2 ω 2 + ω02 t −∞ × sgn −ψ˙ s2 (t) − $ dt. Since −1, $ > −ψ˙ (t) s2 ˙ sgn −ψs2 (t) − $ = 1, $ < −ψ˙ s2 (t)

(3.59)

sign change occurs when the angular bush velocity ψ˙ is equal to pivot velocity $. Applying (3.9) for sign “-” one gets p 2ω0 ω 2 + ω02 cosh ω 2 + ω02 t p , $ = −ψ˙ s2 (t) = − (3.60) ω 2 + ω02 cosh2 ω 2 + ω02 t and hence $ω02 x2 + 2xω0 ω 2 + ω02 + $ω 2 = 0,

(3.61)

where the following formula has been applied q 2 2 x = cosh ω + ω0 t ≥ 1. Since equation (3.61) is of the second order, then the sign of 2 ∆ = 4ω02 ω 2 + ω02 − $2 ω 2 2 determines the roots number. For ∆ < 0 $ > ω 2 + ω 0 ω the analyzed equation has no real solutions. If ∆ ≥ 0 $ ≤ ω 2 + ω02 ω , then equation (3.61) has the following two roots s 2 2 ω 2 + ω02 ω + ω02 x1 = − + − ω2 , $ $ (3.62) s 2 2 2 2 2 ω + ω0 ω + ω0 x2 = − − − ω2 . $ $ Observe that for $ > 0 one gets x1,2 < 1, and hence equation (3.61) has no physically realized solutions. It means that for $ > 0 we have sgn −ψ˙ s2 (t) − $ = −1.

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3.2

SmoothNonsmooth

53

Homoclinic Chaos Criterion

Integral I30 is defined by the formula ∞ ∞ ¯ I30 = −I30 (t) −∞ = −I¯3 (t) −∞ = −2 lim I¯3 (t) = −πλµλ ω 2 + 4ω02 , t→∞

(3.63) where I¯30 (t) = I¯3 (t) is given by (3.51). Integral I is given by the equation 40 ∞ ∞ I40 = − I¯40 (t) −∞ = − I¯4 (t) −∞ = −2 lim I¯4 (t) = −πλµλ ω 2 , (3.64) t→∞ where I¯40 (t) = I¯4 (t) is defined by formula (3.52). Expression I50 is governed by the following formula ∞ ∞ I50 = − I¯50 (t) −∞ = − I¯5 (t) −∞ = −2 lim I¯5 (t) = 0, (3.65) t→∞ where I¯50 (t) = I¯5 (t) is defined by equation (3.55). Substituting (3.42), (3.43) and (3.63), (3.64), (3.65) into (3.58) one gets ! q ! ω02 ω 2 2 Ms2− (t0 ) = −γA12 cos ω1 t0 − 4δ atanh p + ω + ω0 ω ω 2 + ω02 − 2πλµλ ω 2 + 2ω02 . Next, the criterion of homoclinic chaos for the orbit q04 = ψs2 , −ψ˙ s2 is defined by the following inequality! ! q ω02 ω 2 2 2 2 |γA12 | > 4δ atanh p + ω + ω0 + 2πλµλ ω + 2ω0 . ω ω 2 + ω02 (3.66)

Fig. 3.6 Chaotic threshold γ = γ($) generated by the chaos criteria for ω0 = 1; ω = 2; δ = 0.1; λµ = 0.1

To sum up, we have obtained four Melnikov’s criteria of homoclinic chaos. Figure 3.6 shows the chaotic threshold γ = γ($) generated by the chaos criteria (3.26), (3.40), (3.57), (3.66).

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3.3

Numerical Simulations

In order to verify the obtained results, the following parameters are fixed: $ = 0.2 and ω0 = 1; ω = 2; δ = 0.1; λµ = 0.1 and then one gets four critical values of the parameter γ; γcr1 ≈ 1.66; γcr2 ≈ 1.68; γcr3 ≈ 2.22; γcr4 ≈ 2.24. Figure 3.7 presents the bifurcation diagram of the rotated Froude pendulum for ω0 = 1; ω = 2; δ = 0.1; T0 = 0.1; $ = 0.2 with the amplitude γ as a control parameter.

Fig. 3.7

Bifurcation diagram for ω0 = 1; ω = 2; δ = 0.1; T0 = 0.1; $ = 0.2.

The bifurcation diagram confirms that near the critical values of γ, there are bifurcations of homoclinic type and a periodic window between them. For example, taking γ = 1.60 there is a periodic motion of the analyzed system (see Figure 3.8).

Fig. 3.8

Phase portrait for γ = 1.60; ω0 = 1; ω = 1; δ = 0.1; T0 = 0.2; $ = 0.2.

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55

Next, for γ = 1.66 chaotic motion with the fractal attractor occur (Figure 3.9).

Fig. 3.9 Phase portrait and Poincar´ e map for γ = 1.66; ω0 = 1; ω = 2; δ = 0.1; T0 = 0.1; $ = 0.2.

Almost the same situation is repeated above with respect to the periodic window (see Figure 3.6). Figure 3.10 presents periodic motions of the analyzed system for γ = 2.08 (the other parameters are the same as before). A relatively small change of the γ parameter to γ = 2.12 introduces a radical change of the phase portrait generated by the Froude rotated pendulum (see Figure 3.11). To conclude, the unperturbed system associated with the Froude pendulum has two or four homoclinic orbits. Their number depends on the value of the rotating velocity ω. A destruction of the homoclinic orbits due to relatively small forcing and friction causes the homoclinic bifurcation, and chaotic motions of the analyzed system. The applied Melnikov

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Fig. 3.10

Phase portrait for γ = 2.08; ω0 = 1; ω = 2; δ = 0.1; T0 = 0.1; $ = 0.2.

Fig. 3.11 Phase portrait and Poincar´ e map of a chaotic attractor for γ = 2.12; ω0 = 1; ω = 2; δ = 0.1; T0 = 0.1; $ = 0.2.

method enables prediction of mathematical relations of the system parameters describing the homoclinic bifurcation. Analytical thresholds for the occurrence of chaos have been analytically derived and confirmed by numerical simulations.

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Chapter 4

Smooth and Nonsmooth Dynamics of a Quasi-Autonomous Oscillator with Coulomb and Viscous Frictions In this chapter the results of investigations of occurrence of a chaotic stickslip and slip-slip dynamics in a very weakly forced oscillator, using both the Melnikov’s technique and the so called wandering trajectories approach are compared. A good agreement of the analytical chaotic threshold and numerical simulation are demonstrated. 4.1

Stick-Slip Oscillator with Periodic Excitation

Consider a mechanical system which consists of a mass m riding on a driving belt, as shown in Figure 4.1 (see also [3–6]). The belt is moving at constant velocity v∗ . The mass m is attached to inertial space by a Duffing-type spring, where k1 and k2 are stiffness coefficients. A friction force θ, which depends on the relative velocity, acts between the mass m and belt. Additionally, the mass m is driven by a small periodic external excitation Γ cos ωt. Γ and ω are the amplitude and frequency of excitation, respectively. The one-degree-of-freedom stick-slip oscillations are governed by the following second-order differential equation m¨ x − k1 x + k2 x3 = εΓ cos ωt − εθ (x˙ − v∗ ) ,

(4.1)

where ε > 0 is a perturbation parameter, 3

θ (x˙ − v∗ ) = θ0 sgn (x˙ − v∗ ) − A (x˙ − v∗ ) + B (x˙ − v∗ ) ,

(4.2)

and it corresponds to the ratio of the friction characteristic to the relative velocity. This function model is presented in Figure θ0 , A and B are friction coefficients. It is possible to rewrite this equation in the dimensionless form x ¨ − ax + bx3 = ε (γ cos ωt − T (x˙ − v∗ )) , 57

(4.3)

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Fig. 4.1

One-degree-of-freedom mechanical system exhibiting stick-slip oscillations.

where 3

T (x˙ − v∗ ) = T0 sgn (x˙ − v∗ ) − α (x˙ − v∗ ) + β (x˙ − v∗ ) , and a = k1 /m, b = k2 /m, γ = Γ/m, T = θ/m, T0 = θ0 /m, α = A/m β = B/m. In the paper [7] the Melnikov function for mechanical system (4.1)

Fig. 4.2

has been obtained.

Friction model.

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59

Then after computing the corresponding integrals, the Melnikov criterion has been applied to obtain the following inequality πω 16βa4 4a2 √ > − √ α − 3βv∗2 πγω sech 2 35b 2ab 3 2ab ! r r  q q  bv∗2 bv∗2 1 1 1 1 a 2To a for v∗ < √2ab , 2 + 4 − 2a2 − 2 − 4 − 2a2 +   a 0 for v∗ ≥ √2ab .

(4.4)

This inequality gives the possibility of chaotic threshold estimation in the forced stick-slip oscillator (4.1). 4.2

Analysis of the Wandering Trajectories

The chaotic behaviour of nonlinear deterministic systems supposes that the trajectories of motion wander around the various equilibrium states. They are characterized by unpredictability and sensitive dependence on the initial conditions. By analyzing trajectories of motion of these systems, it is possible to find the regions of chaotic vibrations in control parameters space. Let a dynamical system be expressed as the following set of ordinary differential equations x˙ = f (t, x) ,

(4.5)

where x ∈ Rn is the state vector, f is defined in R × Rn and describes the time derivative of the state vector. It is supposed, that f is smooth enough to guarantee existence and uniqueness of a solution of the equations (4.3). The right-hand side can be discontinuous while the solution of the set of differential equations (4.3) remains continuous. The continuous dependence property on the initial conditions x(0) = x (t0 ) of a solution of the equations (4.3) will be used: for every initial condition x(0) , x ˜(0) ∈ Rn , for every number T > 0, no matter how large, and for every preassigned arbitrary small ε > 0 it is possible to find a positive number δ > 0 such that if the distance ρ between x(0) and x ˜(0) , ρ x(0) , x ˜(0) and |t| ≤ T , the following inequality is satisfied ρ x(0) , x ˜(0) < ε. (4.6) That is if the initial points are chosen close enough, then during the preassigned arbitrary large time interval −T ≤ t ≤ T , the distance between

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simultaneous positions of moving points will be less given positive number ε. The metric ρ on Rn can be determined in various ways, for example v n n u X uX 2 ρ1 x(0) , x ˜(0) = t (xi − x ˜i ) , ρ2 x(0) , x ˜(0) = |xi − x ˜i |, i=1

i=1

or ρ3 x(0) , x ˜(0) = max |xi − x ˜i |, where x = (x1 , x2 , . . . , xn ) ∈ Rn , x ˜ = (˜ x1 , x ˜2 , . . . , x ñ ) ∈ Rn . It is assumed that the trajectories x (t) remain in a closed bounded domain of the space Rn , i.e. ∃Ci ∈ R : max |xi (t) | ≤ Ci . t

To analyze trajectories of the set (4.3), we introduce the characteristic vibration amplitudes Ai of components of the motion xi (t): 1 Ai = | max xi (t) − min xi (t) |, i = 1, 2, . . . , n. t1 ≤t≤T 2 t1 ≤t≤T Here [t1 , T ] ⊂ [t0 , T ] and [t0 , T ] is the time interval, in which the trajectory is considered. The interval [t0 , t1 ] is the time interval, in which all transient processes are damped. The characteristic vibration amplitudes Ai can be calculated simultaneously with the integration of the trajectory. By the embedding theorem if Sε (x) = {˜ x ∈ Rn : ρ (x, x ˜) < ε} is a hyper-sphere with center at the point x and with radius ε and Pε1 ,ε2 ,...,εn (x) = {˜ x ∈ Rn : |xi − x ˜i | < εi } is the n-dimensional parallelepiped then for any ε > 0 there is parallelepiped Pε1 ,ε2 ,...,εn (x) such that Pε1 ,ε2 ,...,εn (x) ⊂ Sε (x). And conversely, for any parallelepiped Pε1 ,ε2 ,...,εn (x) it is possible to indicate ε > 0 such that Sε (x) ⊂ Pε1 ,ε2 ,...,εn (x) . Let us choose in the parallelepiped Pδ1 ,δ2 ,...,δn x(0) two neighboring (0) (0) initial points x(0) and x ˜(0) , such that |xi − x ˜i | < δi , where δi is small in comparison with Ai . In the case of regular motion, it is expected that the εi (0) (0) in the inequality |xi − x ˜i | < εi is also small in comparison with Ai . The wandering orbits attempt to fill some bounded domain of the phase space. The neighboring trajectories at the instant t0 diverge exponentially on the (0) (0) average afterwards. Hence, the absolute values of differences |xi − x ˜i | for some instant t1 can take values in the interval [0, 2Ai ]. By analyzing the equilibrium states of (4.3) it is easy to choose an α parameter, 0 < α < 1, such that from the truth of the statement ∃t∗ ∈ [t1 , T ] : |x (t∗ )i − x ˜ (t∗ )i | > αAi ,

i = 1, 2, . . . , n

(4.7)

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61

it follows that there is a time interval or set of time intervals, for which the affixes of the closed at the initial instant trajectories x (t) and x ˜ (t) move around various equilibrium states of these trajectories are sensitive to changing of the initial conditions. Thus, these trajectories are wandering. Indeed, as it has already been mentioned, all trajectories are in the closed bounded domain of the space Rn . We choose the measure of divergence of the trajectories, which is inadmissible for the case of regularity of the motion. When the characteristic vibration amplitudes Ai are found, the divergence measures αAi of the observable trajectories in the directions of the generalized coordinates xi are determined by α. Let us briefly discuss the choice of the α parameter. Note that this choice is non-unique and the α parameter can take various values of the interval (0, 2). There are values of the α parameter, which a priori correspond to the inadmissible divergence measures αAi of the trajectories in the sense of regularity. For example, 1 1 2 3 α ∈ { , , , }. 3 2 3 4 Other choices are possible. If the representative points of the observable trajectories move chaotically, then for another choice α from the set of a priori appropriate α, the divergence of the trajectories will be recorded at another time t∗ .As numerical experiments show, the domains of chaotic behaviour obtained with various a priori appropriate values of the parameter α are practically congruent. Therefore, figures for different values of α are not presented. A similar non-unique choice of parameters occurs when applying another criteria for the chaotic oscillations. For instance, according to the procedure for calculation of the Lyapunov exponents d (t) = d0 2λt . Here λ is a Lyapunov exponent, d0 is the initial distance measure between the starting points, d (t) is the distance between trajectories at instant t. The base 2 is chosen for a convenience. In all other respects the parameter α > 1 in the relation d (t) = d0 αλt is arbitrary. That is α can take values, for example α = {2, 3, 4, 5} or other choices are possible. In general, the specificity required by numerical methods is such that, all parameters have to be concrete and most of them can be non-unique. The parameter α might have another physical interpretation. Assume that for the nonlinear dynamical system under investigation, it is possible to identify the singular points (equilibria). In the case, for instance, of two-well potential systems we have two nodes and one saddle. An external periodic excitation applied to such one-degree-of-freedom system may

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cause a chaotic response. Chaos is characterized by unpredictable switches between the two potential wells. A phase point may wander between all three singular points. Consider two neighboring nodes. As a result of a switch, representative points close at the initial instant of the phase trajectories, are in motion about different equilibrium states afterward. Hence a choice of α, in the relation αAi ' d2 , is related to the distance d between the two nodes separated by a saddle. However, many nonlinear dynamical systems do not have analytical solutions, and sometimes it is laborious to find the singular points and hence it is recommended the parameter α be taken from a priori appropriate values. Our approach has been successfully applied in the case of smooth and nonsmooth systems. By varying parameters and using condition (4.4), it is possible to find domains of chaotic motion (including transient and alternating chaos) and domains of regular motion. Remark. All inequalities (4.4) do not have to be checked for the case, when the equations of motion under investigation can be transformed to a normal form. It means that the inequalities related to velocities xj = x˙ i may be canceled. In other words, solutions related to regular motion with respect to xi are also regular in relation to xj = x˙ i . Here i, j ∈ {1, n}. 4.3

Comparison of Analytical and Numerical Results

The mechanical system (4.1) was investigated using both Melnikov’s technique [7] and approach based on the wandering trajectories analysis. The results obtained for a = b = 1, α = β = T0 = 0.3, ω = 2, x (0) = 1, v (0) = 0.4 are shown in the (v∗ , γ) plane in Figure 4.3. The γ (v∗ ) solid-curve presented in Figure 4.3 is plotted using an analytical prediction (4.2) and separates the graph into two parts. Above this curve, chaos can appear, because near the line, the stable and unstable manifolds intersect transversally. The domains of the stick-slip chaotic dynamics of the system (4.1), obtained on the basis of the wandering trajectories analysis are marked by dots. The condition (4.4) reads ∃t∗ ∈ [t1 , T ] : |x (t∗ ) − x ˜ (t∗ ) | > αA, where A=

1 | max x (t) − min x (t) |. t1 ≤t≤T 2 t1 ≤t≤T

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63

Fig. 4.3 Domains of stick-slip chaos in the (v∗ , γ) plane (a = b = 1, α = β = T0 = 0.3, ω = 2, x(0) = 1, v(0) = 0.4). The smooth chaotic threshold is obtained using Melnikov’s technique.

The time period for the simulation is 300 non-dimensional time units. During computations, a half period corresponds to the time interval [t1 , t0 ], where transitional processes are damped. The integration step size is equal to 3×10−3 in non-dimensional time units. The plane of parameters (v∗ , γ) is uniformly sampled in rectangle parallelepiped (0 < v∗ ≤ 1.1; 0 < γ ≤ 1.5) by 100 × 100 nodal points. Initial conditions of the closed trajectories are distinguished by 0.5 percent with ratio to characteristic vibration amplitudes A and α parameter is equal 1/3. The results obtained show a good agreement between the analytical chaotic threshold and numerical simulation. According to the approach applied in this paper, chaotic motion of the oscillator is observed before the cusp at √ a 2 v∗ = √ = ' 0.71. 2 2b Systems with friction are nonsmooth and cause some difficulties in both the theoretical and numerical analyses. We have obtained a more precise definition to the domains of the stick-slip and slip-slip chaotic dynamics of a one-degree-of-freedom very weakly forced oscillator in the (v∗ , γ) plane using a new approach based on analysis of the wandering trajectories. A comparison with analytical prediction, obtained using Melnikov’s technique

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[7], demonstrates a good agreement with the results presented. The standard numerical methods, in particular the direct calculations of Lyapunov exponents, are widely used in the literature, but they are timeconsuming. Our approach is effective, convenient to use, requires much less computational time in comparison with other approaches, and can be applied to an investigation of a wide class of problems. According to this approach the characteristic vibration amplitudes Ai ’s are produced simultaneously to the integration of the trajectory. Thus, it is sufficient to integrate two equations only for each selected trajectory.

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Chapter 5

Application of the Melnikov-Gruendler Method to Mechanical Systems The Melnikov-Gruendler method (MGM) is a generalization of the Melnikov method for mechanical systems with finite degrees-of-freedom. In this chapter we introduce fundamental assumptions and steps of the MGM approach with respect to a few degrees-of-freedom. Then we focus on 2-DOF mechanical systems. We show how this general MGM approach reduces to classical Melnikov’s algorithm for the case of 1-DOF. These considerations can be easily generalized for other classes of coupled oscillators.

5.1

Mechanical Systems with Finite Number of Degrees-ofFreedom

Below, we consider a n− DOF mechanical system governed by the equation x ¨ + S (x, x, ˙ a) = εG (x, x, ˙ t, b) ,

(5.1)

T

where x = [x1 , x2 , . . . xn ] denotes the vector of motion (trajectory), and n ∈ N denotes a number of DOF, S (x, x, ˙ a) is the matrix, and G (x, x, ˙ t, b) is the vector of perturbation depending on the system positions and velocities and is T -periodical in time t. A dynamic system governed by (5.1) de pends on the parameters: ai = {ai1 , ai2 , . . . aini }, bj = bj1 , bj2 , . . . bmj , i, j = 1 . . . n, ni , mj ∈ N. According to the general assumption associated with the application of any perturbation technique, we consider a small perturbation manifested by an introduction of small parameter ε ≥ 0. Equation (5.1) is given in the following equivalent form x˙ = v, v˙ = −S (x, v, a) + εG (x, v, t, b) . 65

(5.2)

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For ε = 0 one gets a Hamiltonian (by the assumption) unperturbed system of the form x ¨ + S (x, v, a) = 0,

(5.3)

x˙ = v, v˙ = −S (x, v, a) .

(5.4)

or

Let system (5.4) have in the phase coordinates (x, v) origin a singular point p0 associated with a differentiable homoclinic orbit described by the following parametric equation x0 (x (t) , v (t) , a) q0 (x (t) , v (t) , a) = (5.5) v0 (x (t) , v (t) , a) and satisfying the relation

lim q0 (t) = 0. If a saddle point is associ-

t→±∞

ated with more than one periodic orbit, then the consideration should be repeated for each of them. Define now the Poincaré transformation: Pε : Σt0 → Σt0 mapping (in a general case) a hyperspace Σt0 = {(x, v, t)|t = t0 ∈ [0, T ]} into itself. During this transformation the homoclinic orbit q0 (x (t) , v (t) , a) is mapped in a discrete manner by mapping Pε . Owing to non-zero perturbation (ε > 0), singular point p0 is shifted maximally from pε on plane Σt0 at a distance proportional to the first power of small parameter ε, and the homoclinic orbits split into stable W s (pε ) and unstable W n (pε ) differential manifolds defined by the following formulas, respectively s t0 n W (pε ) = (x, v) ∈ Σ lim Pε = pε , n→+∞ n t0 −n W (pε ) = (x, v) ∈ Σ lim Pε = pε . n→+∞ The main goal of both Melnikov and Melnikov-Gruendler methods is to estimate a distance between W s (pε ) and W n (pε ) in the form of the so called Melnikov function (vector), and then to estimate parameters a and b of the investigated dynamic system, for which a transversal intersection of two manifolds on plane Σt0 appears. According to the Smale-Birkhoff theorem, on the homoclinic trajectories [67], a set of intersection points of W s (pε ) and W n (pε ) is an empty set ∅ or its power is equal to the cardinal number ℵ0 . This property allows us to separate parameter space into two separable classes applying the concept of a hyper surface defined

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Application of the Melnikov-Gruendler Method to Mechanical Systems

67

by the Melnikov criterion. One may analytically predict parameters of the analyzed oscillator in order to realize its regular or chaotic dynamics. Note that in the classical Melnikov method one may estimate a distance between stable and unstable differential manifolds directly by equations (5.2) for n=1. According to the theory of perturbation, one may prove [33] that high order estimation of a distance between stable and unstable differential manifolds is given by a variational equation associated with homoclinic orbits and generated by equations (5.4). Therefore, in the MG method, linear homogeneous differential equations with non-constant coefficients are studied x˙ = v v˙ = −Dv S (x, v)|q0 (t) v − Dx S (x, v)|q0 (t) x,

(5.6)

or in the equivalent form of the second order ODEs x ¨ + Dx˙ S (x, x)| ˙ q0 (t) x˙ + Dx S (x, x)| ˙ q0 (t) x = 0,

(5.7)

or in the matrix form X˙ = S (t) · X,

(5.8)

T

where: X = [v1 , . . . , vn , x1 , . . . , xn ] and the matrix of 2n × 2n order is " # U 0 S (t) = , (5.9) −Dv S (x, v)|q0 (t) −Dx S (x, v)|q0 (t) where Un×n has unit elements on the second diagonal and remaining elements are equal to zero. According to the theorem of block matrices [64], matrix S (t) has the eigenvalue equal to 1 of multiplicity: n/2 for even n and (n + 1)/2 for odd n and the value of -1 of multiplicity: n/2 for even n and (n − 1)/2 for odd n. In addition, this matrix has n eigenvalues equal to the eigenvalues of matrix Dx S (x, v)|q0 (t) . Denote by γ (t) = ψ (1) (t) , . . . , ψ (2n) (t) a fundamental solutions matrix associated with equation (5.8) having vectors ψ (i) (t) with the following properties lim t−ki e−λi t ψ (i) (t) = c(i) ∧

t→∞

lim t−kσ(i) e−λσ(i) t ψ (i) (t) = c(σ(i)) ,

t→−∞

where i = 1, . . . , 2n, ki ∈ N, c(i) ∈ R2n is the constant, σ (i) denotes a certain permutation of the set of indexes i, which couples them into pairs, and λi are the eigenvalues of matrix S (0) exhibiting the earlier described properties of matrix S (t). Owing to an asymptotical behavior of vectors ψ (i) (t) they can be divided into one of the following four groups:

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(1) ss:

lim ψ (i) (t) = 0;

t→±∞

(i) (2) sn: lim ψ (t) = 0, lim ψ (t) = ∞; t→+∞ t→−∞ (3) ns: lim ψ (i) (t) = ∞, lim ψ (i) (t) = 0; t→+∞ t→−∞ (4) nn: lim ψ (i) (t) = ∞. t→±∞ (i)

Now, depending on which of the mentioned groups it belongs to, a measure Mi (t0 ) is defined to enable distance estimation between manifolds W s (pε ) and W n (pε ) on the plane Σt0 with the accuracy of higher order with respect to small parameterε. Next, the associated Melnikov functions are defined in the following way R∞ ss: Mi (t0 ) = Ki (t, t0 ) dt; sn: Mi (t0 ) = ns: Mi (t0 ) =

−∞ R0

Ki (t, t0 ) dt;

−∞ R∞

Ki (t, t0 ) dt;

0

nn: not defined; where: Ki (t, t0 ) n o = det ψ (1) (t) , . . . , ψ (i−1) (t) , G (q0 , t + t0 , b) , ψ (i+1) (t) , . . . , ψ (2n) (t) . (5.10) Knowledge of the Melnikov function allows us to define a homoclinic chaos criterion in the form of appropriate relations between parameters of the considered mechanical systems for which the Melnikov function has simple zeros.

5.2

2-DOFs Mechanical Systems

Let us consider the following 2-DOF mechanical systems (n = 2) governed by the equations x ¨1 + S1 (x1 , x2 , a1 ) = εG1 (x1 , x˙ 1 , x2 , x˙ 2 , t, b1 ) , (5.11) x ¨2 + S2 (x1 , x2 , a2 ) = εG2 (x1 , x˙ 1 , x2 , x˙ 2 , t, b2 ) . For ε = 0 one gets the following unperturbed system x ¨1 + S1 (x1 , x2 , a1 ) = 0, x ¨2 + S2 (x1 , x2 , a2 ) = 0,

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or equivalently x˙ 1 = v1 , v˙ 1 = −S1 (x1 , x2 , a1 ) , x˙ 2 = v2 , v˙ 2 = −S2 (x1 , x2 , a2 ) .

(5.12)

Let system (5.12) have in the phase coordinates origin a singular point p0 associated with a homoclinic orbit governed by the following parametric equation q0 (x1 (t) , x˙ 1 (t) , x2 (t) , x˙ 2 (t) , a1 , a2 )  x10 (x1 (t) , x˙ 1 (t) , x2 (t) , x˙ 2 (t) ,  x˙ 10 (x1 (t) , x˙ 1 (t) , x2 (t) , x˙ 2 (t) , =  x20 (x1 (t) , x˙ 1 (t) , x2 (t) , x˙ 2 (t) , x˙ 20 (x1 (t) , x˙ 1 (t) , x2 (t) , x˙ 2 (t) , that satisfies the following property

a1 , a1 , a1 , a1 ,

 a2 ) a2 )  , a2 )  a2 )

(5.13)

lim q0 (t) = 0. According to the

t→±∞

introduced assumptions and carrying out a linearization procedure along the homoclinic orbit, the following variational equations are obtained ∂S1 ∂S1 x1 + x2 = 0, (5.14) x ¨1 + ∂x1 ∂x2 q0

and

q0

∂S2 ∂S2 x ¨2 + x1 + x2 = 0, ∂x1 q0 ∂x2 q0

(5.15)

x˙ 1 = v1 ,

∂S1 ∂x1

x˙ 2 = v2 ,

∂S1 ∂x2

2 v˙ 2 = − ∂S ∂x1 x1 −

∂S2 ∂x2 x2 ,

v˙ 1 = −

q0

x1 −

q0

or in the equivalent matrix form

q0

x2 , (5.16)

q0

x˙ = Sx, T

where: x = [x1 , v1 , x2 , v2 ] and  0  ∂S1  − ∂x 1  q0 S= 0  ∂S − ∂x12 q0

10

0

1 0 − ∂S ∂x2

q0

2 0 − ∂S ∂x2

q0

00



 0  . 1  0

Then, we consider two particular cases of variational equations.

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Lack of Coupling Assume that variational equations computed along a homoclinic orbit q0 (t) are uncoupled, i.e. the following condition is satisfied ∂S1 ∂S2 = = 0. ∂x2 q0 ∂x1 q0

Equations (5.14) and (5.15) take the following forms, respectively ∂S1 x ¨1 + x1 = 0, ∂x1 q0

and

∂S2 x ¨2 + x2 = 0, ∂x2 q0

(5.17)

(5.18)

x˙ 1 = v1 , v˙ 1 = −

∂S1 ∂x1

q0

x1 , (5.19)

x˙ 2 = v2 ,

2 v˙ 2 = − ∂S ∂x2 x2 . q0

In the considered case, system (5.16) has the following fundamental solutions matrix   xs1 xs2 0 0  x˙ s1 x˙ s2 0 0   γ (t) =  (5.20)  0 0 ys1 ys2  , 0 0 y˙ s1 y˙ s2 where xs1 , xs2 are the linearly independent particular solutions to equation (5.17), whereas ys1 , ys2 are the solutions to equation (5.18). Distance components of stable and unstable manifolds are defined by the formulas 0 xs2 0 0 G (q (t) , t + t0 ) x˙ s2 0 0 K1 (t, t0 ) = 1 0 0 0 ys1 ys2 G (q (t) , t + t ) 0 y˙ y˙ 2 0 0 s1 s2

and

xs1 0 0 x˙ s1 G1 (q0 (t) , t + t0 ) 0 K2 (t, t0 ) = 0 ys1 0 0 G (q (t) , t + t ) y˙ 2 0 0 s1

0 0 , ys2 y˙ s2

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xs1 x˙ K3 (t, t0 ) = s1 0 0

xs2 0 0 x˙ s2 G1 (q0 (t) , t + t0 ) 0 , 0 0 ys2 0 G2 (q0 (t) , t + t0 ) y˙ s2

xs1 x˙ K4 (t, t0 ) = s1 0 0

xs2 x˙ s2 0 0

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0 0 0 G1 (q0 (t) , t + t0 ) . ys1 ys2 y˙ s1 G2 (q0 (t) , t + t0 )

Expansion of K1 (t, t0 ) with respect to the first column gives K1 (t, t0 ) = −G1 (q0 (t) , t + t0 ) K11 (t, t0 ) − G2 (q0 (t) , t + t0 ) K12 (t, t0 ) , (5.21) where: xs2 0 0 K11 (t, t0 ) = 0 ys1 ys2 = Axs2 , 0 y˙ y˙ s1 s2 xs2 0 0 K12 (t, t0 ) = x˙ s2 0 0 = 0, 0 y y s1 s2

and

y y A = s1 s2 = const . y˙ s1 y˙ s2

The last equation is yielded by the following theorem. Theorem 5.1. If x1 (t) = r1 (t) and x2 (t) = C(t)r(t) are particular solutions of the following ODE x ¨ + p(t)x = 0, and C¯ ∈ R, then x1 (t) x2 (t) r(t) C(t)r(t) = C¯ . = r(t) x˙ 1 (t)x˙ 2 (t) ˙ (C(t)r(t))˙

Proof.

Substitution of x2 (t) to (5.22) gives

¨ ˙ r(t) C(t)r(t) + 2C(t) ˙ + C(t)¨ r(t) + p(t)C(t)r(t) = 0 or equivalently ¨ ˙ r(t) C(t)r(t) + 2C(t) ˙ + C(t) (¨ r(t) + p(t)r(t)) = 0 .

(5.22)

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Since r(t) is a solution to (5.22), then ¨ ˙ r(t) C(t)r(t) + 2C(t) ˙ =0. Multiplying both sides of the last equation by r(t) gives 2 ¨ ˙ r(t)r(t) C(t)r (t) + 2C(t) ˙ =0.

Observe that

2 ˙ C(t)r (t) ˙ = 0 ,

and hence 2 ˙ C(t)r (t) = C¯ ,

where C¯ is the integration constant. Adding and extracting the expression C(t)r(t)r(t) ˙ one gets 2 ˙ C(t)r (t) + C(t)r(t)r(t) ˙ − C(t)r(t)r(t) ˙ = C¯ .

After a simple transformation one obtains, (C(t)r(t))˙r(t) − C(t)r(t)r(t) ˙ = C¯ , which is equivalent to r(t) C(t)r(t) = C¯ . r(t) ˙ (C(t)r(t))˙

Substitution of the above formulas to (5.21) gives K1 (t, t0 ) = −Axs2 G1 (q0 (t) , t + t0 ) . Expansion of K2 (t, t0 ) with respect to a second column yields K2 (t, t0 ) = G1 (q0 (t) , t + t0 ) K21 (t, t0 ) + G2 (q0 (t) , t + t0 ) K22 (t, t0 ) , (5.23) where: xs1 0 0 K21 (t, t0 ) = 0 ys1 ys2 = Axs1 , 0 y˙ y˙ s1 s2 xs1 0 0 K22 (t, t0 ) = x˙ s1 0 0 = 0. 0 y y s1 s2

Substitution of the above equations to (5.23) gives K2 (t, t0 ) = Axs1 G1 (q0 (t) , t + t0 ) .

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Expanding K3 (t, t0 ) as regards the third column gives K3 (t, t0 ) = −G1 (q0 (t) , t + t0 ) K31 (t, t0 ) − G2 (q0 (t) , t + t0 ) K32 (t, t0 ) , (5.24) where: xs1 xs2 0 K31 (t, t0 ) = 0 0 ys2 = 0, 0 0 y˙ s2 xs1 xs2 0 K32 (t, t0 ) = x˙ s1 x˙ s2 0 = Bys2 , 0 0 y s2

and by the Theorem 5.1 the following notation is applied x x B = s1 s2 = const . x˙ s1 x˙ s2 Substitution of the above relations to (5.24) gives

K3 (t, t0 ) = −Bys2 G2 (q0 (t) , t + t0 ) . Expansion of K4 (t, t0 ) with respect to the fourth column yields K4 (t, t0 ) = G1 (q0 (t) , t + t0 ) K41 (t, t0 ) + G2 (q0 (t) , t + t0 ) K42 (t, t0 ) , (5.25) where xs1 xs2 0 K31 (t, t0 ) = 0 0 ys1 = 0, 0 0 y˙ s1 xs1 xs2 0 K32 (t, t0 ) = x˙ s1 x˙ s2 0 = Bys1 . 0 0 y s1

Substitution of the obtained results into (5.25) allows us to obtain K4 (t, t0 ) = Bys1 G2 (q0 (t) , t + t0 ) . Owing to the obtained results and depending on asymptotical behavior of vectors composed of columns of the matrix of fundamental solutions one may efficiently define the Melnikov functions.

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Symmetry Assume that the variational equations along homoclinic orbit q0 (t) are characterized by symmetry defined via the relation ∂S2 ∂S1 ∂S2 ∂S1 = , = . ∂x1 ∂x2 ∂x2 ∂x1 q0

q0

q0

q0

Summation of equations (5.14) and (5.15) gives ∂ (S1 + S2 ) x ¨+ x = 0, ∂x1 q0

(5.26)

where: x = x1 + x2 . Subtraction of both sides of equations (5.14) and (5.15) gives ∂ (S1 − S2 ) y¨ + (5.27) y = 0, ∂x1 q0

where y = y1 − y2 , and hence

x1 = x2 =

1 2 1 2

(x + y) , (x − y) .

Note that system (5.16) has in the considered case the following matrix of fundamental solutions   ys2 ys1 xs2 xs1  y˙ s2 y˙ s1 x˙ s2 x˙ s1   γ (t) =  (5.28)  −ys2 −ys1 xs2 xs1  , −y˙ s2 −y˙ s1 x˙ s2 x˙ s1 where xs1 , xs2 are particular solutions to equation (5.26), whereas ys1 , ys2 are particular solutions to equations (5.27). Components of a distance between stable and unstable manifolds are defined as follows 0 ys1 xs2 xs1 G (q (t) , t + t0 ) y˙ s1 x˙ s2 x˙ s1 , K1 (t, t0 ) = 1 0 0 −ys1 xs2 xs1 G (q (t) , t + t ) −y˙ x˙ x˙ 2 0 0 s1 s2 s1 and

ys2 0 xs2 xs1 y˙ G1 (q0 (t) , t + t0 ) x˙ s2 x˙ s1 K2 (t, t0 ) = s2 , −y 0 xs2 xs1 s2 −y˙ G (q (t) , t + t ) x˙ x˙ s2 2 0 0 s2 s1 ys2 ys1 0 xs1 y˙ y˙ s1 G1 (q0 (t) , t + t0 ) x˙ s1 , K3 (t, t0 ) = s2 −y 0 xs1 s2 −ys1 −y˙ −y˙ G (q (t) , t + t ) x˙ s2 s1 2 0 0 s1

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ys2 y˙ K4 (t, t0 ) = s2 −ys2 −y˙ s2

ys1 y˙ s1 −ys1 −y˙ s1

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xs2 0 x˙ s2 G1 (q0 (t) , t + t0 ) . xs2 0 x˙ s2 G2 (q0 (t) , t + t0 )

Expansion of K1 (t, t0 ) with respect to the first column gives K1 (t, t0 ) = −G1 (q0 (t) , t + t0 ) K11 (t, t0 ) − G2 (q0 (t) , t + t0 ) K12 (t, t0 ) , (5.29) where: ys1 xs2 xs1 K11 (t, t0 ) = −ys1 xs2 xs1 , −y˙ x˙ x˙ s1 s2 s1 ys1 xs2 xs1 K12 (t, t0 ) = y˙ s1 x˙ s2 x˙ s1 . −y x x s1 s2 s1

Expansion of K11 (t, t0 ) with respect to the first column gives K11 (t, t0 ) = 2ys1 K111 (t, t0 ) ,

(5.30)

where: xs2 xs1 . K111 (t, t0 ) = x˙ s2 x˙ s1

Since functions xs1 , xs2 are the linearly independent particular solutions to equations (5.26), then one gets K111 (t, t0 ) = A = const . Substitution of the above equations to (5.30) gives K11 (t, t0 ) = 2Ays1 . Expansion of K12 (t, t0 ) with respect to the first column yields K12 (t, t0 ) = −2Ays1 . Substitution of the above relations to (5.29) gives K1 (t, t0 ) = −2A (G1 (q0 (t) , t + t0 ) − G2 (q0 (t) , t + t0 )) ys1 . Expansion of K2 (t, t0 ) with respect to the second column gives K2 (t, t0 ) = G1 (q0 (t) , t + t0 ) K21 (t, t0 ) + G2 (q0 (t) , t + t0 ) K22 (t, t0 ) , (5.31)

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where: ys2 xs2 xs1 K21 (t, t0 ) = −ys2 xs2 xs1 , −y˙ x˙ x˙ s2 s2 s1 ys2 xs2 xs1 K22 (t, t0 ) = y˙ s2 x˙ s2 x˙ s1 . −y x x s2 s2 s1

Expansion of K21 (t, t0 ) with respect to the first column yields K21 (t, t0 ) = 2Ays2 , whereas expansion of K22 (t, t0 ) with respect to the first column gives K22 (t, t0 ) = −2Ays2 . Substitution of the above relations to (5.31) gives K2 (t, t0 ) = 2A (G1 (q0 (t) , t + t0 ) − G2 (q0 (t) , t + t0 )) ys2 . Expansion of K3 (t, t0 ) with respect to the third column gives K3 (t, t0 ) = −G1 (q0 (t) , t + t0 ) K31 (t, t0 ) − G2 (q0 (t) , t + t0 ) K32 (t, t0 ) , (5.32) where ys2 ys1 xs1 K31 (t, t0 ) = −ys2 −ys1 xs1 , −y˙ −y˙ x˙ s2 s1 s1 ys2 ys1 xs1 K32 (t, t0 ) = y˙ s2 y˙ s1 x˙ s1 . −y −y x s2 s1 s1

Expansion of K31 (t, t0 ) with respect to the third column gives K31 (t, t0 ) = 2xs1 K311 (t, t0 ) , where: y y K311 (t, t0 ) = s2 s1 = B = const. y˙ s2 y˙ s1

After substitution of the above relations to (5.33) one gets K31 (t, t0 ) = 2Bxs1 ,

(5.33)

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whereas expansion of K32 (t, t0 ) with respect to the third column gives K32 (t, t0 ) = 2Bxs1 .

(5.34)

Substitution into (5.32) yields K3 (t, t0 ) = −G1 (q0 (t) , t + t0 ) K31 (t, t0 ) − G2 (q0 (t) , t + t0 ) K32 (t, t0 ) . Expansion of K4 (t, t0 ) with respect to the fourth column yields K4 (t, t0 ) = −G1 (q0 (t) , t + t0 ) K41 (t, t0 ) + G2 (q0 (t) , t + t0 ) K42 (t, t0 ) , (5.35) where: ys2 ys1 xs2 K41 (t, t0 ) = −ys2 −ys1 xs2 , −y˙ −y˙ x˙ s2 s1 s2 ys2 ys1 xs2 K42 (t, t0 ) = y˙ s2 y˙ s1 x˙ s2 . −y −y x s2 s1 s2 Expansion of K41 (t, t0 ) with respect to the third column gives K41 (t, t0 ) = 2Bxs2 , whereas expansion of K42 (t, t0 ) with respect to the third column yields K42 (t, t0 ) = 2Bxs2 . Substitution into (5.35) gives K4 (t, t0 ) = −2B (G1 (q0 (t) , t + t0 ) − G2 (q0 (t) , t + t0 )) xs2 . Therefore, it has been shown that depending on asymptotical behavior of the vectors composed of columns of the fundamental solutions matrix, one may define the Melnikov functions. The analysis of two types of variational equations with symmetry or without coupling regarding 2-DOF mechanical systems exhibited essential simplification of relations defining the Melnikov functions.

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5.3

Reduction of the Melnikov-Gruendler Method for 1DOF Systems

In the case n = 1 we deal with 1-DOF mechanical system governed by the following second order ODE x ¨1 + S (x1 , x˙ 1 , a) = εG (x1 , x˙ 1 , t, b) , or in the form of the first order ODEs x˙ 1 = v1 + ε · 0, v˙ 1 = −S (x1 , v1 , a) + εG (x1 , v1 , t, b) , where x1 ∈ R. The homoclinic orbit is defined via the parametric equation of the form x10 (x1 (t) , v1 (t) , a) q10 (x1 (t) , v1 (t) , a) = . v10 (x1 (t) , v1 (t) , a) Since function q˙10 (t) satisfies a variational equation on the homoclinic orbit q10 (t) one gets: T

ψ (1) (t) = [x˙ 10 (t) , x ¨10 (t)] , ψ (2) (t) = [Q (t) x˙ 10 (t) , (Q (t) x˙ 10 (t))˙]

T

and function Q (t) is given by the Lagrange constant variation method. Fundamental solutions matrix follows Qx˙ 10 x˙ 10 Qv10 v10 γ1 (t) = = . (Qx˙ 10 )˙ x ¨10 (Qv10 )˙ v˙ 10 Substituting the first column of the fundamental solutions matrix by perturbation vector one obtains a matrix whose determinant is defined as follows 0 v10 K (t, t0 ) = = v10 G (x10 , v10 , t + t0 , b) . G (x10 , v10 , t + t0 , b) v˙ 10 According to the introduced assumption that

lim q10 (t) = 0 one may

t→±∞

conclude that vector ψ (1) (t) is of ss-type. Therefore, the Melnikov function is defined in the following way Z∞ Z∞ M (t0 ) = K (t, t0 ) dt = v10 G (x10 , v10 , t + t0 , b) dt. −∞

−∞

Note that the obtained result is identical with the one defining the Melnikov function for an oscillator with 1-DOF and with relatively small periodic time-dependent excitation, which has been shown in references [7, 9]. Owing to this result, one may conclude that the Melnikov-Gruendler approach is a generalization of the classical Melnikov method.

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Chapter 6


A self-excited spherical pendulum with dry friction is analysed analytically and numerically. The Melnikov-Gruendler method is applied yielding criterion of chaos occurrence, and then the obtained analytical results are verified numerically. 6.1

Analytical Prediction of Chaos

A spherical pendulum with mass m and length l is considered. Its motion is defined using two general coordinates θ and ψ (Figure 6.1). Motion along coordinate θ is realized via movable bush-shaft coupling, where dry Coulomb friction characterized by coefficient µ occurs. The bush has radius r and rotates at constant angular velocity $ = const > 0. The system is excited harmonically with amplitude Γ and frequency ω. The system motion is damped along θ and ψ linearly, and the damping is characterized by viscosity coefficient c. The studied system has two-degrees-of-freedom, and hence the Melnikov-Gruendler technique is applied to predict its chaotic dynamics. Dynamics of the analyzed system is governed by the equations ml2 θ¨ − 21 ml2 ψ˙ 2 sin 2θ + mgl sin θ = εΓ cos ωτ − εcθ˙ − εT r sgn θ˙ − $ , ˙ ml2 ψ¨ sin2 θ + ml2 ψ˙ θ˙ sin 2θ = −εcψ, where ε is the small parameter, and frictional force is µ g T = ml p θ˙2 + ψ˙ 2 sin2 θ + cos θ , l 1 + µ2 (1 − λ2 ) where µ.= tan ρ is the friction coefficient, and λ = r/l < 1. Denoting by p µλ = µ 1 + µ2 (1 − λ2 ) geometrically modified friction coefficient, the 79

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Fig. 6.1

Physical model of a spherical pendulum.

friction force is defined in the following way ˙ ψ˙ = mlµλ θ˙2 + ψ˙ 2 sin2 θ + g cos θ . T˜ θ, θ, l Equations of the pendulum motion are as follows

(6.1)

˙ ψ˙ ml2 θ¨ − 12 ml2 ψ˙ 2 sin 2θ + mgl sin θ = εΓ cos ωτ − εcθ˙ − εrT˜ θ, θ, × sgn θ˙ − $ , ˙ ml2 ψ¨ sin2 θ + ml2 ψ˙ θ˙ sin 2θ = εΓ1 cos ωτ − εcψ. For θ 6= 0, θ 6= π the equations have the form ˙ ψ˙ sgn θ˙ − $ , θ¨ − 21 ψ˙ 2 sin 2θ + ω02 sin θ = εγ cos ωτ − εδ θ˙ − εT θ, θ, cos ωτ ψ˙ − εδ 2 , ψ¨ + 2ψ˙ θ˙ cot θ = εγ1 2 sin θ sin θ p wherethe following notation is applied: ω0 = g/l, λ = r/l, δ = c ml2 , γ = Γ ml2 , γ1 = Γ1 ml2 , and ˙ ψ˙ rT θ, θ, ˙ ψ˙ = T θ, θ, = λµλ θ˙2 + ψ˙ 2 sin2 θ + ω02 cos θ . 2 ml In the case of θ = 0 or θ = π one gets ml2 θ¨ = εΓ1 cos ωτ − εcθ˙ − εrT˜ sgn θ˙ − $ , (6.2) Γ2 cos ωτ − cψ˙ = 0,

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where   mlµλ θ˙2 + g for θ=0, l T˜ =  mlµλ θ˙2 − g for θ=π. l

The second equation of 6.2 yields1

Γ2 ψ˙ = cos ωτ. c In the above the formula defines a required angular velocity ψ˙ for the time instant τ to be in one of the vertical pendulum configurations. Introducing a new variable ϕ = θ − π one gets ϕ¨ − 12 ψ˙ 2 sin 2ϕ − ω02 sin ϕ = εγ cos ωτ − εδ ϕ˙ − εT ϕ, ϕ, ˙ ψ˙ sgn (ϕ˙ − $) , cos ωτ ψ˙ ψ¨ + 2ψ˙ ϕ˙ cot ϕ = εγ1 − εδ 2 , 2 sin ϕ sin ϕ (6.3) and T ϕ, ϕ, ˙ ψ˙ = λµλ ϕ˙ 2 + ψ˙ 2 sin2 ϕ − ω02 cos ϕ .

(6.4)

The so far obtained equations are presented in the form of the first order ODEs ϕ˙ = σ, σ˙ = 21 ζ 2 sin 2ϕ + ω02 sin ϕ + εγ cos ωτ − εδσ − εT (ϕ, σ, ζ) sgn (σ − $) , ψ˙ = ζ, cos ωτ ζ ζ˙ = −2ζσ cot ϕ + εγ1 − εδ 2 . sin2 ϕ sin ϕ (6.5) For ε = 0 one gets ϕ¨ − 21 ψ˙ 2 sin 2ϕ − ω02 sin ϕ = 0, ψ¨ + 2ψ˙ ϕ˙ cot ϕ = 0,

(6.6)

ϕ˙ = σ, σ˙ = 21 ζ 2 sin 2ϕ + ω02 sin ϕ, ψ˙ = ζ, ζ˙ = −2ζσ cot ϕ.

(6.7)

or equivalently

1 For

c → 0 we have ψ˙ → ∞.

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Equilibria are defined by the equation sin ϕ0 = 0. Integrating the second equation of (6.6) one obtains the momentum conservation principle ψ˙ sin2 ϕ = C, where C is the constant, and for ϕ 6= 0, ϕ 6= π one gets C ψ˙ = . sin2 ϕ

(6.8)

Substituting (6.8) to the first equation of (6.6) one obtains cos ϕ ϕ¨ − C 2 3 − ω02 sin ϕ = 0. (6.9) sin ϕ Multiplying both sides of the equations by ϕ˙ one gets dϕ˙ ϕ˙ cos ϕ ϕ˙ − C 2 − ω02 ϕ˙ sin ϕ = 0. dt sin3 ϕ Integration with respect to time yields the law of energy conservation in the form ϕ˙ 2 C2 1 + + ω02 cos ϕ = K, 2 2 sin2 ϕ where K is the constant. A phase trajectory is defined by s C2 ϕ˙ = ± 2K − − 2ω02 cos ϕ. sin2 ϕ Note that for C = 0 particular trajectories are defined crossing the point ϕ = 0, and they are determined by the following differential equation q ϕ˙ = ± 2K − 2ω02 cos ϕ. Substituting ϕ = 0 one gets K = ω02 , and hence q ϕ˙ = ± 2ω02 (1 − cos ϕ). Separating the variables √

dϕ , 1 − cos ϕ and integrating both sides of this equation one gets r cos ϕ + 1 ω0 t = ± atanh , 2 2ω0 dt = ± √

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and hence ϕ¯0 (t) = ±2 arccos (tanh ω0 t) . Differentiation with respect to time yields ϕ¯˙ 0 (t) = ∓2ω0 sech (ω0 t) . For a homoclinic orbit C = 0, formula (6.8) gives ψ˙ = 0, and after integration one obtains ψ = ψ0 , where ψ0 = const. Owing to these considerations, two one-parameter families of the homoclinic orbits are found:   ϕ0 (t)  ϕ˙ 0 (t)   q01 (t, ψ0 ) =  (6.10)  ψ0  , 0 

 ϕ0 (t)  −ϕ˙ 0 (t)   q02 (t, ψ0 ) =   ψ0  , 0

(6.11)

ϕ0 (t) = |ϕ¯0 (t)| = 2 arccos (tanh (ω0 t)) ,

(6.12)

ϕ˙ 0 (t) = |ϕ¯˙ 0 (t)| = 2ω0 sech (ω0 t) .

(6.13)

where

and

They are shown in Figure 6.2 for ω0 = 1 in the (ϕ, ϕ) ˙ plane. Then, quantities necessary for application of the method are computed. Linearization of (6.6) in the vicinity of the origin yields ϕ¨ − ψ˙ 02 cos 2ϕ0 + ω02 cos ϕ0 ϕ − ψ˙ 0 sin 2ϕ0 ψ˙ = 0, 2ψ˙ 0 ϕ˙ 0 ψ¨ + ϕ + 2ψ˙ 0 cot ϕ0 ϕ˙ + 2ϕ˙ 0 cot ϕ0 ψ˙ = 0. sin2 ϕ0 Using (6.12), (6.13) the linearized (variational) equations along orbit (6.10) are ϕ¨ − ω02 2 tanh2 (ω0 t) − 1 ϕ = 0, (6.14) ψ¨ − 2ω0 (coth (ω0 t) − 2 tanh (ω0 t)) ψ˙ = 0,

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Fig. 6.2

Homoclinic orbits for ω0 = 1 in the (ϕ, ϕ) ˙ plane.

and ϕ˙ = σ, σ˙ = ω02 2 tanh2 (ω0 t) − 1 ϕ, ψ˙ = ζ, ζ˙ = 2ω0 (coth (ω0 t) − 2 tanh (ω0 t)) ζ.

(6.15)

The first of four linearly independent solutions of equations (6.15) has the form   ϕ˙ 0 (t)  ϕ¨0 (t)   ψ (4) (t) = q˙01 (t, ψ0 ) =  (6.16)  0 . 0 Therefore, a particular solution of the first equation of (6.14) is defined by ϕs1 (t) = ϕ˙ 0 (t) = 2ω0 sech (ω0 t) . The second linearly independent integral of the first equation of (6.14) is obtained according to the Lagrange constant variation method ϕ (t) = C (t) ϕ˙ 0 (t) . The following relation is obtained and then is substituted to the first equation of (6.14) to give 2C˙ ϕ¨0 +C¨ ϕ˙ 0 = −Cϕ0 +ω02 2 tanh2 (ω0 t) − 1 C ϕ˙ 0 = ϕ−ω ¨ 02 2 tanh2 (ω0 t) − 1 ϕ. Taking into account the first equation of (6.14) one obtains 2C˙ ϕ¨0 + C¨ ϕ˙ 0 = 0,

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and a general solution is 1 C (t) = C2 + C1 t + sinh (ω0 t) , ω0 where C1 and C2 are the integration constants. Therefore, a general solution of the first equation of (6.14) is 1 ϕ (t) = C1 ϕs1 +C2 ϕs2 = C1 t + sinh (2ω0 t) sech (ω0 t)+C2 sech (ω0 t) , 2ω0 where ϕs1 and ϕs2 are the particular solutions. Hence, the successive independent solution of (6.15) has the form   Q (t) ϕ˙ 0 (t)  (Q (t) ϕ˙ 0 (t))·  , ψ (2) (t) =  (6.17)   0 0

where Q (t) = t + 2ω1 0 sinh (2ω0 t). A general solution of the second equation of (6.14) has the form ψ = ψ0 + C tanh3 (ω0 t) . The two remaining linearly independent solutions of (6.15) have the form     0 0  0   0  (1) (3)    ψ (t) =   ψs (t)  , ψ (t) =  ψ0  , 0 ψ˙ s (t) where the following notation is applied ψs = tanh3 (ω0 t) ,

ψ˙ s = 3ω0 sech2 (ω0 t) tanh2 (ω0 t) .

A fundamental matrix of the solution of system (6.15) is as follows   0 Q (t) ϕ˙ 0 (t) 0 ϕ˙ 0 (t)  0 (Q (t) ϕ˙ 0 (t))· 0 ϕ¨0 (t)  . γ1 (t) =  (6.18)  ψs (t) 0 ψ0 0  ψ˙ s (t) 0 0 0 A linear variational equation along the orbit (6.11), according to (6.12) and (6.13), is defined as follows ϕ¨ − ω02 2 tanh2 (ω0 t) − 1 ϕ = 0, (6.19) ψ¨ + 2ω0 (coth (ω0 t) − 2 tanh (ω0 t)) ψ˙ = 0,

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and ϕ˙ = σ, σ˙ = ω02 2 tanh2 (ω0 t) − 1 ϕ, (6.20) ψ˙ = ζ, ζ˙ = −2ω0 (coth (ω0 t) − 2 tanh (ω0 t)) ζ. Two first solutions of (6.20) are defined by (6.16) and (6.17). A solution to the second equation of (6.19) has the form 1 1 3t ψ = ψ0 + Cψs = ψ0 + C − coth (ω0 t) + sinh (2ω0 t) . 2 ω0 4ω0 The two remaining independent solutions of (6.20) have the form     0 0     0 (3)   0  ψ (1) (t) =   P (t)  , ψ (t) =  ψ0  , 0 P˙ (t) where: P (t) =

1 1 3t − coth (ω0 t) + sinh (2ω0 t) , 2 ω0 4ω0 P˙ (t) = coth2 (ω0 t) cosh2 (ω0 t) .

A fundamental matrix of solutions of (6.20) is defined as follows   0 Q (t) ϕ˙ 0 (t) 0 ϕ˙ 0 (t)  0 (Q (t) ϕ˙ 0 (t))· 0 ϕ¨0 (t)  . γ2 (t) =  (6.21)  P (t) 0 ψ0 0  P˙ (t) 0 0 0 Next, we consider (6.5) for ε > 0. A perturbation vector along the homoclinic orbit (6.10) has the form   0    γ cos ωτ − δ ϕ˙ 0 − T ϕ0 , ϕ˙ 0 , ψ˙ 0 sgn (ϕ˙ 0 − $)  . h (q01 , t) =    0   ˙ ψ0 cos ωτ γ1 sin 2 ϕ − δ sin2 ϕ 0 0 Substituting the first column of the fundamental matrix of solutions (6.18) by vector h (q01 , t) one obtains a matrix whose determinant is K11 (t, t0 ) = Q (t) ϕ˙ 0 (t) 0 ˙ γ cos ωτ − δ ϕ˙ 0 − T ϕ0 , ϕ˙ 0 , ψ0 sgn (ϕ˙ 0 − $) (Q (t) ϕ˙ 0 (t))· 0 0 ψ˙ 0 cos ωτ γ1 sin − δ 0 2ϕ sin2 ϕ0 0

0 ϕ˙ 0 (t) 0 ϕ¨0 (t) . ψ0 0 0 0

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87

Substituting the second column of the fundamental solution matrix (6.18) by the vector h (q01 , t) one obtains the following determinant K12 (t, t0 ) = 0 0 0 ϕ˙ 0 (t) 0 γ cos ωτ − δ ϕ˙ 0 − T ϕ0 , ϕ˙ 0 , ψ˙ 0 sgn (ϕ˙ 0 − $) 0 ϕ¨0 (t) . ψ (t) 0 ψ0 0 s ˙ ψ˙ 0 cos ωτ ψs (t) γ1 sin 0 0 2 ϕ − δ sin2 ϕ 0 0

Substituting the third column of the mentioned fundamental solution matrix by vector h (q01 , t) one obtains a matrix with the corresponding determinant K13 (t, t0 ) = 0 Q (t) ϕ˙ 0 (t) 0 ϕ˙ 0 (t) 0 (Q (t) ϕ˙ 0 (t))· γ cos ωτ − δ ϕ˙ 0 − T ϕ0 , ϕ˙ 0 , ψ˙ 0 sgn (ϕ˙ 0 − $) ϕ¨0 (t) . ψ (t) 0 0 0 s ˙ ψ˙ 0 cos ωτ ψs (t) 0 γ1 sin 0 2 ϕ − δ sin2 ϕ 0 0

Substituting the fourth column of the quoted matrix by vector h (q01 , t) one gets the following determinant K14 (t, t0 ) = 0 Q (t) ϕ˙ 0 (t) 0 (Q (t) ϕ˙ 0 (t))· ψ (t) 0 s ˙ ψs (t) 0

0 0 γ cos ωτ − δ ϕ˙ 0 − T ϕ0 , ϕ˙ 0 , ψ˙ 0 sgn (ϕ˙ 0 − $) . ψ0 0 ˙ ψ cos ωτ 0 0 γ1 sin2 ϕ0 − δ sin2 ϕ0

0

Expanding K11 (t, t0 ) with respect to the third column one obtains cos ωτ ϕ˙ 0 −δ 2 , (6.22) K11 (t, t0 ) = A γ1 2 sin ϕ0 sin ϕ0 where (see Theorem 5.1) Q (t) ϕ˙ 0 (t) ϕ˙ 0 (t) = const. A= · (Q (t) ϕ˙ 0 (t)) ϕ¨0 (t)

Expanding K12 (t, t0 ) with respect to the third column we get K12 (t, t0 ) = −ψ0 ψ˙ s (t) ϕ˙ 0 (t) × γ cos ωτ − δ ϕ˙ 0 − T ϕ0 , ϕ˙ 0 , ψ˙ 0 sgn (ϕ˙ 0 − $) .

(6.23)

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Expanding K13 (t, t0 ) with respect to the third row one obtains ϕ˙ 0 cos ωτ K13 (t, t0 ) = Aψs (t) δ 2 − γ1 2 . sin ϕ0 sin ϕ0 Expanding K14 (t, t0 ) with respect to the third column we get K14 (t, t0 ) = −ψ˙ s (t) Q (t) ϕ˙ 0 (t) × γ cos ωτ − δ ϕ˙ 0 − T ϕ0 , ϕ˙ 0 , ψ˙ 0 sgn (ϕ˙ 0 − $) .

(6.24)

The perturbation vector along the homoclinic orbit (6.11) is   0    γ cos ωτ + δ ϕ˙ 0 − T ϕ0 , −ϕ˙ 0 , ψ˙ 0 sgn (−ϕ˙ 0 − $)    h (q02 , t) =   0   ˙ ψ0 cos ωτ γ1 sin 2 ϕ − δ sin2 ϕ 0 0 Substituting the first column of the fundamental matrix of solutions (6.21) by vector h (q02 , t), one obtains a matrix associated with the following determinant K21 (t, t0 ) = 0 Q (t) ϕ˙ 0 (t) 0 ϕ˙ 0 (t) γ cos ωτ + δ ϕ˙ 0 − T ϕ0 , −ϕ˙ 0 , ψ˙ 0 sgn (−ϕ˙ 0 − $) (Q (t) ϕ˙ 0 (t))˙ 0 ϕ¨0 (t) . 0 0 ψ0 0 cos ωτ ˙ γ1 sin2 ϕ − δ sinψ20ϕ 0 0 0 0 0

Substituting the second column of the fundamental matrix of solutions (6.21) by the vector h (q02 , t), one obtains a matrix whose determinant is K22 (t, t0 ) 0 0 0 ϕ˙ 0 (t) 0 γ cos ωτ + δ ϕ˙ 0 − T ϕ0 , −ϕ˙ 0 , ψ˙ 0 sgn (−ϕ˙ 0 − $) 0 ϕ¨0 (t) . = 0 ψ0 0 P (t) ˙ ψ˙ 0 cos ωτ P (t) 0 0 γ1 sin 2 ϕ − δ sin2 ϕ 0 0

Substituting the third column of the fundamental matrix of solutions (6.21) by vector h (q02 , t) one obtains the following matrix K23 (t, t0 ) = 0 Q (t) ϕ˙ 0 (t) 0 ϕ˙ 0 (t) 0 (Q (t) ϕ˙ 0 (t))˙ γ cos ωτ + δ ϕ˙ 0 − T ϕ0 , −ϕ˙ 0 , ψ˙ 0 sgn (−ϕ˙ 0 − $) ϕ¨0 (t) . P (t) 0 0 0 ˙ ψ˙ 0 cos ωτ P (t) 0 γ1 sin 0 2 ϕ − δ sin2 ϕ 0 0

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89

Substituting the fourth column of the fundamental matrix of solutions (6.21) by the vector h (q02 , t) one gets the following determinant K24 (t, t0 ) = 0 Q (t) ϕ˙ 0 (t) 0 0 0 (Q (t) ϕ˙ 0 (t))˙ 0 γ cos ωτ + δ ϕ˙ 0 − T ϕ0 , −ϕ˙ 0 , ψ˙ 0 sgn (−ϕ˙ 0 − $) . P (t) 0 ψ0 0 ˙ ψ˙ 0 cos ωτ P (t) 0 0 γ1 sin 2 ϕ − δ sin2 ϕ 0 0

Expanding K21 (t, t0 ) with respect to the third column one obtains ! cos ωτ ψ˙ 0 K21 (t, t0 ) = Aψ0 γ1 2 −δ 2 . (6.25) sin ϕ0 sin ϕ0 Expanding K22 (t, t0 ) with respect to the third column we get K22 (t, t0 ) = −ψ0 γ cos ωτ + δ ϕ˙ 0 − T ϕ0 , −ϕ˙ 0 , ψ˙ 0 sgn (−ϕ˙ 0 − $) × P˙ (t) ϕ˙ 0 (t) .

(6.26) Expanding K23 (t, t0 ) with respect to the third row we obtain ! cos ωτ ψ˙ 0 K23 (t, t0 ) = −δA γ1 2 −δ 2 P (t) . sin ϕ0 sin ϕ0 Expanding K24 (t, t0 ) with respect to the third column we get K24 (t, t0 ) = ψ0 P˙ (t) Q (t) ϕ˙ 0 (t) (6.27) × γ cos ωτ + δ ϕ˙ 0 − T ϕ0 , −ϕ˙ 0 , ψ˙ 0 sgn (−ϕ˙ 0 − $) . According to (6.23), the Melnikov function is M12 (t0 ) = −M121 (t0 ) + M122 + M123 ,

(6.28)

where M121 (t0 ) = ψ0 γ

Z∞

cos ω (t + t0 ) ϕ˙ 0 (t) ψ˙ s (t) dt

−∞

=

6ω02 ψ0 γ

Z∞

cos ω (t + t0 ) sech3 (ω0 t) tanh2 (ω0 t) dt,

−∞

M122 = ψ0 δ

Z∞

−∞

ϕ˙ 20 (t) ψ˙ s (t) dt = 12ω03 ψ0 δ

Z∞

−∞

sech4 (ω0 t) tanh2 (ω0 t) dt,

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M123 = ψ0

Z∞

ψ˙ s (t) ϕ˙ 0 (t) T ϕ0 , ϕ˙ 0 , ψ˙ 0 sgn (ϕ˙ 0 − $) dt

−∞

=

6ω02 ψ0

Z∞

sech3 (ω0 t) tanh2 (ω0 t) T ϕ0 , ϕ˙ 0 , ψ˙ 0 sgn (ϕ˙ 0 − $) dt.

−∞

Note that M121 (t0 ) can be given in the form2 M121 (t0 ) = M1211 (t0 ) − M1212 (t0 ) , where M1211 (t0 ) =

6ω02 ψ0 γ

Z∞

cos ωt0

cos ωt sech3 (ω0 t) tanh2 (ω0 t) dt,

−∞

M1212 (t0 ) =

6ω02 ψ0 γ

sin ωt0

Z∞

sech3 (ω0 t) tanh2 (ω0 t) sin ωtdt.

−∞

Relation M1211 (t0 ), consisting of an improper integral within the interval of (−∞,∞), can be found in the table of integrals: 2 πψ0 γ πω 2 2 2 M1211 (t0 ) = 3ω − ω ω + ω sech cos ωt0 . 0 0 4ω03 2ω0 In order to compute M1212 (t0 ) observe first that the under integral expression is an odd function, and therefore M1212 (t0 ) = 0. Therefore, M121 (t0 ) =

πψ0 γ 3ω02 − ω 2 ω02 + ω 2 sech 3 4ω0

πω 2ω0

cos ωt0 .

(6.29)

Computing an original function in relation M122 (t0 ), one gets ∞ 4 16 2 3 2 M122 = ψ0 δω0 (4 + cosh (2ω0 t)) sech (ω0 t) tanh (ω0 t) = ψ0 δω02 . 5 5 −∞ (6.30) According to (6.4), one has M123 = M1231 − M1232 , 2 The

following identity is used: cos (a + b) = cos a cos b − sin a sin b.

(6.31)

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91

where the following notations are applied M1231 =

24ω04 ψ0 λµλ

Z∞

sech5 (ω0 t) tanh2 (ω0 t) sgn (ϕ˙ 0 − $) dt,

−∞

M1232 = 6ω04 ψ0 λµλ Z∞ 2 tanh2 (ω0 t) − 1 sech3 (ω0 t) tanh2 (ω0 t) sgn (ϕ˙ 0 − $) dt. × −∞

Let us introduce some necessary relations for computation of M231 . Consider first the relation −1, ϕ˙ 0 < $ sgn (ϕ˙ 0 (t) − $) = 1, ϕ˙ 0 > $ defining explicitly the under integral function with respect to $. A change of sign of the under integral expression occurs, when $ = ϕ˙ 0 (t) = 2ω0 sech (ω0 t) . Computing the inverted function for $ ≤ 2ω0 , one gets 1 $ t1,2 = ± arc sech . ω0 2ω0

(6.32)

(6.33)

In the case $ > 2ω0 , we have t1,2 ∈ / R. Coming back to the computation of integral M1231 , note that its value depends qualitatively on the ratio $/2ω0 . Consider first the case $ > 2ω0 . Since equation (6.32) does not have real roots in the considered integral, then M1231 =

−24ω04 ψ0 λµλ

Z∞

sech5 (ω0 t) tanh2 (ω0 t) dt.

−∞

Integration gives M1231 = −24ω04 ψ0 λµλ

˜ 1231 (t) − lim M ˜ 1231 (t) = − 3 πω 3 ψ0 λµλ , lim M t→∞ t→−∞ 2 0

where the following notation has been applied ω 0 ˜ 1231 (t) = 1 M 6 arctan tanh t 48ω0 2 + 3 + (cosh (2ω0 t) − 7) sech4 (ω0 t) sech (ω0 t) tanh (ω0 t) .

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In the second case, i.e. for $ ≤ 2ω0 one gets first M1231 = 24ω04 ψ0 λµλ ×

−

Zt1

5

2

sech (ω0 t) tanh (ω0 t) dt +

−∞

−

Z∞

Zt2

sech5 (ω0 t) tanh2 (ω0 t) dt

t1

sech5 (ω0 t) tanh2 (ω0 t) dt

!

t2

and after integration M1231 = 24ω04 ψ0 λµλ

t1 ˜ 1231 (t) −M

−∞

t2 ∞ ˜ 1231 (t) − M ˜ 1231 (t) +M t1

t2

= 24ω04 ψ0 λµλ ˜ 1231 (t2 ) − M ˜ 1231 (t1 ) + lim M ˜ 1231 (t) − lim M ˜ 1231 (t) . × 2 M t→−∞

t→∞

To sum up the considerations carried out so far, we get 3 M1231 = − πω03 ψ0 λµλ 2 ( (6.34) ˜ 1231 (t2 ) − M ˜ 1231 (t1 ) for $ ≤ 2ω0 48ω04 ψ0 λµλ M + , 0 for $ > 2ω0 For $ > 2ω0 Z∞ 4 M1232 = −6ω0 ψ0 λµλ 2 tanh2 (ω0 t) − 1 sech3 (ω0 t) tanh2 (ω0 t) dt, −∞

and after integration M1232 = −6ω04 ψ0 λµλ

˜ 1232 (t) − lim M ˜ 1232 (t) = 0, lim M

t→∞

t→−∞

where the following notation is applied ˜ 1232 (t) = − 1 sech3 (ω0 t) tanh3 (ω0 t) . M 3ω0 In the second case, i.e. for $ ≤ 2ω0 , one obtains  Zt1 4  M1232 = 6ω0 ψ0 λµλ − 2 tanh2 (ω0 t) − 1 sech3 (ω0 t) tanh2 (ω0 t) dt −∞

+

Zt2

2 tanh2 (ω0 t) − 1 sech3 (ω0 t) tanh2 (ω0 t) dt

t1

−

Z∞ t2

 2 tanh2 (ω0 t) − 1 sech3 (ω0 t) tanh2 (ω0 t) dt .

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93

The integration procedure gives t1 t2 ∞ ˜ 1232 (t) ˜ 1232 (t) − M ˜ 1232 (t) M1232 = 6ω04 ψ0 λµλ −M +M −∞ t1 t2 4 ˜ ˜ = 12ω0 ψ0 λµλ M1232 (t2 ) − M1232 (t1 ) , where t1 , t2 are defined by (6.33). To conclude ( ˜ 1232 (t2 ) − M ˜ 1232 (t1 ) for $ ≤ 2ω0 12ω04 ψ0 λµλ M M1232 = . 0 for $ > 2ω0

(6.35)

Substituting the result to (6.31) we get M123 = −18πω07 ψ02 λ2 µ2λ  4 M ˜ 1231 (t2 ) − M ˜ 1231 (t1 ) − M ˜ 1232 (t2 ) − M ˜ 1232 (t1 ) for $ ≤ 2ω0 + 0 for $ > 2ω0 (6.36)

Taking into account (6.28), one gets 2 πω πψ0 γ 2 2 2 3ω − ω ω + ω sech cos ωt0 M12 (t0 ) = − 0 0 4ω03 2ω0 16 3 + δ − πλµλ ω0 ψ0 ω02 − 12ω04 ψ0 λµλ 5 2  M ˜ 1232 (t2 ) − M ˜ 1232 (t1 ) − 4 M ˜ 1231 (t2 ) − M ˜ 1231 (t1 ) for $ ≤ 2ω0 × . 0 for $ > 2ω0

Finally, the following chaos criterion is obtained πψ0 γ 2 πω 2 2 2 ω0 + ω sech > 4ω 3 3ω0 − ω 2ω0 0 16 3 2 4 5 δ − 2 πλµλ ω0 ψ0 ω0 − 12ω0 ψ0 λµλ  M ˜ 1232 (t2 ) − M ˜ 1232 (t1 ) − 4 M ˜ 1231 (t2 ) − M ˜ 1231 (t1 ) ×  0

for $ ≤ 2ω0 for $ > 2ω0 (6.37)

According to (6.24), the second Melnikov function is M14 (t0 ) = −M141 (t0 ) + M142 + M143 ,

(6.38)

.

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where: M141 (t0 ) = γ

Z∞

cos ω (t + t0 )ψ˙ s (t) Q (t) ϕ˙ 0 (t)

−∞

=

Z∞

6γω02

−∞

1 cos ω (t + t0 ) t + sinh (2ω0 t) sech3 (ω0 t) tanh2 (ω0 t) dt, 2ω0

M142 = ψ0 δ

Z∞

Q (t) ϕ˙ 20 (t) ψ˙ s (t) dt

−∞

=

12ψ0 δ4ω03

Z∞

−∞

M143 =

Z∞

1 sinh (2ω0 t) sec h4 (ω0 t) tanh2 (ω0 t) dt, t+ 2ω0

ψ˙ s (t) Q (t) ϕ˙ 0 (t) T ϕ0 , ϕ˙ 0 , ψ˙ 0 sgn (ϕ˙ 0 − $) dt

−∞

= 6ω02

Z∞

−∞

1 sinh (2ω0 t) T ϕ0 , ϕ˙ 0 , ψ˙ 0 sech3 (ω0 t) tanh2 (ω0 t) t + 2ω0

× sgn (ϕ˙ 0 − $) dt. Formula M141 (t0 ) is defined by the relation M141 (t0 ) = M1411 (t0 ) − M1412 (t0 ) ,

(6.39)

where: M1411 (t0 ) = 6γω02 cos ωt0 Z∞ 1 × t+ sinh (2ω0 t) sech3 (ω0 t) tanh2 (ω0 t) cos ωtdt, 2ω0 −∞

M1412 (t0 ) = 6γω02 sin ωt0 Z∞ 1 × t+ sinh (2ω0 t) sech3 (ω0 t) tanh2 (ω0 t) sin ωtdt. 2ω0 −∞

Observe that the under integral functionM1411 (t0 ) is even, and the integral computed from −∞ to ∞ is equal to zero, and therefore M1411 (t0 ) = 0.

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95

M1412 (t0 ) is defined by M1412 (t0 ) = M14121 (t0 ) + M14122 (t0 ) ,

(6.40)

where: M14121 (t0 ) =

6γω02

sin ωt0

Z∞

t sech3 (ω0 t) tanh2 (ω0 t) sin ωtdt

−∞

3πγ πω sech 4ω02 2ω0 πω × 8ω0 ω ω 2 − ω02 + π 3ω02 − ω 2 ω02 + ω 2 tanh sin ωt0 , 2ω0 =

M14122 (t0 ) = 3γω0 sin ωt0

Z∞

sinh (2ω0 t) sec h3 (ω0 t) tanh2 (ω0 t) sin ωtdt

−∞

ω 5ω02 − ω 2 πω = 3πγ sech sin ωt0 . ω02 2ω0 Substituting the relations obtained so far to (6.40), one gets 3πγ πω M1412 (t0 ) = sech 4ω 2 2ω0 0 πω sin ωt0 . × 32ωω03 + π 3ω04 + 2ω02 ω 2 − ω 4 tanh 2ω0 Substituting the obtained relation into (6.39), one gets 3πγ πω M141 (t0 ) = − 2 sech 4ω 2ω0 0 πω 3 4 2 2 4 × 32ωω0 + π 3ω0 + 2ω0 ω − ω tanh sin ωt0 . 2ω0 Observe that the under integral functionM142 (t0 ) is odd and hence M142 = 0. Applying (6.4), one gets M143 = M1431 − M1432 ,

(6.41)

where: M1431 =

24ω04 ψ0 λµλ

Z∞

−∞

1 sech5 (ω0 t) tanh2 (ω0 t) t + sinh (2ω0 t) 2ω0

× sgn (ϕ˙ 0 − $) dt,

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M1432 = 6ω04 ψ0 λµλ

Z∞

−∞

2 tanh2 (ω0 t) − 1 sech3 (ω0 t) tanh2 (ω0 t)

1 × t+ sinh (2ω0 t) sgn (ϕ˙ 0 − $) dt. 2ω0 For $ > 2ω0 expression M1431 yields Z∞ 1 5 2 4 M1431 = −24ω0 ψ0 λµλ sech (ω0 t) tanh (ω0 t) t + sinh (2ω0 t) dt, 2ω0 −∞

Since the under integral function is odd, then M1431 = 0. In the second case, i.e. for $ ≤ 2ω0 , additive property of the defined integral gives  Zt1 1 5 2 4  M1431 = 24ω0 ψ0 λµλ − sech (ω0 t) tanh (ω0 t) t + sinh (2ω0 t) dt 2ω0 −∞

+

Zt2

1 sech5 (ω0 t) tanh2 (ω0 t) t + sinh (2ω0 t) dt 2ω0

t1

−

Z∞ t2

 1 sinh (2ω0 t) dt . sech5 (ω0 t) tanh2 (ω0 t) t + 2ω0

Carrying out the integration, one gets t1 t2 ∞ 4 ˜ ˜ ˜ M1431 = 24ω0 ψ0 λµλ −M1431 (t) + M1431 (t) − M1431 (t) −∞ t t2 1 4 ˜ ˜ = 48ω0 ψ0 λµλ M1431 (t2 ) − M1431 (t1 ) , where t1 , t2 are defined by (6.33), where ˜ 1431 (t) = M

1 −18 ω0 t= ln 1 + ie−ω0 t + = Li2 ie−ω0 t 2 144ω0

+ sech (ω0 t) 9 (1 + ω0 t tanh (ω0 t)) + sech4 (ω0 t) × (1 − 23 cosh (2ω0 t) + 3ω0 t (cosh (2ω0 t) − 7) tanh (ω0 t)))) and M1431 =

(

˜ 1431 (t2 ) − M ˜ 1431 (t1 ) for $ ≤ 2ω0 48ω04 ψ0 λµλ M 0

for $ > 2ω0

.

(6.42)

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97

Li2 (z) is a generalization of the logarithmic function, and it is defined by df

Li2 (z) =

Z0

∞

ln

X zk 1−t dt = . t k2 k=1

z

It has, among other, the property: Li2 (¯ z ) = Li2¯(z), and in a general case it is defined in the following way ∞ k df X z Lin (z) = . kn k=1

Observe that for n = 1 one gets: Li1 (z) = − ln(1 − z), and ∂Lin (z) Lin−1 (z) = . ∂z z For $ > 2ω0 relation M1432 is Z∞ 4 M14324 = −6ω0 ψ0 λµλ 2 tanh2 (ω0 t) − 1 sech3 (ω0 t) tanh2 (ω0 t) −∞

1 × t+ sinh (2ω0 t) dt. 2ω0 Oddness of the integrand gives M1432 = 0. In the second case, i.e. for $ ≤ 2ω0 , one gets  Zt1 4  M1432 = 6ω0 ψ0 λµλ − 2 tanh2 (ω0 t) − 1 sech3 (ω0 t) tanh2 (ω0 t) dt −∞

+

Zt2

2 tanh2 (ω0 t) − 1 sech3 (ω0 t) tanh2 (ω0 t) dt

t1

−

Z∞ t2

 2 tanh2 (ω0 t) − 1 sech3 (ω0 t) tanh2 (ω0 t) dt .

After integration we have t1 t2 ∞ 4 ˜ ˜ ˜ M1432 = 6ω0 ψ0 λµλ −M1432 (t) + M1432 (t) − M1432 (t) −∞ t t2 1 4 ˜ ˜ = 12ω0 ψ0 λµλ M1432 (t2 ) − M1432 (t1 ) ,

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where: sech6 (ω0 t) ˜ M1432 (t) = − 42 cosh (ω0 t) + 13 cosh (3ω0 t) + 9 cosh (5ω0 t) 144ω02 3 + 48ω0 t sinh (ω0 t) , and t1 , t2 are defined by relations (6.33). Finally, the above considerations lead to the following estimation ( 4 ˜ ˜ 12ω0 ψ0 λµλ M1432 (t2 ) − M1432 (t1 ) for $ ≤ 2ω0 M1432 = . (6.43) 0 for $ > 2ω0 Substituting the above relations to (6.41), one gets M143 = 12ω04 ψ0 λµλ  4 M ˜ 1431 (t2 )− M ˜ 1431 (t1 ) − M ˜ 1432 (t2 ) − M ˜ 1432 (t1 ) for $ ≤ 2ω0 × . 0 for $ > 2ω0 (6.44)

Substituting the above relations to (6.38), one obtains πω 3πγ sech M14 (t0 ) = 4ω02 2ω0 πω 3 4 2 2 4 × 32ωω0 + π 3ω0 + 2ω0 ω − ω tanh sin ωt0 + 12ω04 ψ0 λµλ 2ω0  4 M ˜ 1431 (t2 ) − M ˜ 1431 (t1 ) − M ˜ 1432 (t2 ) − M ˜ 1432 (t1 ) for $ ≤ 2ω0 × . 0 for $ > 2ω0

Finally, the chaos criterion is as follows 3πγ πω πω 3 4 2 2 4 sech 32ωω + π 3ω + 2ω ω − ω tanh > 0 0 0 4ω02 2ω0 2ω0

12ω04 ψ0 λµλ   4 M ˜ 1431 (t2 ) − M ˜ 1431 (t1 ) − M ˜ 1432 (t2 ) − M ˜ 1432 (t1 ) for $ ≤ 2ω0 × . 0 for $ > 2ω0 (6.45)

Since

and

lim P˙ (t) ϕ˙ 0 (t) = ∞ t−>±∞ lim P˙ (t) ϕ˙ 0 (t) Q (t) = ∞, t−>±∞

the Melnikov functions defined by (6.26) and (6.27) are divergent.

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6.2

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99

Numerical Results

In Figure 6.3 chaotic threshold in the parameters plane (γ1 , $) for fixed ω0 = 1, ω = 0, 5, δ = 0, 15, T0 = 0, 1, γ2 = 0, 3 is reported. Taking $ = 0, 2 and ω0 = 1, ω = 0, 5, δ = 0, 15, T0 = 0, 1, and γ2 = 0, 3 one may easy compute a critical value of the external amplitude excitation γ1 ≈ 0, 466. In order to verify the obtained analytical results via the Melnikov-Gruendler approach the bifurcation diagrams, shown in Figure 6.5, have been computed. The amplitude of harmonic excitation γ1 served as the control parameter.

Fig. 6.3 Bifurcation diagrams for $ = 0, 2 and ω0 = 1, ω = 0, 5, δ = 0, 15, T0 = 0, 1, γ2 = 0, 3.

One may observe a system transition into chaos in the neighborhood of γ1 ≈ 0, 5. Figure 6.6 presents phase plots (up) and Poincaré maps (down) of the investigated pendulum for the following fixed parameters: γ1 = 0, 49, ω0 = 1, ω = 0, 5, δ = 0, 15, T0 = 0, 1, γ2 = 0, 3, $ = 0, 2.

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Fig. 6.4 Phase portrait and Poincar´ e map for γ1 = 0, 49, ω0 = 1, ω = 0, 5, δ = 0, 15, T0 = 0, 1, γ2 = 0, 3, $ = 0, 2.

Fig. 6.5 Phase portrait and Poincar´ e map for γ1 = 0, 5, ω0 = 1, ω = 0, 5, δ = 0, 15, T0 = 0, 1, γ2 = 0, 3, $ = 0, 2.

Our system moves in a periodic manner with a frequency locked with the frequency of excitation. On the other hand in Figure 6.7 also the phase portrait and the Poincaré maps are reported for the same fixed parameters but increasing the amplitude of excitation up to the value of γ1 = 0, 5. The obtained irregular dynamics after the homoclinic bifurcation validates our theoretical prediction.

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101

Fig. 6.6 Phase portrait (up) and Poincar´ e map (down) of the investigated self-excited spherical system (γ1 = 0.49).

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Fig. 6.7 Phase portrait (up) and Poincar´ e map (down) of the investigated self-excited spherical system (γ1 = 0.5).

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Chapter 7

A Double Self-excited Duffing-type Oscillator

Two-degrees-of-freedom mechanical self-excited autonomous and nonautonomous systems with Duffing-type stiffnesses and polynomial-type friction are studied. The general Melnikov-Gruendler formalism is introduced and the associated homoclinic intersection functions are derived for both slip and stick-slip events. A numerical verification of the obtained analytical results is also addressed.

7.1

Analytical Prediction of Chaos

We study a Duffing-type 2-DOF mechanical system with discontinuities (see Figure 7.1). It consists of two masses m lying on a rigid belt moving at constant speed vˆ. It is assumed that between both masses and the belt polynomial-type friction occurs. The masses m are coupled by springs with stiffness satisfying the following conditions: k0 (−z) = −k0 (z). Each of the masses is linked to a basis by non-linear elastic elements of Duffingtype characteristics as well as elements modeling internal, aerodynamic and hydrodynamic friction. One of the masses is driven harmonically with frequency Ω and relatively small amplitude Γ. It is assumed that all frictional forces are relatively small, which is formally exhibited by an introduction of small (perturbation) parameter ε (see [8]). Dimensional equations of motion have the form m¨ z1 − k · z1 + k1 · z13 + εc1 (z1 , z˙1 ) + k0 (z1 − z2 ) + εθ (z˙1 − vˆ) = εΓ cos Ωτ, m¨ z2 − k · z2 + k1 · z23 + εc2 (z2 , z˙2 ) − k0 (z1 − z2 ) + εθ (z˙2 − vˆ) = 0, (7.1) where dry friction characteristic is as follows 3 θ (z˙ − vˆ) = θ0 sgn (z˙ − vˆ) − α ˆ (z˙ − vˆ) + βˆ (z˙ − vˆ) .

103

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Fig. 7.1

Model of the considered system.

Elements modeling internal, aerodynamic and hydrodynamic friction are defined by the formula cj (zj , z˙j ) = kj1 zj + cj1 z˙j + cj2 z˙j2 sgn z˙j + cj3 z˙j3 , j = 1, 2. According to the introduced general form, one may simultaneously analyze oscillations of various 2-DOF mechanical systems. This is easily realized by a proper choice of parameters. For example, in order to study a self-excited 2-DOF Duffing-type oscillator with polynomial-type friction ˆ whereas kj1 , cj1 , cj2 , cj3 should one has to assign non-zero values of θ0 , α ˆ , β, ˆ kj1 , cj1 , cj3 equal be equal to zero. On the other hand, taking θ0 , α ˆ , β, to zero, and assuming cj2 6= 0, it is possible to predict thresholds of chaos in our 2-DOF mechanical system with aerodynamic friction, etc. Since the studied system has two degrees of freedom, the Melnikov-Gruendler approach is applied to study homoclinic chaos occurrence. Introducing the following variables p p p z1 = x1 k/k1 , z2 = x2 k/k1 , τ = t m/k1 , the following non-dimensional equations are obtained x ¨1 − x1 + x31 + a · k0 (b · (x1 − x2 )) = ε (γ cos ωt − δ1 (x1 , x˙ 1 ) − T (x˙ 1 − v∗ )) , x ¨2 − x2 + x32 − a · k0 (b · (x1 − x2 )) = −ε (T (x˙ 2 − v∗ ) + δ2 (x2 , x˙ 2 )) , (7.2) where 3

T (x˙ − v∗ ) = T0 sgn (x˙ − v∗ ) − α (x˙ − v∗ ) + β (x˙ − v∗ )

(7.3)

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105

and δj (xj , x˙ j ) = αj1 xj + δj1 x˙ j + δj2 x˙ 2j sgn (x˙ j ) + δj3 x˙ 3j . (7.4) q q In the above, the following notation is applied: a = k1 kk1 , b = kk1 , γ = q √ √ p ˆ 2 v ˆ mk1 θ0 k1 ˆ √α Γ kk1 , ω = Ω m and v = , T = , β = mkβk√mk , ∗ 0 k k k k , α = mk 1 q 3 cj1 cj2 kj1 cj3 k 2 k αj1 = √mk , δj1 = √mk , δj2 = m k1 , δj3 = k1 m . The analyzed system is given in the first order ODEs form x˙ 1 = v1 , v˙ 1 = x1 − x31 − a · k0 (b · (x1 − x2 )) + ε (γ cos ωt − δ1 (x1 , v1 ) − T (v1 − v∗ )) , x˙ 2 = v2 , v˙ 2 = x2 − x32 + a · k0 (b · (x1 − x2 )) − ε (T (v2 − v∗ ) + δ2 (x2 , v2 )) . (7.5) For ε = 0 one gets x ¨1 − x1 + x31 + a · k0 (b · (x1 − x2 )) = 0, x ¨2 − x2 + x32 − a · k0 (b · (x1 − x2 )) = 0,

(7.6)

x˙ 1 = v1 , v˙ 1 = x1 − x31 − a · k0 (b · (x1 − x2 )) , x˙ 2 = v2 , v˙ 2 = x2 − x32 + a · k0 (b · (x1 − x2 )) .

(7.7)

or equivalently

Since the analyzed system has an equilibrium in the origin of phase coordinates, the following homoclinic solution is assumed q0 (x1 , x2 ) (t) = (Ar (Bt) , Ar (Bt)) , where r (t) = sech (t). Constants A and B are defined by a substitution of the homoclinic orbit into equations (7.6) from the following algebraic equation sech (Bt) 1 − B 2 + A2 − 2B 2 sech2 (Bt) = 0. (7.8) Assuming B = 1 and substituting it into (7.8) one obtains √ A = ± 2. Furthermore, we assume that the critical point being the origin of our coordinates system is associated with a homoclinic solution of the form q0 (x1 , x2 ) (t) = (Ar (Bt) , −Ar (Bt)) .

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Constants A and B are determined in a way similar to the previous one (substituting the homoclinic orbit equation to equations (7.6)): a · k0 (2Ab sech (Bt)) + A sech (Bt) B 2 − 1 + A2 − 2B 2 sech2 (Bt) = 0. Since this equation is satisfied for k0 (z) = k0 · z, hence A sech (Bt) B 2 − 1 + 2ak0 b + A2 − 2B 2 sech2 (Bt) = 0. √ Taking B = ± 1 − 2abk0 (abk0 < 1/2) one gets

(7.9)

A2 − 2 (1 − 2abk0 ) = 0, and finally 1 . 2 Note that irrespective of the coupling characteristic k0 (z1 − z2 ) there are two homoclinic orbits associated with the considered equilibrium and defined by the formula  √  ±√2r (t)  ± 2r˙ (t)   √ q01,2 (x1 , x˙ 1 , x2 , x˙ 2 ) (t) =  (7.10)  ± 2r (t)  . √ ± 2r˙ (t) p A = ± 2 (1 − 2abk0 ),

abk0
0. A perturbation vector associated with homoclinic orbits (7.10) and sign “+” is as follows   0  γ cos ωt − δ1 (x10 , x˙ 10 ) − T (x˙ 01 − v∗ )  . h (q01 , t) =    0 −T (x˙ 02 − v∗ ) − δ2 (x20 , x˙ 20 ) Substituting the first column of the fundamental solution matrix (7.20) by vector h (q01 , t) one obtains the matrix whose determinant is i h (1) (2) (3) (4) (7.27) K1 (t, t0 ) = det K1 , K1 , K1 , K1 , where: (1)

K1



 0  γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) − T (x˙ 01 − v∗ )  , =   0 −T (x˙ 02 − v∗ ) − δ2 (x20 , x˙ 20 )

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(2)

K1



 Q (t) r˙ (t)  (Q (t) r˙ (t))˙  =  Q (t) r˙ (t)  , (Q (t) r˙ (t))˙

(3)

K1

 √ P (−t) exp − 1 − 2ct √  P (−t) exp − 1 − 2ct ˙   √ =  −P (−t) exp − 1 − 2ct  , √ − P (−t) exp − 1 − 2ct ˙ 

(4)

K1



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r¨ (t)

Its expansion with respect to the first column gives √ K1 (t, t0 ) = 2P (−t) exp − 1 − 2ct × (γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) + δ2 (x20 , x˙ 20 ) − T (x˙ 01 − v∗ ) + T (x˙ 02 − v∗ )) . Since in the considered case x01 = x02 , one gets √ K1 (t, t0 ) = 2P (−t) exp − 1 − 2ct

(γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) + δ2 (x10 , x˙ 10 ))

(7.28)

Substitution of the second column of the fundamental matrix solution (7.20) by vector h (q01 , t) gives a matrix whose determinant is i h (1) (2) (3) (4) K2 (t, t0 ) = det K2 , K2 , K2 , K2 , (7.29) where: (1)

K2

(2)

K2

(3)

K2

 √ P (t) exp 1 − 2ct √  P (t) exp 1 − 2ct ˙   √ =  −P (t) exp 1 − 2ct  , √ − P (t) exp 1 − 2ct ˙ 





(4)

K2



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r¨ (t)

Its expansion with respect to the second column gives K2 (t, t0 ) = 2A γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) − δ2 (x20 , x˙ 20 ) − T (x˙ 01 − v∗ ) − T (x˙ 02 − v∗ ) r˙ (t) ,

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where (see Theorem 7.1) √ √ P (t) exp 1 + 2at P (−t) exp − √1 − 2at = const. √ A = P (t) exp 1 + 2at ˙ P (−t) exp − 1 − 2at ˙

Since x01 = x02 , one gets K2 (t, t0 ) = 2A

× (γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) − δ2 (x10 , x˙ 10 ) − 2T (x˙ 01 − v∗ )) r˙ (t) . (7.30) Substituting the third column of the fundamental solution matrix (7.20) by vector h (q01 , t) one obtains a matrix whose determinant is h i (1) (2) (3) (4) K3 (t, t0 ) = det K3 , K3 , K3 , K3 , (7.31) where: (1)

K3

(3)

K3


(2)

K3







(4)

K3



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r¨ (t)

Expansion with respect to the third column provides √ K3 (t, t0 ) = −2P (t) exp 1 − 2ct × (γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) − δ2 (x20 , x˙ 20 ) − T (x˙ 01 − v∗ ) − T (x˙ 02 − v∗ )) . Since x01 = x02 , one obtains K3 (t, t0 ) = −2 (γ cos ω (t + t0 )) − δ1 (x10 , x˙ 10 ) − δ2 (x10 , x˙ 10 ) √ − 2T (x˙ 01 − v∗ ) P (t) exp 1 − 2ct .

(7.32)

Substituting the fourth column of the fundamental solution matrix (7.20) by vector h (q01 , t) a new matrix is obtained, whose determinant is as follows h i (1) (2) (3) (4) K4 (t, t0 ) = det K4 , K4 , K4 , K4 , (7.33) where: (1)

K4


(2)

K4




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(3)

K4

(4)

K4




 0  γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) − T (x˙ 01 − v∗ )  . =   0 −T (x˙ 02 − v∗ ) − δ2 (x20 , x˙ 20 )

Expansion with respect to the fourth column gives | limt→±∞ K4 (t, t0 )| = ∞. A similar procedure is applied for sign “-”, where the perturbation vector is as follows   0  γ cos ωt + δ1 (x10 , x˙ 10 ) − T (−x˙ 01 − v∗ )  . h (q02 , t) =    0 −T (−x˙ 02 − v∗ ) + δ2 (x20 , x˙ 20 ) Substituting the first column of the fundamental solution matrix (7.20) by vector h (q02 , t) one obtains a matrix whose determinant is given below: i h (4) (1) (2) (3) (7.34) K1− (t, t0 ) = det K1− , K1− , K1− , K1− , where: 

0



 γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) − T (−x˙ 01 − v∗ )  (1) , K1− =    0

(3)

K1−

−T (−x˙ 02 − v∗ ) + δ2 (x20 , x˙ 20 )   Q (t) r˙ (t)  (Q (t) r˙ (t))˙ (2)  K1− =   Q (t) r˙ (t)  , (Q (t) r˙ (t))˙  √    P (−t) exp − 1 − 2ct r˙ (t) √  P (−t) exp − 1 − 2ct ˙   r¨ (t)  (4)    √ =  −P (−t) exp − 1 − 2ct  , K1− =  r˙ (t)  . √ − P (−t) exp − 1 − 2ct ˙ r¨ (t)

Expansion with respect to the first column of the determinant yields √ K1− (t, t0 ) = 2P (−t) exp − 1 − 2ct × (γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) − δ2 (x20 , x˙ 20 ) −T (−x˙ 01 − v∗ ) + T (−x˙ 02 − v∗ )) .

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In our case x01 = x02 and therefore √ K1− (t, t0 ) = 2P (−t) exp − 1 − 2ct

× (γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) − δ2 (x10 , x˙ 10 )) .

(7.35)

Substitution of the second column of the solution matrix (7.20) by vector h (q02 , t) gives the following determinant h i (1) (2) (3) (4) K2− (t, t0 ) = det K2− , K2− , K2− , K2− , (7.36) where: (1)

K2−






0

 γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) − T (−x˙ 01 − v∗ )  (2) , K2− =    0 −T (−x˙ 02 − v∗ ) + δ2 (x20 , x˙ 20 )

(3)

K2−


(4)

K2−



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r¨ (t)

Expansion with respect to the second column gives K2− (t, t0 ) = 2A (γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) + δ2 (x20 , x˙ 20 ) −T (−x˙ 01 − v∗ ) − T (−x˙ 02 − v∗ )) r˙ (t) , where

√ √ P (t) exp 1 + 2at P (−t) exp − √1 − 2at = const. √ A = P (t) exp 1 + 2at ˙ P (−t) exp − 1 − 2at ˙

Since x01 = x02 , therefore K2− (t, t0 ) = 2A

× (γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) + δ2 (x10 , x˙ 10 ) − 2T (−x˙ 01 − v∗ )) r˙ (t) (7.37) Substituting the third column of the fundamental solution matrix (7.20) by vector h (q02 , t) gives the following determinant: h i (1) (2) (3) (4) K3− (t, t0 ) = det K3− , K3− , K3− , K3− , (7.38)

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where (1)

K3−




(2)

K3−



 Q (t) r˙ (t)  (Q (t) r˙ (t))˙  =  Q (t) r˙ (t)  , (Q (t) r˙ (t))˙ 

0

 γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) − T (−x˙ 01 − v∗ )  (3) , K3− =    0

(4)

K3−

−T (−x˙ 02 − v∗ ) + δ2 (x20 , x˙ 20 )



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r¨ (t)

Similarly, expansion with respect to the third column yields √ K3− (t, t0 ) = −2P (t) exp 1 − 2ct × γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) + δ2 (x20 , x˙ 20 ) −T (−x˙ 01 − v∗ ) − T (−x˙ 02 − v∗ ) , and for x01 = x02 we have

√ K3− (t, t0 ) = −2P (t) exp 1 − 2ct × (γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) + δ2 (x10 , x˙ 10 ) − 2T (−x˙ 01 − v∗ )) . (7.39) When substituting the fourth column of the fundamental solution matrix (7.20) by vector h (q02 , t) one gets a matrix with the following determinant h i (1) (2) (3) (4) K4− (t, t0 ) = det K4− , K4− , K4− , K4− , (7.40) where: (1)

K4−

 √   P (t) exp 1 − 2ct Q (t) r˙ (t) √  P (t) exp 1 − 2ct ˙   (Q (t) r˙ (t))˙ (2)    √ =  −P (t) exp 1 − 2ct  , K4− =  Q (t) r˙ (t)  , √ − P (t) exp 1 − 2ct ˙ (Q (t) r˙ (t))˙ √   P (−t) exp − 1 − 2ct √  P (−t) exp − 1 − 2ct ˙  (3)  √ K4− =   −P (−t) exp − 1 − 2ct  , √ − P (−t) exp − 1 − 2ct ˙ 



0



 γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) − T (−x˙ 01 − v∗ )  (4) . K4− =    0 −T (−x˙ 02 − v∗ ) + δ2 (x20 , x˙ 20 )

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117

Expansion with respect to the fourth column gives lim K4− (t, t0 ) = ∞. t→±∞ Perturbation vector along homoclinic orbits (7.11) for sign “+” is as follows   0  γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) − T (x˙ 01 − v∗ )  . h (q03 , t) =    0 −T (x˙ 02 − v∗ ) − δ2 (x20 , x˙ 20 )

Substitution of the first column of the fundamental solution matrix (7.26) by vector h (q03 , t) gives the following determinant: i h (4) (3) (1) (2) (7.41) L1 (t, t0 ) = det L1 , L1 , L1 , L1 , where (1)

L1




(2)

L1

(3)

L1

√ Q 1 − 2ct r˙ √  Q 1 − 2ct r˙ √ =  Q 1 − 2ct r˙ √ Q 1 − 2ct r˙ 

 √ e−t R − 1 − 2ct √  e−t R − 1 − 2ct ˙   √ =  −e−t R − 1 − 2ct  , √ − e−t R − 1 − 2ct ˙ 

 √ 1 − 2ct √ 1 − 2ct ˙ , √ 1 − 2ct  √ 1 − 2ct ˙ (4)

L1

 √ r˙ 1 − 2ct √  r¨ 1 − 2ct  . √ =  r˙ 1 − 2ct  √ r¨ 1 − 2ct 

Expansion with respect to the first column gives √ L1 (t, t0 ) = 2e−t R − 1 − 2ct γ cos ω (t + t0 ) − δ1(x10 , x˙ 10 ) +δ2 (x20 , x˙ 20 ) − T (x˙ 01 − v∗ ) + T (x˙ 02 − v∗ ) , and because x01 = −x02 , one gets √ L1 (t, t0 ) = 2e−t R − 1 − 2ct γ cos ω (t + t0 ) − δ1 (x 10 , x˙ 10 ) +δ2 (x10 , x˙ 10 ) − T (x˙ 01 − v∗ ) + T (−x˙ 01 − v∗ ) .

(7.42)

Substitution of the second column of the fundamental solution matrix (7.26) by the vector h (q03 , t) gives a matrix and the associated determinant of the form: h i (1) (2) (3) (4) L2 (t, t0 ) = det L2 , L2 , L2 , L2 , (7.43)

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where (1)

L2

(2)

L2

(3)

L2

 √ et R 1 − 2ct √  et R 1 − 2ct ˙   √ =  −et R 1 − 2ct  , √ − et R 1 − 2ct ˙ 





(4)

L2

 √ r˙ 1 − 2ct √  r¨ 1 − 2ct  . √ =  r˙ 1 − 2ct  √ r¨ 1 − 2ct 

Expansion with respect to the second column gives √ L2 (t, t0 ) = 2Ar˙ 1 − 2ct γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) − δ2 (x20 , x˙ 20 ) √ − T (x˙ 01 − v∗ ) − 2AT (x˙ 02 − v∗ ) r˙ 1 − 2ct . Since x01 = −x02 , one gets √ L2 (t, t0 ) = 2Ar˙ 1 − 2ct γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) +δ2 (x10 , x˙ 10 ) − T (x˙ 01 − v∗ ) − T (−x˙ 01 − v∗ ) .

(7.44)

Substitution of the third column of the fundamental solution matrix (7.26) by vector h (q03 , t) gives the following determinant h i (1) (2) (3) (4) L3 (t, t0 ) = det L3 , L3 , L3 , L3 , (7.45) where: (1)

L3

(2)

L3



 √ 1 − 2ct √ 1 − 2ct ˙ , √ 1 − 2ct  √ 1 − 2ct ˙

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(3)

L3

SmoothNonsmooth

119



 0  γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) − T (x˙ 01 − v∗ )  , =   0 −T (x˙ 02 − v∗ ) − δ2 (x20 , x˙ 20 )   √ r˙ 1 − 2ct √  r¨ 1 − 2ct  (4) , √ L3 =   r˙ 1 − 2ct  √ r¨ 1 − 2ct

Expansion with respect to the third column gives

L3 (t, t0 ) = −2 γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) − δ2 (x20 , x˙ 20 ) − T (x˙ 10 − v∗) √ − T (x˙ 20 − v∗) et R 1 − 2ct .

Since x01 = −x02 , one gets √ L3 (t, t0 ) = −2et R 1 − 2ct γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) +δ2 (x10 , x˙ 10 ) − T (x˙ 01 − v∗ ) − T (−x˙ 01 − v∗ ) .

(7.46)

Substituting the fourth column of the fundamental solution matrix (7.26) by vector h (q03 , t) a matrix with the following determinant is obtained h i (1) (2) (3) (4) L4 (t, t0 ) = det L4 , L4 , L4 , L4 , (7.47) where:

 √ et R 1 − 2ct √  et R 1 − 2ct ˙  (1)  √ L4 =   −et R 1 − 2ct  , √ − et R 1 − 2ct ˙ √  √  Q 1 − 2ct r˙ 1 − 2ct √ √   Q 1 − 2ct r˙ (2) √1 − 2ct ˙ , √ L4 =   Q 1 − 2ct r˙  √ 1 − 2ct √ Q 1 − 2ct r˙ 1 − 2ct ˙  √  e−t R − 1 − 2ct √  e−t R − 1 − 2ct ˙  (3)  √ L4 =   −e−t R − 1 − 2ct  , √ − e−t R − 1 − 2ct ˙   0  γ cos ω (t + t0 ) − δ1 (x10 , x˙ 10 ) − T (x˙ 01 − v∗ )  . =   0 −T (x˙ 02 − v∗ ) − δ2 (x20 , x˙ 20 ) 

(4)

L4

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Expansion with respect to the third column gives: lim L4 (t, t0 ) = ∞. t→±∞ The perturbation vector along homoclinic orbits (7.11) for interior sign is as follows:   0  γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) − T (−x˙ 01 − v∗ )  . h (q04 , t) =    0 −T (−x˙ 02 − v∗ ) + δ2 (x20 , x˙ 20 )

Substituting the first column of the fundamental solution matrix (7.26) by vector h (q04 , t) gives the following determinant: i h (3) (4) (2) (1) (7.48) L1− (t, t0 ) = det L1− , L1− , L1− , L1− , where: 



0

 γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) − T (−x˙ 01 − v∗ )  (1) , L1− =    0 −T (−x˙ 02 − v∗ ) + δ2 (x20 , x˙ 20 )

(2)

L1−

(3)

L1−



 √ 1 − 2ct √ 1 − 2ct ˙ , √ 1 − 2ct  √ 1 − 2ct ˙

(4)

L1−

 √ r˙ 1 − 2ct √  r¨ 1 − 2ct  . √ =  r˙ 1 − 2ct  √ r¨ 1 − 2ct 

Owing to the first column expansion, we get √ L1− (t, t0 ) = 2e−t R − 1 − 2ct γ cos ω (t + t0 ) + δ1 (x 10 , x˙ 10 ) −δ2 (x20 , x˙ 20 ) − T (−x˙ 01 − v∗ ) + T (−x˙ 02 − v∗ ) , and because x01 = −x02 we have √ L1− (t, t0 ) = 2e−t R − 1 − 2ct γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) +δ2 (x10 , x˙ 10 ) − T (−x˙ 01 − v∗ ) + T (x˙ 01 − v∗ ) . Substituting the second column of (7.26) by h (q04 , t) we obtain h i (1) (2) (3) (4) L2− (t, t0 ) = det L2− , L2− , L2− , L2− ,

(7.49)

(7.50)

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121

where: (1)

L2−






0


(3)

L2−


(4)

L2−

 √ r˙ 1 − 2ct √  r¨ 1 − 2ct  . √ =  r˙ 1 − 2ct  √ r¨ 1 − 2ct 

Expansion with respect to the second column gives √ L2− (t, t0 ) = 2Ar˙ 1 − 2ct γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) +δ2 (x20 , x˙ 20 ) − T (−x˙ 01 − v∗ ) − T (−x˙ 02 − v∗ ) . Since x01 = −x02 , one gets √ L2− (t, t0 ) = 2Ar˙ 1 − 2ct γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) −δ2 (x10 , x˙ 10 ) − T (−x˙ 01 − v∗ ) − T (x˙ 01 − v∗ ) .

(7.51)

Substituting the third column of the fundamental solution matrix (7.26) by vector h (q04 , t) yields a matrix, and then the following determinant h i (1) (2) (3) (4) L3− (t, t0 ) = det L3− , L3− , L3− , L3− , (7.52) where: (1)

L3−

(2)

L3−



 √ 1 − 2ct √ 1 − 2ct ˙ , √ 1 − 2ct  √ 1 − 2ct ˙

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



0


(4)

L3−

 √ 1 − 2ct r˙ √  r¨ 1 − 2ct  . √ =  r˙ 1 − 2ct  √ r¨ 1 − 2ct 

A similar operation with respect to the third column gives √ L3− (t, t0 ) = −2et R 1 − 2ct γ cos ω (t + t0 ) + δ1 (x10 , x˙ 10 ) +δ2 (x20 , x˙ 20 ) − T (−x˙ 01 − v∗ ) − T (−x˙ 02 − v∗ )

and owing to x01 = −x02 one gets √ L3− (t, t0 ) = −2et R 1 − 2ct γ cos ω (t + t0 ) + δ1 (x 10 , x˙ 10 ) −δ2 (x10 , x˙ 10 ) − T (−x˙ 01 − v∗ ) − T (x˙ 01 − v∗ ) . Application of the described algorithm to column four gives h i (1) (2) (3) (4) L4− (t, t0 ) = det L4− , L4− , L4− , L4− , where: (1)

L4−

(2)

L4−

(3)



 √ et R 1 − 2ct √  et R 1 − 2ct ˙   √ =  −et R 1 − 2ct  , √ − et R 1 − 2ct ˙

√ Q 1 − 2ct r˙ √  Q 1 − 2ct r˙ √ =  Q 1 − 2ct r˙ √ Q 1 − 2ct r˙

L4−

(7.54)





 √ 1 − 2ct √ 1 − 2ct ˙ , √ 1 − 2ct  √ 1 − 2ct ˙


0




(7.53)

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SmoothNonsmooth

123

and lim L4− (t, t0 ) = ∞. According to (7.28), the Melnikov function t→±∞ has the following form M1 (t0 ) = M11 (t0 ) + M12 ,

(7.55)

where: M11 (t0 ) = 2γ

Z∞

√ P (−t) exp − 1 − 2ct cos ω (t + t0 ) dt

0

= 2γ

Z∞

− 23 c +

√

√ 1 − 2c + tanh t tanh t exp − 1 − 2ct cos ω (t + t0 ) dt

0

(7.56) and M12 = Z∞ √ 2 (−δ1 (x10 (t) , x˙ 10 (t)) + δ2 (x10 (t) , x˙ 10 (t))) P (−t) exp − 1 − 2ct dt 0

=2

Z∞

(−δ1 (x10 (t) , x˙ 10 (t)) + δ2 (x10 (t) , x˙ 10 (t)))

0

× − 23 c +

√

√ 1 − 2c + tanh t tanh t exp − 1 − 2ct dt.

(7.57)

On the other hand, M11 (t0 ) = M111 (t0 ) − M112 (t0 ) ,

(7.58)

where: M111 (t0 ) = 2γ cos ωt0 Z∞ √ √ × − 32 c + 1 − 2c + tanh t tanh t exp − 1 − 2ct cos ωtdt 0

and M112 (t0 ) = 2γ sin ωt0 Z∞ √ √ × − 32 c + 1 − 2c + tanh t tanh t exp − 1 − 2ct sin ωtdt. 0

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Upon integration we get

√ √ 1 12 1 − 2c 12ω 2 8a 1 − 2c γ cos (ωt0 ) + + 6 1 − 2c + ω 2 1 − 2c + ω 2 1 − 2c + ω 2 1 √ 1 √ 1 − 2c − ωi + 3ωiψ 1 − 2c + ωi − 3ωiψ 4 4 √ √ 1 1 +3ωiψ 2 + 1 − 2c − ωi − 3ωiψ 2 + 1 − 2c + ωi , 4 4

M111 (t0 ) =

√ 1 12 1 − 2c 12 8a M112 (t0 ) = γω sin (ωt0 ) − − 6 1 − 2c + ω 2 1 − 2c + ω 2 1 − 2c + ω 2 1 √ 1 √ + 3ψ 1 − 2c − ωi + 3ψ 1 − 2c + ωi 4 4 √ √ 1 1 −3ψ 2 + 1 − 2c − ωi − 3ψ 2 + 1 − 2c + ωi , 4 4 where ψ (z) denotes the ”digamma” function. The “digamma” function ψ(z) is a logarithmic derivative to the Euler function Γ(z) defined in the following way Γ0 (z) . Γ(z)

ψ(z) =

¯ It has the property: ψ(¯ z ) = ψ(z). It may be understood as a particular case (for n = 0) of the ”polygamma” function defined as n + 1 derivative of the function ψ(z): ψ (n) (z) =

dn ψ(z) . dz n

It can be expanded into the series: ψ

(n)

n+1

(z) = (−1)

n!

∞ X

k=0

1 , (z + k)n+1

and its main properties follow: ∂ψ (n) (z) = ψ (n+1) (z), ∂z

Z

C

ψ (n) (z)dz = ψ (n−1) (z).

Since ψ (¯ z ) = ψ (z), one gets √ (6 − 4c) 1 − 2c + 6ω 2 M111 (t0 ) = γ cos (ωt0 ) 3 (1 − 2c + ω 2 ) √ 1 √ 1 − ω =ψ 1 − 2c + ωi −=ψ 2 + 1 − 2c + ωi , 4 4

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125

√

1 − 2c − 1 − 4c M112 (t0 ) = γ sin (ωt0 ) 3 (1 − 2c + ω 2 ) √ 1 1 √ 1 − 2c + ωi − 2 ! √ √ 2 4 4 4A (δ11 + δ21 ) + (δ12 + δ22 ) + 2 (δ13 + δ23 ) 3 15 35 √ √ 16 2Aβ 8 2 (7.96) + A 3βv∗2 − α + 3 35 ! r r  q q √ 2 2  v v 1 1 1 1 8AT ∗ ∗ for v∗ < 22 0 2 + 4 − 2 − 2 − 4 − 2 − √   2 0 for v∗ ≥ 2 The Melnikov function (taking into account (7.39)) is defined as follows M3− (t0 ) = −M31− (t0 ) − M32− + M33−

(7.97)

where M31− (t0 ) = M31 (t0 ) = 2γ

Z0

−∞

P (t) exp

√

1 − 2ct cos ω (t + t0 ) dt

and M32− = M32 = 2 × exp

√

Z0

(δ1 (x10 (t) , x˙ 10 (t)) + δ2 (x10 (t) , x˙ 10 (t))) P (t)

−∞

1 − 2ct dt,

M33− = −M33 = 4

Z0

−∞

T (−x˙ 01 − v∗ ) P (t) exp

√

1 − 2ct dt.

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145

Using (7.3), formula M33− takes the form M33− = M331− + M332− + M333− − M334− + M335−

(7.98)

where the following notation is applied M331− = 4T0

Z0

sgn (−x˙ 10 (t) − v∗ ) − 23 c −

−∞

√

× exp

Z0

− 23 c −

√

−∞

M333− ×

Z0

1 − 2c − tanh t tanh t

1 − 2ct dt,

M332− = 4v∗ 1 − βv∗2 ×

√

√ 1 − 2c − tanh t tanh t exp 1 − 2ct dt,

√ = 4 2 3βv∗2 − α √

− 23 c −

−∞

√ 1 − 2c − tanh t tanh t exp 1 − 2ct sech t tanh tdt,

M334− = 24βv∗ ×

Z0

− 23 c −

√

√ 1 − 2c − tanh t tanh t exp 1 − 2ct sech2 t tanh2 tdt,

√

√ 1 − 2c − tanh t tanh t exp 1 − 2ct sech3 t tanh3 tdt.

−∞

√ M335− = 8 2β ×

Z0

− 23 c −

−∞

Since for −∞ < t ≤ 0 we have sgn (−x˙ 10 (t) − v∗ ) = −1, one gets M331− = −4T0

Z0

−∞

− 32 c −

√

√ 1 − 2c − tanh t tanh t exp 1 − 2ct dt.

Integration gives 4T0 (3 − 2c) √ , 3 1 − 2c 1 − βv∗2 (3 − 2c) √ , 3 1 − 2c

M331− = M331 = −

M332− = M332 =

4v∗

(7.99)

(7.100)

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M333− = M333 =

M334− = M334 =

2+

√

√ 4 2 3βv∗2 − α , 3

(7.101)

256βv∗ √ √ 1 − 2c 4 + 1 − 2c 6 + 1 − 2c

1 √ 1 − 2c 10 + 1 − 2c √ √ × 4c2 + 297 1 + 1 − 2c + 6c 46 + 7 1 − 2c 1√ 1√ × F 6, 1 + 1 − 2c; 4 + 1 − 2c; −1 2 2 √ + (−2c − 3) 42 − 4c + 24 1 − 2c 1√ 1√ × F 6, 2 + 1 − 2c; 5 + 1 − 2c; −1 2 2 √ 1√ 1√ + −2c − 3 + 3 1 − 2c F 6, 3 + 1 − 2c; 6 + 1 − 2c; −1 , 2 2 (7.102) ×

8+

√

√ −2048 2β √ √ √ √ M335− = M335 = 3 + 1 − 2c 5 + 1 − 2c 7 + 1 − 2c 9 + 1 − 2c 1 √ √ × 11 + 1 − 2c 13 + 1 − 2c √ √ × 2c2 + 252 1 + 1 − 2c − 3c 73 + 9 1 − 2c √ √ 1 1 × F 8, 3 + 1 − 2c ; 11 + 1 − 2c ; −1 2 2 √ + 2 (−2c − 3) 20 − c + 8 1 − 2c √ √ 1 1 × F 8, 5 + 1 − 2c ; 13 + 1 − 2c ; −1 2 2 √ + 2c2 (−5c − 12) −1 + 1 − 2c √ √ 1 1 × F 8, 7 + 1 − 2c ; 15 + 1 − 2c ; −1 . 2 2 (7.103) Substituting (7.99), (7.100), (7.101), (7.102) and (7.103) to (7.98) one gets √ 4 2 4 (3 − 2c) 2 M33 = √ v∗ 1 − βv∗ − T0 − 3βv∗2 − α − M334 − M335 . 3 3 1 − 2c (7.104)

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147

Substitution of (7.77), (7.78) and (7.104) to (7.75) gives the Melnikov function defined as follows 4 (3 − 2c) M3− (t0 ) = −A˜ sin (ωt0 + κ) + √ v∗ 1 − βv∗2 − T0 − M323 3 1 − 2c √ √ 2 2 2 2 δ11 + δ21 + 2 3βv∗ − α + (α21 + α11 ) − M334 − M324 + 3 3 √ √ √ 1 1 − M335 3 1 − 2c − cψ 1 + 1 − 2c + cψ 3 + 1 − 2c . 4 4 The chaos criterion is √ 4 (3 − 2c) 2 2 2 δ11 + δ21 + 2 3βv∗2 − α A˜ > √ v∗ 1 − βv∗ − T0 + 3 3 1 − 2c √ 2 − M323 − M324 − M334 − M335 + (α21 + α11 ) 3 √ √ √ 1 1 . × 3 1 − 2c − cψ 1 + 1 − 2c + cψ 3 + 1 − 2c 4 4 (7.105)

According to (7.42), the Melnikov function is N1 (t0 ) = N11 (t0 ) − N12 − N13 + N14

(7.106)

where N11 (t0 ) = 2γ

Z∞

√ R − 1 − 2ct e−t cos ω (t + t0 ) dt

0

= 2γ

Z∞ 0

×e

−t

! √ √ tanh − 1 − 2ct 3 − 4c √ − sech2 − 1 − 2ct − 3 (1 − 2c) 1 − 2c

cos ω (t + t0 ) dt

and N12 = 2

Z∞

√ (δ1 (x10 (t) , x˙ 10 (t)) − δ2 (x10 (t) , x˙ 10 (t))) e−t R − 1 − 2ct dt,

0

Z∞

N13 = 2

√ e−t R − 1 − 2ct T (x˙ 01 − v∗ ) dt

0

=2

Z∞ 0

−t

e

! √ √ tanh − 1 − 2ct 3 − 4c 2 √ − sech − 1 − 2ct − 3 (1 − 2c) 1 − 2c

× T (x˙ 01 − v∗ ) dt,

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Z∞

N14 = 2

√ e−t R − 1 − 2ct T (−x˙ 01 − v∗ ) dt

0

=2

Z∞ 0

−t

e


× T (−x˙ 01 − v∗ ) dt, Formula N11 (t0 ) is defined as follows N11 (t0 ) = N111 (t0 ) − N112 (t0 ) ,

(7.107)

where N111 (t0 ) = 2γ cos ωt0 ! √ Z∞ √ tanh − 1 − 2ct 3 − 4c 2 √ − sech − 1 − 2ct − e−t cos ωtdt × 3 (1 − 2c) 1 − 2c 0

(7.108) and N112 (t0 ) = 2γ sin ωt0 ! √ Z∞ √ tanh − 1 − 2ct 3 − 4c √ × − sech2 − 1 − 2ct − e−t sin ωtdt. 3 (1 − 2c) 1 − 2c 0

(7.109) Upon integration we have √ 2ω (4c − 3) 1 − 2c − 3 (2c − 1) γ cos ωt0 √ N112 (t0 ) = − 1 + ω2 3 (2c − 1) 1 − 2c 3 √ 1 + ωi 1 − ωi √ √ + ω 1 − 2c ψ +ψ 2 4 1 − 2c 4 1 − 2c 1 1 1 + ωi 1 − ωi −ψ . −ψ 2+ √ 2+ √ 4 4 1 − 2c 1 − 2c Owing to the property: ψ (¯ z ) = ψ (z) one gets √ 6 1 − 2cω 2 − 2 (4c − 3) N111 (t0 ) = γ cos ωt0 3 (2c − 1) (1 + ω 2 ) 1 ω 1 + ωi 1 + ωi √ −=ψ , + =ψ 2+ √ 2c − 1 4 4 1 − 2c 1 − 2c (7.110)

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149

√ 2ω (4c − 3) + 3 1 − 2c N112 (t0 ) = γ sin ωt0 − 3 (2c − 1) (1 + ω 2 ) 1 + ωi 1 1 − ωi ω √ − (1 − 2c)/ 2 the studied equation does not possess real solutions. This observation leads to N131 = −2T0 ! √ Z∞ √ tanh − 1 − 2ct 3 − 4c 2 −t √ × e − sech − 1 − 2ct − dt 3 (1 − 2c) 1 − 2c 0

=− where

¯131 (t) − N ¯131 (0) = 2T0 (4c − 3) lim N t→∞ 6c − 3

√ √ 2T0 e−t 3 − 4c + 3 1 − 2c tanh 1 − 2ct ¯ N131 (t) = . 3 (2c − 1)

(7.115)

Integral N132 is N132 = 2 v∗ − βv∗3 ! √ Z∞ √ tanh − 1 − 2ct 3 − 4c 2 −t √ × e − sech − 1 − 2ct − dt 3 (1 − 2c) 1 − 2c 0

=

2 (4c − 3) v∗ − βv∗3 . 6c − 3 (7.116)

On the other hand, integral N133 is equal to √ 2 (1 − 2c) 3βv∗2 − α N133 = 6c − 3 √ √ √ √ ∞ 3 −t × e sech 1 − 2ct sinh 2 1 − 2ct + 2 1 − 2ct cosh 2 1 − 2ct 0 ,

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SmoothNonsmooth


and therefore 2p 2 (1 − 2c) 3βv∗2 − α . 3 is defined as follows N133 =

Integral N134

(7.117)

2

384 (1 − 2c) βv∗ √ √ √ N134 = √ 3 1 − 2c 1 + 2 1 − 2c 1 + 4 1 − 2c 1 + 6 1 − 2c 1 √ √ × 1 + 8 1 − 2c 1 + 10 1 − 2c √ √ × 297 1 + 1 − 2c − 6 191 + 152 1 − 2c c √ + 16 69 + 40 1 − 2c c2 1 1 (7.118) × F 6, 1 + √ ;4 + √ ; −1 2 1 − 2c 2 1 − 2c √ √ − (8c − 3) −6 4 + 7 1 − 2c + 80c 1 + 1 − 2c 1 1 × F 6, 2 + √ ;5 + √ ; −1 2 1 − 2c 2 1 − 2c √ √ + 3 1 − 2c − 1 − 2c 4 1 − 2c − 3 1 1 × F 6, 3 + √ ;6 + √ ; −1 , 2 1 − 2c 2 1 − 2c whereas integral N135 is

√ 3 1024 2 (1 − 2c) β √ √ √ 1 + 3 1 − 2c 1 + 5 1 − 2c 1 + 7 1 − 2c 1 + 9 1 − 2c 1 √ √ × 1 + 11 1 − 2c 1 + 13 1 − 2c √ √ × 252 1 + 1 − 2c − 3 327 + 263 1 − 2c c √ + 954 + 572 1 − 2c c2 1 1 1 1 ; ; −1 × F 8, 3+ √ 11 + √ 2 2 1 − 2c 1 − 2c √ √ − 2 (8c − 3) −8 − 20 1 − 2c + 16c + 39c 1 − 2c 1 1 1 1 √ √ × F 8, 5+ ; 13 + ; −1 2 2 1 − 2c 1 − 2c √ √ √ + 12 1 − 2c − 1 − 53c 1 − 2c − 1 + c2 60 1 − 2c − 58 1 1 1 1 ; ; −1 . × F 8, 7+ √ 15 + √ 2 2 1 − 2c 1 − 2c (7.119)

N135 =

√

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155

Substitution of the above formulas to (7.114) gives 2p 2 (4c − 3) v∗ − βv∗3 + T0 − 2 (1 − 2c) 3βv∗2 − α −N134 −N135 . 6c − 3 3 (7.120) Applying (7.3), the formula for N14 takes the form N13 =

N14 = N141 + N142 + N143 − N144 + N145 ,

(7.121)

where N141 = 2T0

Z∞ 0

−t

e


× sgn (−x˙ 01 − v∗ ) dt, N142 = 2 v∗ − βv∗3 ! √ Z∞ √ tanh − 1 − 2ct 3 − 4c 2 −t √ dt, − sech − 1 − 2ct − × e 3 (1 − 2c) 1 − 2c 0

√ N143 = 2 2 (1 − 2c) 3βv∗2 − α ! √ Z∞ √ tanh − 1 − 2ct 3 − 4c 2 −t √ − sech − 1 − 2ct − × e 3 (1 − 2c) 1 − 2c 0 √ √ × sech 1 − 2ct tanh 1 − 2ct dt, 2

N144 = 12 (1 − 2c) βv∗ ! √ Z∞ √ tanh − 1 − 2ct 3 − 4c 2 −t √ × e − sech − 1 − 2ct − 3 (1 − 2c) 1 − 2c 0 √ √ × sech2 1 − 2ct tanh2 1 − 2ct dt, √ 3 N145 = 4 2 (1 − 2c) β ! √ Z∞ √ tanh − 1 − 2ct 3 − 4c 2 −t √ × e − sech − 1 − 2ct − 3 (1 − 2c) 1 − 2c 0 √ √ × sech3 1 − 2ct tanh3 1 − 2ct dt.

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√ Integral N141 for |v∗ | > (1 − 2c) 2 is defined as follows N141 = −2T0 Z∞ × e−t 0

=−

! √ √ tanh − 1 − 2ct 3 − 4c 2 √ − sech − 1 − 2ct − dt 3 (1 − 2c) 1 − 2c

¯141 (t) − N ¯141 (0) = 2T0 (4c − 3) , lim N t→∞ 6c − 3 (7.122)

where

√ √ 2T0 e−t 3 − 4c + 3 1 − 2c tanh 1 − 2ct ¯ N141 (t) = . 3 (2c − 1) √ In the second case, i.e. for |v∗ | < (1 − 2c) 2 one obtains ¯141 (t) t3 + N ¯141 (t) t4 − N ¯141 (t) ∞ N141 = 2T0 −N 0 t t4 3 ¯ ¯ ¯ ¯ = −2T0 2 N141 (t3 ) − N141 (t4 ) + lim N141 (t) − N141 (0) , t→∞

where t3 , t4 are defined as earlier. Taking into account (7.122) one obtains ¯141 (t3 ) − N ¯141 (t4 ) − 4c − 3 . N141 = −2T0 2 N 6c − 3 This part of our consideration provides the following main result  − 2c  4T N ¯ (t ) − N ¯141 (t4 ) for v∗ < 1 √ 2T0 (4c − 3)  0 141 3 2 N141 = − 1 − 2c  6c − 3 0 for v∗ ≥ √ 2 (7.123) Taking into account (7.116), integral N142 is equal to N142 = N132 =

2 (4c − 3) v∗ − βv∗3 . 6c − 3

(7.124)

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157

On the other hand, according to (7.117), integral N133 is as follows 2p N143 = N133 = 2 (1 − 2c) 3βv∗2 − α . (7.125) 3 Taking into account (7.118), integral N144 is defined in the following way 12βv∗ √ N144 = 2 4725 − 18216c + 17536c + 30 1 − 2c (189 + 8c (64c − 79)) 1 × 360 (2c − 1) × (30 (2c − 1) (7695 + 4c (−13587 + 8c (4419 + 128c (16c − 39)))) √ − 4 1 − 2c (69255 + c (−555705 + 8c (209115 + 32c (4392c − 8747)))) + (8c − 3) − 1485 + 8394c − 15712c2 + 9728c3 √ + 1 − 2c (−1485 + 4c (2751 + 16c (320c − 411))) 1 1 2 × F 1, 1 + √ ;4 + √ ; −1 + 2 (3 − 8c) (8c − 5) 2 1 − 2c 2 1 − 2c √ × −12 + 24c + 1 − 2c (40c − 21) 1 1 × F 1, 2 + √ ;5 + √ ; −1 + (8c − 3) 2 1 − 2c 2 1 − 2c √ × 45 + 102c − 896c2 + 1024c3 + 1 − 2c × (−45 + 4c (207 + 16c (32c − 39))) 1 1 ;6 + √ ; −1 . × F 1, 3 + √ 2 1 − 2c 2 1 − 2c (7.126) According to (7.119), integral N145 is √ 3 1024 2 (1 − 2c) β √ √ √ √ N145 = 1 + 3 1 − 2c 1 + 55 1 − 2c 1 + 7 1 − 2c 1 + 9 1 − 2c 1 √ √ × 1 + 11 1 − 2c 1 + 13 1 − 2c √ √ √ × 252 1 + 1 − 2c − 3 327 + 263 1 − 2c c + 954 + 572 1 − 2c c2 1 1 1 1 ; ; −1 × F 8, 3+ √ 11 + √ 2 2 2 1 − 2c 2 1 − 2c √ √ − 2 (8c − 3) −8 − 20 1 − 2c + 16c + 39c 1 − 2c (7.127)

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1 1 1 1 × F 8, 5+ √ ; 13 + √ ; −1 2 2 2 1 − 2c 2 1 − 2c √ √ √ + 12 1 − 2c − 1 − 53c 1 − 2c − 1 + c2 60 1 − 2c − 58 1 1 1 1 ; ; −1 . × F 8, 7+ √ 15 + √ 2 2 2 1 − 2c 2 1 − 2c Substituting the above formulas to (7.121) one obtains 2p 2 (4c − 3) N14 = v∗ − βv∗3 + T0 + 2 (1 − 2c) 3βv∗2 − α − N144 6c − 3 3  − 2c  ¯141 (t3 ) − N ¯141 (t4 ) for v∗ < 1 √  4T0 N 2 + N145 − 1 − 2c .  0 for v∗ ≥ √ 2 (7.128) Substitution of the above formulas to (7.106) gives p 2 2 (1 − 2c) ˜ N1 (t0 ) = A sin (ωt0 − κ) − (δ11 − δ21 ) + N123 + N124 3 √ 2 − (α21 − α11 ) 3 (2c − 1) √ 1 1 1 1 × 3 1 − 2c + cψ 1+ √ − cψ 3+ √ 4 4 1 − 2c 1 − 2c 4p + 2 (1 − 2c) 3βv∗2 − α + N134 + N135 − N144 + N145 3  q  4T0 −N ¯141 (t3 ) + N ¯141 (t4 ) for v∗ < 1−2c q 2 . + 0 for v∗ ≥ 1−2c 2 Finally, the next criterion of chaos is defined in the following way 2p2 (1 − 2c) ˜ A > − (δ11 − δ21 ) + N123 + N124 3 √ 2 − (α21 − α11 ) 3 (2c − 1) √ 1 1 1 1 √ √ − cψ × 3 1 − 2c + cψ 1+ 3+ 4 4 1 − 2c 1 − 2c p 4 + 2 (1 − 2c) 3βv∗2 − α + N134 + N135 − N144 + N145 3  q  4T0 −N ¯141 (t3 ) + N ¯141 (t4 ) for v∗ < 1−2c q 2 + 0 for v∗ ≥ 1−2c 2

(7.129)

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159

According to (7.44) the Melnikov function is N2 (t0 ) = N21 (t0 ) − N22 − N23 − N24 ,

(7.130)

where N21 (t0 ) = 2Aγ

Z∞

√

r˙

−∞

√

= −2Aγ 1 − 2c

1 − 2ct cos ω (t + t0 ) dt

Z∞

sech

√

−∞

√ 1 − 2ct tanh 1 − 2ct cos ω (t + t0 ) dt

and N22 = 2A

Z∞

(δ1 (x10 , x˙ 10 ) − δ2 (x10 , x˙ 10 )) r˙

√

−∞

N23 = 2A

Z∞

r˙

√

−∞

1 − 2ct T (x˙ 01 − v∗ ) dt

√ = −2A 1 − 2c

Z∞

sech

√

−∞

N24 = 2A

Z∞

−∞

r˙

√

1 − 2ct dt,

√ 1 − 2ct tanh 1 − 2ct T (x˙ 01 − v∗ ) dt,

1 − 2ct T (−x˙ 01 − v∗ ) dt

√ = −2A 1 − 2c

Z∞

−∞

sech

√

√ 1 − 2ct tanh 1 − 2ct T (−x˙ 01 − v∗ ) dt.

The formula for N21 (t0 ) is as follows N21 (t0 ) = N211 (t0 ) − N212 (t0 ) , where

√ N211 (t0 ) = −2Aγ 1 − 2c cos ωt0 Z∞ √ √ × sech 1 − 2ct tanh 1 − 2ct cos ωtdt −∞

and

√ N212 (t0 ) = −2Aγ 1 − 2c sin ωt0 Z∞ √ √ × sech 1 − 2ct tanh 1 − 2ct sin ωtdt. −∞

(7.131)

July 19, 2007

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Upon integration we have N211 (t0 ) = 0, 2πAγω N212 (t0 ) = − √ sech 1 − 2c

πω √ 2 1 − 2c

sin ωt0 .

Substitution of the above formulas to (7.131) gives 2πAγω πω √ N21 (t0 ) = √ sech sin ωt0 . 1 − 2c 2 1 − 2c

(7.132)

Formula N22 is defined by N22 = −N221 + N222 + N223 + N224 ,

(7.133)

where N221 = 2

p

2 (1 − 2c)A (α11 − α21 )

Z∞

sech2

√

−∞

N222

√ 1 − 2ct tanh 1 − 2ct dt,

Z∞ √ √ √ = 2 2 (1 − 2c) A (δ11 − δ21 ) sech2 1 − 2ct tanh2 1 − 2ct dt, −∞

2

N223 = 4 (1 − 2c) A (δ12 − δ22 ) Z∞ √ √ × 1 − 2ct tanh3 1 − 2ct sech3 −∞

√ 1 − 2ct tanh 1 − 2ct dt Z∞ √ √ 2 = 8 (1 − 2c) A (δ12 − δ22 ) sech3 1 − 2ct tanh3 1 − 2ct dt,

× sgn sech

√

0

√

3

N224 = 4 2 (1 − 2c) A (δ13 − δ23 )

Z∞

sech4

−∞

√

√ 1 − 2ct tanh4 1 − 2ct dt.

Carrying out the computations one gets N221 = 0, N222

p 4A 2 (1 − 2c) = (δ11 − δ21 ) , 3

July 19, 2007

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161

3

N223

16A (1 − 2c) 2 (δ12 − δ22 ) , = 15

5 √ 16 2A (1 − 2c) 2 N224 = (δ13 − δ23 ) . 15 Substitution of the above formulas to (7.133) gives 3 4 4A p 2 (1 − 2c) (δ11 − δ21 ) + (1 − 2c) 2 (δ12 − δ22 ) N22 = 3 5 √ 5 4 2 + (1 − 2c) 2 (δ13 − δ23 ) . 5

(7.134)

According to (7.3), the formula for N23 takes the form N23 = −N231 − N232 + N233 + N234 + N235 ,

(7.135)

where √

N231 = 2AT0 1 − 2c

Z∞

√

sech

−∞

N232 = 2A v∗ − βv∗3

√

N233 = 2 2A

3βv∗2

√

1 − 2c

Z∞

sech

− α (1 −

Z∞

3 2c) 2

sech2

−∞

Z∞

5 2c) 2

sech3

−∞

N235

√

−∞

N234 = 12Aβv∗ (1 −

√ 1 − 2ct tanh 1 − 2ct sgn (x˙ 01 − v∗ ) dt,

√

√ 1 − 2ct tanh 1 − 2ct dt, √

√ 1 − 2ct tanh2 1 − 2ct dt,

√ 1 − 2ct tanh3 1 − 2ct dt,

Z∞ 7 √ √ √ 2 = 4 2Aβ (1 − 2c) sech4 1 − 2ct tanh4 1 − 2ct dt. −∞

√ Integral N231 for |v∗ | > (1 − 2c) 2 is defined in the following way √ N231 = −2AT0 1 − 2c

Z∞

−∞

sech

√

√ 1 − 2ct tanh 1 − 2ct dt = 0.

July 19, 2007

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SmoothNonsmooth


√ For |v∗ | < (1 − 2c) 2 one has  Zt1 √ √ √  N231 = 2AT0 1 − 2c − sech 1 − 2ct tanh 1 − 2ct dt −∞

+

Zt2

−

t1 Z∞

sech

sech

√

√ 1 − 2ct tanh 1 − 2ct dt

√

! √ 1 − 2ct tanh 1 − 2ct dt ,

t2

where t1 , t2 are defined as earlier. Carrying out the integration one gets √ √ t1 t2 N231 = 2AT0 sech 1 − 2ct −∞ − sech 1 − 2ct t1 √ ∞ + sech 1 − 2ct t 2 √ √ = 4AT0 sech 1 − 2ct1 − sech 1 − 2ct2 . According to (7.1) one has v v  s s u u u1 u1 2 2 1 v 1 v ∗ ∗ . − −t − − N231 = 4AT0 t + 2 4 2 (1 − 2c)2 2 4 2 (1 − 2c)2 The following main result of this part of consideration is reported: N231 = r  q  4AT 1 1 + 0 2 4 −  0

v∗2 2(1−2c)2

−

r

1 2

−

q

1 4

−

v∗2 2(1−2c)2

for v∗
(1 − 2c) 2, one gets √ N241 = −2AT0 1 − 2c

Z∞

√

sech

−∞

√ 1 − 2ct tanh 1 − 2ct dt = 0.

(7.141) √ In the second case, i.e. for |v∗ | < (1 − 2c) 2 one obtains  Zt3 √ √ √  N241 = 2AT0 1 − 2c − sech 1 − 2ct tanh 1 − 2ct dt −∞

+

Zt4

−

t3 Z∞

sech

sech

√

√ 1 − 2ct tanh 1 − 2ct dt

√

! √ 1 − 2ct tanh 1 − 2ct dt ,

t4

where t3 , t4 are defined as earlier. Upon integration one has √ √ t3 t4 N241 = 2AT0 sech 1 − 2ct −∞ − sech 1 − 2ct t 3 √ ∞ + sech 1 − 2ct t4 √ √ = 4AT0 sech 1 − 2ct3 − sech 1 − 2ct4 . Taking into account (7.1), one obtains r r q q v∗2 1 1 1 1 N241 = 4AT0 − − − 2 4 2 + 4 − 2(1−2c)2

v∗2 2(1−2c)2

,

and finally N241 = r  q  4AT 1 1 − 0 2 4 −  0

v∗2 2(1−2c)2

−

r

1 2

+

q

1 4

−

v∗2 2(1−2c)2

for v∗
− 2 (1 − 2c) (δ11 − δ21 ) 3 1 − 2c 2 1 − 2c √ 5 3 4 4 2 + (1 − 2c) 2 (δ12 − δ22 ) + (1 − 2c) 2 (δ13 − δ23 ) 5 5 r r  q q v∗2 v∗2  8AT 1 1 1 1 for v∗ < 0 2 − 4 − 2(1−2c)2 − 2 + 4 − 2(1−2c)2 +  0 for v∗ ≥

1−2c √ 2 1−2c √ 2

1−2c √ 2 1−2c √ 2

(7.146)

According to (7.46), the Melnikov function is defined as follows N3 (t0 ) = −N31 (t0 ) + N32 + N33 + N34 ,

(7.147)

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SmoothNonsmooth


where N31 (t0 ) = 2γ

Z0

R

−∞

= 2γ

√

1 − 2ct et cos ω (t + t0 ) dt

! √ √ tanh 1 − 2ct 3 − 4c 2 √ − sech 1 − 2ct − 3 (1 − 2c) 1 − 2c

Z0

−∞ t

× e cos ω (t + t0 ) dt and N32 = 2

Z0

(δ1 (x10 , x˙ 10 ) − δ2 (x10 , x˙ 10 )) et R

−∞

N33 = 2

Z0

et R

√

−∞

=2

Z0

e

t

−∞

√

1 − 2ct dt,

1 − 2ct T (x˙ 01 − v∗ ) dt =


× T (x˙ 01 − v∗ ) dt,

N34 = 2

Z0

et R

√

−∞

=2

Z0

−∞

e

t

1 − 2ct T (−x˙ 01 − v∗ ) dt


× T (−x˙ 01 − v∗ ) dt. Formula for N31 (t0 ) is as follows N31 (t0 ) = N311 (t0 ) − N312 (t0 ) ,

(7.148)

where N311 (t0 ) = 2γ cos ωt0 ×

Z0

−∞

! √ √ tanh 1 − 2ct 3 − 4c 2 √ − sech 1 − 2ct − et cos ωtdt, 3 (1 − 2c) 1 − 2c

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167

and N312 (t0 ) = 2γ sin ωt0 ×

Z0

−∞

! √ √ tanh 1 − 2ct 3 − 4c 2 √ − sech 1 − 2ct − et sin ωtdt. 3 (1 − 2c) 1 − 2c

Comparing with (7.108) and taking into account (7.110) one gets √ 6 1 − 2cω 2 − 2 (4c − 3) N311 (t0 ) = N111 (t0 ) = γ cos ωt0 3 (2c − 1) (1 + ω 2 ) ω 1 + ωi 1 1 + ωi √ + =ψ −=ψ 2+ √ . 2c − 1 4 4 1 − 2c 1 − 2c Comparing with (7.109) and according to (7.111) one has √ 2ω (4c − 3) + 3 1 − 2c N312 (t0 ) = −N112 (t0 ) = −γ sin ωt0 − 3 (2c − 1) (1 + ω 2 ) ω 1 + ωi 1 1 − ωi √ + 2 (1 − 2c) (δ11 − δ21 ) − N323 − N324 + (α11 − α21 ) 3 3 (1 − 2c) √ 1 1 1 1 × 3 1 − 2c + cψ 1+ √ − cψ 3+ √ 4 4 1 − 2c 1 − 2c 4 (4c − 3) + v∗ − βv∗3 − T0 − N334 − N335 − N344 + N345 6c − 3 ( ¯331 (t1 ) − N ¯331 (t2 ) for v∗ < 1−2c √ 4T0 N 2 − √ 0 for v∗ ≥ 1−2c 2

(7.164)

According to (7.49), the Melnikov function is defined by N1− (t0 ) = N11− (t0 ) + N12− + N13− − N14− ,

(7.165)

where N11− (t0 ) = N11 (t0 ) = 2γ

Z∞

√ R − 1 − 2ct e−t cos ω (t + t0 ) dt,

0

N12− = 2

Z∞

√ (δ1 (x10 , x˙ 10 ) + δ2 (x10 , x˙ 10 )) e−t R − 1 − 2ct dt,

0

and N13− = N13 = 2

Z∞ 0

√ e−t R − 1 − 2ct T (x˙ 01 − v∗ ) dt,

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SmoothNonsmooth


N14− = N14 = 2

Z∞

√ e−t R − 1 − 2ct T (−x˙ 01 − v∗ ) dt.

0

The integral for N12− is defined as follows N12− = N121− − N122− − N123− − N124− , where: N121− = 2

p

2 (1 − 2c) (α11 + α21 )

Z∞

sech

√

√ 1 − 2ct e−t R − 1 − 2ct dt,

0

√

N122− = 2 2 (1 − 2c) (δ11 + δ21 ) √ × e R − 1 − 2ct dt,

Z∞

sech

√

√ 1 − 2ct tanh 1 − 2ct

0

−t

2

N123− = 4 (1 − 2c) (δ12 + δ22 ) Z∞ √ √ √ × sech2 1 − 2ct tanh2 1 − 2ct e−t R − 1 − 2ct 0

× sgn sech √

√

√ 1 − 2ct tanh 1 − 2ct dt,

3

N124− = 4 2 (1 − 2c) (δ13 + δ23 ) √ × e−t R − 1 − 2ct dt.

Z∞

sech3

√

√ 1 − 2ct tanh3 1 − 2ct

0

The computations give √ 2 N121− = (α21 + α11 ) 3 (2c − 1) √ 1 1 1 1 − cψ , × 3 1 − 2c + cψ 1+ √ 3+ √ 4 4 1 − 2c 1 − 2c

N122−

p 2 2 (1 − 2c) =− (δ11 + δ21 ) , 3

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177

3

128 (1 − 2c) 2 (δ22 + δ12 ) √ √ √ N123− = − 3 1 + 2 1 − 2c 1 + 4 1 − 2c 1 + 6 1 − 2c 1 √ √ × 1 + 8 1 − 2c 1 + 10 1 − 2c √ √ √ × −6a 191 + 152 1 − 2c + 297 1 + 1 − 2c + 16 69 + 40 1 − 2c c2 1 1 1 1 2+ √ 8+ √ × F 6, ; ; −1 2 2 1 − 2c 1 − 2c √ √ + ((3 + 8c) −6 4 + 7 1 − 2c + 16c 3 + 5 1 − 2c 1 1 1 1 × F 6, 4+ √ ; 10 + √ ; −1 2 2 1 − 2c 1 − 2c √ √ + −3 + 3 1 − 2c + 6c − 8c 1 − 2c 1 1 1 1 6+ √ 12 + √ × F 6, ; ; −1 , 2 2 1 − 2c 1 − 2c √ 3 1024 2 (1 − 2c) (δ23 + δ13 ) √ √ 1 + 3 1 − 2c 1 + 5 1 − 2c 1 + 7 1 − 2c 1 √ √ √ × 1 + 9 1 − 2c 1 + 11 1 − 2c 1 + 13 1 − 2c √ √ √ × 252 1 + 1 − 2c − 3c 327 + 263 1 − 2c + 954 + 572 1 − 2c c2 1 1 1 1 3+ √ 11 + √ × F 8, ; ; −1 2 2 1 − 2c 1 − 2c √ √ + 2 (3 + 8c) −8 − 20 1 − 2c + 16c + 39c 1 − 2c 1 1 1 1 ; ; −1 × F 8, 5+ √ 13 + √ 2 2 1 − 2c 1 − 2c √ √ √ + 12 −1 + 1 − 2c − 53c −1 + 1 − 2c + −58 + 60 1 − 2c c2 1 1 1 1 √ √ ; ; −1 , × F 8, 7+ 15 + 2 2 1 − 2c 1 − 2c √

N124− = −

and hence √

√ 2 1 1 N12− = (α21 + α11 ) 3 1 − 2c + cψ 1+ √ 3 (2c − 1) 4 1 − 2c 1 1 2p + − cψ 3+ √ 2 (1 − 2c) (δ11 + δ21 ) − N123 − N124 . 4 3 1 − 2c (7.166)

July 19, 2007

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SmoothNonsmooth


Substitution of (7.112), (7.120), (7.128) and (7.166) into (7.165) yields 2p N1− (t0 ) = A˜ sin (ωt0 − κ) + 2 (1 − 2c) (δ11 + δ21 ) − N123 − N124 3 √ 2 (α21 + α11 ) + 3 (2c − 1) √ 1 1 1 1 1+ √ × 3 1 − 2c + cψ − cψ 3+ √ 4 4 1 − 2c 1 − 2c 4p 2 (1 − 2c) 3βv∗2 − α − N134 − N135 + N144 − N145 − 3  q  4T0 −N ¯141 (t3 ) + N ¯141 (t4 ) for v∗ < 1−2c q 2 − 0 for v∗ ≥ 1−2c 2

The criterion of chaos is defined by the following inequality √ 2p 2 A˜ > 2 (1 − 2c) (δ11 + δ21 ) − N123 − N124 + (α21 + α11 ) 3 3 (2c − 1) √ 1 1 1 1 × 3 1 − 2c + cψ 1+ √ − cψ 3+ √ 4 4 1 − 2c 1 − 2c 4p − 2 (1 − 2c) 3βv∗2 − α − N134 − N135 + N144 − N145 3  q  4T0 −N ¯141 (t3 ) + N ¯141 (t4 ) for v∗ < 1−2c q 2 − 0 for v∗ ≥ 1−2c 2

(7.167)

According to (7.51), the Melnikov function is N2− (t0 ) = N21− (t0 ) + N22− − N23− − N24− ,

(7.168)

where N21− (t0 ) = N21 (t0 ) = 2Aγ

Z∞

r˙

−∞

N22− = N22 = 2A

Z∞

√

1 − 2ct cos ω (t + t0 ) dt,

(δ1 (x10 , x˙ 10 ) − δ2 (x10 , x˙ 10 )) r˙

−∞

√

1 − 2ct dt,

and N23− = N23 = 2A

Z∞

−∞

r˙

√

1 − 2ct T (x˙ 01 − v∗ ) dt,

July 19, 2007

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SmoothNonsmooth


N24− = N24 = 2A

Z∞

r˙

−∞

√

179

1 − 2ct T (−x˙ 01 − v∗ ) dt.

Substitution of (7.132), (7.134), (7.139) and (7.145) to (7.168) yields the next Melnikov function defined in the following way 2πAγω πω 4A p √ N2− (t0 ) = √ sec h sin ωt0 + 2 (1 − 2c) (δ11 − δ21 ) 3 1 − 2c 2 1 − 2c √ 3 5 4 4 2 + (1 − 2c) 2 (δ12 − δ22 ) + (1 − 2c) 2 (δ13 − δ23 ) 5 5 r r  q q v∗2 v∗2  8AT 1 1 1 1 √ for v∗ < 1−2c 0 2 − 4 − 2(1−2c)2 − 2 + 4 − 2(1−2c)2 2 . +  √ 0 for v∗ ≥ 1−2c 2 The condition of intersection of stable and unstable manifolds is as follows 4A p 2π |A| γω πω √ √ sec h > 2 (1 − 2c) (δ11 − δ21 ) 3 1 − 2c 2 1 − 2c √ 3 5 4 4 2 2 2 + (1 − 2c) (δ12 − δ22 ) + (1 − 2c) (δ13 − δ23 ) 5 5 r  q  8AT 1 1 − 0 2 4 − +  0

v∗2 2(1−2c)2

−

r

1 2

+

q

1 4

−

v∗2 2(1−2c)2

for v∗
− 3 (1 − 2c) 3 √ 1 1 1 1 × 3 1 − 2c + cψ 1+ √ − cψ 3+ √ 4 4 1 − 2c 1 − 2c 4 (4c − 3) + v∗ − βv∗3 − T0 − N334 − N335 − N344 + N345 6c − 3 ( ¯331 (t1 ) − N ¯331 (t2 ) + N ¯341 (t3 ) − N ¯341 (t4 ) for v∗ < 1−2c √ 4T0 N 2 − 1−2c 0 for v∗ ≥ √2

(7.171)

7.2

Numerical Simulations

In Figure 7.2 chaotic thresholds γ = γ(v∗ ) generated by chaos criteria governed by formulas (7.61), (7.74), (7.86), (7.88), (7.96), and (7.105) and associated with collapsing of the homoclinic orbits q01,2 (t) governed by (7.10) are reported. For the parameters δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1, critical values of the parameter γ are as follows: γcr1 ≈ 0, 071, γcr2 ≈ 0, 47, γcr ≈ 1, 59, γcr4 ≈ 3, 76. We assume that the coupling stiffness has the form either k0 (z) = z 5 or k0 (z) = z 3 . In Figure 7.3, bifurcation diagrams of the investigated system in the case k0 (z) = z 5

July 19, 2007

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SmoothNonsmooth

181

Fig. 7.2 Thresholds of chaos γ = γ(v∗ ) for δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1.

Fig. 7.3 Bifurcation diagrams for k0 (z) = z 5 and for δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1.

and for δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1 are shown. Amplitude of excitation γ serves as the bifurcation parameter.

July 19, 2007

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SmoothNonsmooth


One may observe bifurcation in vicinity of the critical value γcr2 , which takes place in the interval (0, 40, 0, 44). Figure 7.4 displays phase plots and Poincaré maps for k0 (z) = z 5 and for γ = 0, 40; δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1. Regular dynamics is observed. Increase of the excitation amplitude up to γ = 0, 44 remaining fixed other parameters results in occurrence of chaotic dynamics.

Fig. 7.4 Phase plots and Poincar´ e maps for k0 (z) = z 5 and for γ = 0, 40; δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1.

In Figure 7.5, phase plots and Poincaré maps for k0 (z) = z 5 and for γ = 0, 44; δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1 are shown. The system moves in a chaotic manner and chaotic attractors with complex and stick-slip structures are exhibited. Let us analyze now dynamics of the investigated system for the coupling term k0 (z) = z 3 . Note that owing to the Melnikov-Gruendler approach applied to the Duffing oscillator, the same transition into chaos is now also predicted as it was for the case of k0 (z) = z 5 . In Figure 7.6 bifurcational diagrams for the parameter γ for the stiffness k0 (z) = z 3 and the fixed parameters δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1 are shown. It is confirmed that the homoclinic bifurcation takes place in the vicinity

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183


Fig. 7.6 Bifurcation diagrams for k0 (z) = z 3 and for δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1.

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of γcr2 . In Figure 7.7, phase plots Poincaré maps for k0 (z) = z 3 and for γ = 0, 44; δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1 are reported.


For γ = 0, 44, the system moves on the periodic orbit with excitation frequency. In Figure 7.8, phase plots, Poincaré maps for k0 (z) = z 3 and for γ = 0, 48; δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1 are shown. Increase of γ up to γ = 0, 48 yields occurrence of a chaotic attractor. In Figure 7.9, chaotic orbits yielded by destruction of the homoclinic orbits q01,2 (t) and q03,4 (t) and obtained by (7.10) and (7.11), respectively, for the coupling k0 (z) = z are displayed. In this case, additional critical values of the γ parameter are found: γcr6 ≈ 0, 072, γcr7 ≈ 1, 19, γcr8 ≈ 1, 32, γcr9 ≈ 1, 48, γcr10 ≈ 1, 54. They correspond to collapse of the homoclinic orbits q03,4 (t). In Figure 7.10, the associated bifurcational diagrams for k0 (z) = z and δ1 = 0, 1; δ2 = 0, 2;

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Fig. 7.9 Chaotic thresholds for γ = γ(v∗ ); k0 (z) = z and for δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1.

Fig. 7.10 Bifurcational diagrams for k0 (z) = z and for δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1.

c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1 are shown. Since in the considered case we have ten different critical values of the parameter γ, one may expect much more complex chaotic dynamics. In Figure 7.11, phase projections and Poincaré maps of the studied for k0 (z) =

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187

z and for γ = 0, 60; δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1 are shown.

Fig. 7.11 Phase plots and Poincar´ e maps for k0 (z) = z and for γ = 0, 60; δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1.

In the beginning, the system moves in a periodic manner in the vicinity of one of its equilibrium position. In Figure 7.12, phase projections and Poincaré maps for k0 (z) = z and for γ = 0, 61; δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1 are reported. Increase

Fig. 7.12 Phase plots and Poincar´ e maps for k0 (z) = z and for γ = 0, 60; δ1 = 0, 1; δ2 = 0, 2; c = 0, 3; T0 = 0, 3; α = 0, 2; β = 0, 1; v∗ = 0, 2; ω = 1.

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of the control parameter γ up to γ = 0, 61 results in a qualitative change of the phase portraits. Amplitudes of oscillations increased, and the mass externally excited moves between previous three equilibrium positions in a complex chaotic manner. Let us finally conclude the so far carried out considerations. The applied Melnikov-Gruendler method of analytical and numerical analysis of the coupled self-excited oscillators included arbitrary type of coupling elements. In general two homoclinic orbits exist which yield six chaos criterions. Furthermore, depending on the chosen type of coupling elements one may also expect additional homoclinic orbits associated with additionally found two more homoclinic orbits which yielded six more chaotic thresholds. In result, a practically homogeneous wide chaotic zone occurred in the bifurcation diagram along with the narrow periodic windows.

7.3

Additional Numerical Example

In this section we are going to predict an intersection of stable and unstable manifolds associated with autonomous mechanical system governed by the following non-dimensional equations (see also [13], and in addition [15]) x˙ = u, u˙ = x − x3 + fξ (x, y) − ε1 T10 (u − w) ,

(7.172)

y˙ = v, v˙ = y − y 3 − fξ (x, y) − ε2 T20 (v − w) , where: 3

0 0 0 Ti0 (X − w) = Ti0 sgn (X − w0 ) − Bi1 (X − w0 ) + Bi2 (X − w0 ) , (7.173)

and 3

fξ (x, y) = ξ (x − y) − ξ (x − y) .

(7.174)

The following relations hold between dimensional and non-dimensional equations s s 2 ˜ ˜ k B km k B i2 i1 0 0 0 , Bi2 =p , w0 = w , i = 1, 2, Ti0 = Ti0 , Bi1 =√ 3 k2 k mk m3 k˜3 (7.175) where w is the tape velocity, and Bi1 , Bi2 , Ti0 are the friction coefficients (see Figure 7.1). For ε1 = ε2 = 0 we obtain the unperturbed system, which possesses a homoclinic orbit of a hyperbolic point a = 0. First,

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a linearization along a homoclinic orbit is applied and the fundamental solutions are formulated. Let us denote by γ (t) the homoclinic orbit of the point a. It has the following form   q (t) q˙ (t)  γ (t) =  q (t) , q˙ (t)

where

q (t) =

√

2 sech (t) .

(7.176)

The linearized system of the unperturbed equations (7.172) in vicinity of the homoclinic orbit γ (t) reads ψ˙ 1 = ψ2 , ψ˙ 2 = 1 + ξ − 3q 2 (t) ψ1 − ξψ3 , ψ˙ 3 = ψ4 , ψ˙ 4 = 1 + ξ − 3q 2 (t) ψ3 − ξψ1 .

(7.177)

We seek the fundamental solution of the above equations. It can be shown that ψ (4) = γ˙ (t) is a solution to Eqs. (7.177). Next, applying the following substitution: q˙ (t) → r (t) q˙ (t) and substituting into (7.177), one gets the second solution

ψ (2)



 r (t) q˙ (t) r˙ (t) q˙ (t) + r (t) q¨ (t) , =   r (t) q˙ (t)

(7.178)

r˙ (t) q˙ (t) + r (t) q¨ (t)

where r (t) =

1 1 3 C1 t − C1 coth (t) + C1 sinh (2t) + C2 . 4 2 8

(7.179)

Two remaining solutions are as follows

ψ (1)

 11  ψ ψ 12   = ψ 13  , ψ 14

ψ (3)

 31  ψ ψ 32   = ψ 33  ψ 34

(7.180)

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where eδt −3δ tanh (t) (1 + tanh (t)) + (δ + 2) (δ + 1) − 3 (δ + 1) sech2 (t) , 2 eδt 12 ψ = −3δ (1 + 2 tanh (t)) sech2 (t) + δ tanh (t) (1 + tanh (t)) 2 + δ (δ + 2) (δ + 1) − 3 (δ + 1) (δ − 2 tanh (t)) sech2 (t) , ψ 11 =

ψ 13 = −ψ 11 , ψ 14 = −ψ 12 , e−δt 3δ tanh (t) (1 + tanh (t)) + (δ − 2) (δ − 1) + 3 (δ − 1) sech2 (t) , 2 3δ e−δt ψ 32 = (1 + 2 tanh (t)) sech2 (t) − δ tanh (t) (1 + tanh (t)) 2 e−δt − δ (δ − 2) (δ − 1) + 3 (1 − δ) (δ − 2 tanh (t)) sech2 (t) , 2 ψ 33 = −ψ 31 , ψ 31 =

ψ 34 = −ψ 32 . The second step includes computation of the Melnikov-Gruendler function,

Fig. 7.13

Critical surface in three-dimensional parameter space.

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which is as follows δ 5δ 3 + 5δ 2 − 20δ + 52 δ (δ − 2) (δ + 1) (δ + 2) 0=− (V11 + V21 ) − 8 8 δ 5δ 3 + 5δ 2 − 20δ − 56 × (V12 + V22 ) − (V13 + V23 ) 4 + 12 1 + 2δ 2 (V14 + V24 ) , (7.181) √ where δ = 1 + 2ξ and √ √ 128 2 0 64 2 T0 1 2 0 Vi1 = Bi1 − 990w0 + 103 Bi2 + i0 θ √ − w0 21 3465 3 2 2 (7.182) × 79 − 3w0 (sech (t1 ) − sech (t2 )) p − 17 1 − 2w0 2 (sech (t1 ) + sech (t2 )) , √ √ 0 64 2 0 64 2 2Ti0 1 02 0 0 Vi2 = B − 99w + 13 Bi2 + θ √ −w 105 i1 3465 3 2 11 5 5 7 7 × sech (t1 ) − sech (t2 ) + 4 sech (t1 ) − sech (t2 ) , 5 √ √ 0 16 2 0 16 2 2Ti0 1 02 0 0 Vi3 = B − 33w + 4 Bi2 + θ √ −w 105 i1 1155 5 2 × sech5 (t1 ) − sech5 (t2 ) ,

(7.183)

(7.184)

π 0 πw0 0 πw0 0 2 0 Ti0 + Bi1 − 8w + 9 Bi2 16 16 128 0 t2 Ti0 1 t1 0 arctan tanh + θ √ −w − arctan tanh . 4 2 2 2 (7.185)

Vi4 = −

In the above θ (x) is the Heaviside function and !! r q p 1 1 1p 2 2 0 0 t1 = ln 1 + 1 − 2w 1− + 1 − 2w , w0 2 2 t2 = ln

1 w0

q

p 1 − 1 − 2w0 2

1−

r

1 1p − 1 − 2w0 2 2 2

!!

.

The third step consists of a graphical representation of the MelnikovGruendler function in three dimensional parameter space (see Figure 7.13),

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192 Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods 0 0 0 0 where we set B11 = T10 = 0.2, B21 = 0.3, T20 = 0.1. The surface consists of two subsurfaces which correspond to stick-slip chaos of type 1 and stick-slip chaos of type 2. Two sets of parameters (two points) correspond to type 1 (Fig.7.14) and type 2 (Fig.7.15) stick-slip chaos. Type 1 stick-slip motion corresponds to the case where a sign of w0 is not changed.

Fig. 7.14

0 = B 0 = 0.09 . Phase portraits of type 1 w0 = 0.87, B22 12

The reported numerical computations have been carried out using the Runge-Kutta method of order 8 with step 0.001.

Fig. 7.15

0 = B 0 = 1.7 . Phase portraits of type 2 w0 = 0.6, B22 12

In conclusion, we have applied the Melnikov-Gruendler technique to predict two types of non-smooth chaotic behaviour in R4 of two self-excited coupled oscillators with friction. The obtained analytically chaotic threshold has been verified numerically showing surprisingly good agreement.

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Chapter 8

A Triple Self-Excited Duffing-type Oscillator

In this chapter a triple self-excited Duffing-type oscillator with dry friction, viscous-type dampings and external excitation is studied. The MelnikovGruendler technique is applied to predict the system homoclinic bifurcation, and the associated MGM functions are derived for both smooth and nonsmooth events.

8.1

Physical and Mathematical Models

A self-excited Duffing-type oscillator with three-degrees-of-freedom shown in Figure 8.1 is studied. It consists of three masses m lying on a rigid bel moving at constant velocity vˆ. Dry friction (approximated by a modified function composed of a “sign” and a polynomial) occurs between masses and the belt. External masses are linked to a basis through elastic Duffingtype elements and viscous dampers with relatively small damping forces. Middle mass is coupled to external masses by non-linear springs with arbitrary characteristics satisfying the condition k0 (−z) = −k0 (z). One of the external masses is harmonically driven with frequency Ω and relatively small amplitude Γ. Since the studied system has three-degrees-of-freedom, the Melnikov-Gruendler technique is applied to predict a homoclinic bifurcation. Dynamics of the analyzed system is described by the following equations m · z¨1 − k · z1 + k1 · z13 + εc1 · z˙1 + k0 (z1 − z3 ) + εθ (z˙1 − vˆ) = εΓ cos Ωτ, m · z¨2 − k · z2 + k1 · z23 + εc2 · z˙2 + k0 (z2 − z3 ) + εθ (z˙2 − vˆ) = 0, m · z¨3 − k · z3 + k1 · z33 − k0 (z1 − z3 ) − k0 (z2 − z3 ) + εθ (z˙3 − vˆ) = 0, (8.1) where friction force is approximated in the following way 193

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Fig. 8.1

Physical model of the system.

3 θ (z˙ − vˆ) = θ0 sgn (z˙ − vˆ) − α ˆ (z˙ − vˆ) + βˆ (z˙ − vˆ) .

Applying the transformations p p z1 = x1 k/k1 , z2 = x2 k/k1 ,

z3 = x3

p k/k1 ,

τ =t

p

m/k1 ,

equations (8.1) are reduced to the following non-dimensional form x ¨1 − x1 + x31 + a · k0 (b · (x1 − x3 )) = ε (γ cos ωt − δ1 x˙ 1 − T (x˙ 1 − v∗ )) , x ¨2 − x2 + x32 + a · k0 (b · (x2 − x3 )) = −ε (T (x˙ 2 − v∗ ) + δ2 x˙ 2 ) , x ¨3 − x3 + x33 − a · k0 (b · (x1 − x3 )) − a · k0 (b · (x2 − x3 )) = −εT (x˙ 3 − v∗ ) , (8.2) where 3

T (x˙ − v∗ ) = T0 sgn (x˙ − v∗ ) − α (x˙ − v∗ ) + β (x˙ − v∗ ) , (8.3) q q √ p 1 2 and δ1 = √cmk , δ2 = √cmk , a = k1 kk1 , b = kk1 , γ = Γ kk1 , ω = Ω m k r √ 2 ˆ βk θ0 k1 α ˆ vˆ mk1 √ , β= . v∗ = , T0 = , α= √ k k k mk mk1 mk 8.2

Analytical Prediction of Homoclinic Intersections

For ε = 0 the following unperturbed system is obtained x ¨1 − x1 + x31 + a · k0 (b · (x1 − x3 )) = 0, x ¨2 − x2 + x32 + a · k0 (b · (x2 − x3 )) = 0, x ¨3 − x3 + x33 − a · k0 (b · (x1 − x3 )) − a · k0 (b · (x2 − x3 )) = 0.

(8.4)

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The system has the following equilibrium a1 = (0, 0, 0) . We assume that point a1 is associated with the following homoclinic solution q0 (x1 , x2 , x3 ) (t) = (Ar (Bt) , Ar (Bt) , Ar (Bt)) , where r (t) = sech (t). Constants A and B are denoted by substitution of the homoclinic orbit equation to equations (8.4) to yield A sech (Bt) B 2 − 1 + A2 − 2B 2 sech2 (Bt) = 0. (8.5) Taking B = ±1 one gets A2 − 2 = 0, and √ A = ± 2. According to the considerations carried out so far, one may conclude that there are two homoclinic orbits associated with the studied equilibrium and defined by the equations  √  ±√2r (t)  ± 2r˙ (t)   √     ±√2r (t)  q01,2 (x1 , x˙ 1 , x2 , x˙ 2 , x3 , x˙ 3 ) (t) =  (8.6) ,  ±√2r˙ (t)     ± 2r (t)  √ ± 2r˙ (t) where r(t) = sech t. Next, we are going to compute the quantities necessary for application of the Melnikov-Gruendler technique. Linearization of system (8.4) in vicinity of the saddle type equilibrium gives    2  dk0 (b · (x1 − x3 ))  x ¨1 +  x1 = x10 − 1 x1 3x10 + a dx1 x3 = x30 dk0 (b · (x1 − x3 )) +a x1 = x10 · x3 = 0, dx3 x3 = x30

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

  2  dk0 (b · (x2 − x3 )) x ¨2 +  − 1 3x20 + a  x2 x2 = x20 dx2 x3 = x30 dk0 (b · (x2 − x3 )) +a x2 = x20 · x3 = 0, dx3 x3 = x30 

 2 dk0 (b·(x1 −x3 )) x ¨3 +  3x − a dx3  30 ×x3 − a

dk0 (b·(x1 −x3 )) dx1 x1

x1 = x10 x3 = x30

= x10 x3 = x30

2 −x3 )) − a dk0 (b·(x dx3

· x1 − a

dk0 (b·(x2 −x3 )) dx2 x2



x2 = x20 x3 = x30

= x20 x3 = x30

 − 1 

· x2 = 0.

(8.7) The linearized equations of motion (variational equations) along two orbits (8.6) are defined by the equations x ¨1 + 6 sech2 t − 1 + c x1 − cx3 = 0, (8.8) x ¨2 + 6 sech2 t − 1 + c x2 − cx3 = 0, x ¨3 + 6 sech2 t − 1 + 2c x3 − c (x1 + x2 ) = 0, where c = abk00 (0) = k00 (0)/k. The analyzed system is given in the following form of the first order ODEs x˙ 1 = v1 , v˙ 1 = − 6 sech2 t − 1 + c x1 + cx3 , x˙ 2 = v2 , (8.9) x ¨2 = − 6 sech2 t − 1 + c x2 + cx3 , x˙ 3 = v3 , v˙ 3 = − 6 sech2 t − 1 + 2c x3 + cx2 + cx1 . Subtraction of the two first equations of system (8.8) yields y¨ + 6 sech2 t − c¯2 y = 0, (8.10)

where y = x1 − x2 , c¯2 = 1 − c. An analogous equation is defined by (7.14). According to (7.16), one gets y = C1 ec¯t 2 + c¯2 − 3 sech2 t − 3 coth t + (8.11) −C2 e−¯ct 2 + c¯2 − 3 sech2 t + 3 coth t .

Since c¯2 = 1−c, the following general solution of equation (8.10) is obtained y (t) = C1 ys1 + C2 ys2 = √ √ (8.12) = C1 e √1−ct − 13 c + 1 − c − tanh t tanh t + √ 1 − 1−ct −C2 e −3c + 1 − c + tanh t tanh t ,

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where ys1 and ys2 are the particular solutions. Summation of all equations (8.8) yields x ¨ + 6 sech2 t − 1 x = 0, (8.13) where x = x1 + x2 + x3 . Note that this equation is identical with that defined by (7.17) and having the following general solution = C3

3 2t

x (t) = C3 xs3 + C4 xs4 = − coth t + 14 sinh 2t sech t tanh t + C4 sech t tanh t,

(8.14)

where xs3 and xs4 are the particular solutions. Multiplying the third equation of (8.8) by −2, and summing the equations we get z¨ + 6 sech2 t − 1 + 3c z = 0, (8.15) where z = x1 + x2 − 2x3 . This is a particular case of equation (8.10) for c¯2 = 1 − 3c. Using (8.11), the following solution is applied z(t) = C1 ec¯t 2 + c¯2 − 3 sech2 t − 3 coth t + −C2 e−¯ct 2 + c¯2 − 3 sech2 t + 3 coth t and √

z (t) = C5 zs5 + C6 zs6 = √ −c + 1 − 3c − tanh t tanh t + √ −c + 1 − 3c + tanh t tanh t ,

= C5 e √1−3ct −C6 e− 1−3ct

(8.16)

where zs5 and zs6 are the particular solutions. Since y = x1 − x2 , x = x1 + x2 + x3 and z = x1 + x2 − 2x3 , a solution to equations (8.8) is a linear combination defined as follows x1 = 31 x + 12 y + 16 z, x2 = 13 x − 12 y + 16 z, x3 = 13 x − 13 z. The system of equations (8.9) has the following fundamental matrix of solutions   zs5 zs6 ys3 ys4 xs1 xs2   z˙  s5 z˙s6 y˙ s3 y˙ s4 x˙ s1 x˙ s2    zs6 −ys3 −ys4 xs1 xs2  z γ (t) =  s5 (8.17)   z˙s5 z˙s6 −y˙ s3 −y˙ s4 x˙ s1 x˙ s2     −zs5 −zs6 0 0 xs1 xs2  −z˙s5 −z˙s6 0 0 x˙ s1 x˙ s2

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= γ (1) , γ (2) , γ (3) , γ (4) , γ (5) , γ (6) , where

 √ R (t) exp 1 − 3ct  R (t) exp √1 − 3ct ˙    √    R (t) exp √ 1 − 3ct  = ,  R (t) exp 1 − 3ct ˙  √    −R (t) exp 1 − 3ct  √ − R (t) exp 1 − 3ct ˙ 

γ (1)

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙    √    R (−t) exp −√ 1 − 3ct  = ,  R (−t) exp − 1 − 3ct ˙  √    −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙ 

γ (2)

 √ P (t) exp 1 − ct  P (t) exp √1 − ct ˙    √    −P (t) exp √1 − ct  = ,  − P (t) exp 1 − ct ˙     0 0 

γ (3)

 √ P (−t) exp − 1 − ct √  P (−t) exp − 1 − ct ˙    √    −P (−t) exp −√1 − ct  = ,  − P (−t) exp − 1 − ct ˙     0 0 

γ (4)



 Q (t) r˙ (t)  (Q (t) r˙ (t))˙      Q (t) r˙ (t)  = ,  (Q (t) r˙ (t))˙    Q (t) r˙ (t)  (Q (t) r˙ (t))˙



 r˙ (t)  r¨ (t)       r˙ (t)  (5) (6) γ γ = ,  r¨ (t)     r˙ (t)  r¨ (t) √ 3 1 1 where Q (t) = 2 t − coth t + 4 sinh 2t, P (t) = − 3 c + 1 − c − tanh t tanh t √ and R (t) = −c + 1 − 3c − tanh t tanh t. Let us analyze now system

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199

(8.2) for ε > 0. A perturbation vector along the homoclinic trajectory (8.6) for sign “+” is given by the formula   0  γ cos ωt − δ x˙ − T (x˙ − v )  1 10 10 ∗     0   h (q01 , t) =  .   −T (x˙ 20 − v∗ ) − δ2 x˙ 20     0 −T (x˙ 30 − v∗ ) Substitution of the first column of the fundamental solutions matrix (8.17) by vector h (q01 , t) gives a matrix whose determinant is defined as follows i h K1 (t, t0 ) = det K1(1) , K1(2) , K1(3) , K1(4) , K1(5) , K1(6) , where 

0



 γ cos ωt − δ x˙ − T (x˙ − v )  1 10 10 ∗     0   (1) K1 =  ,   −T (x˙ 20 − v∗ ) − δ2 x˙ 20     0 −T (x˙ 30 − v∗ )   √ R (−t) exp − 1 − 3ct  R (−t) exp −√1 − 3ct ˙    √    R (−t) exp −√ 1 − 3ct  (2) K1 =  ,  R (−t) exp − 1 − 3ct ˙  √    −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙   √ P (t) exp 1 − ct √  P (t) exp 1 − ct ˙    √    −P (t) exp √1 − ct  (3) K1 =  ,  − P (t) exp 1 − ct ˙     0 0   √ P (−t) exp − 1 − ct  P (−t) exp −√1 − ct ˙    √    −P (−t) exp −√1 − ct  (4) K1 =  ,  − P (−t) exp − 1 − ct ˙     0 0

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

(5)

K1

 Q (t) r˙ (t)  (Q (t) r˙ (t))˙      Q (t) r˙ (t)  = ,  (Q (t) r˙ (t))˙    Q (t) r˙ (t)  (Q (t) r˙ (t))˙



(6)

K1

 r˙ (t)  r¨ (t)       r˙ (t)  = .  r¨ (t)     r˙ (t)  r¨ (t)

Expansion with respect to the first column gives K1 (t, t0 ) = − (γ cos ω (t + t0 ) − δ1 x˙ 10 − T (x˙ 10 − v∗ )) W11 (t, t0 ) + + (T (x˙ 20 − v∗ ) + δ2 x˙ 20 ) W12 (t, t0 ) + T (x˙ 30 − v∗ ) W13 (t, t0 ) , (8.18) where i h (4) (5) (3) (1) (2) W11 (t, t0 ) = det W11 , W11 , W11 , W11 , W11 , and

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct    √  =  R (−t) exp − 1 − 3ct ˙  ,  −R (−t) exp −√1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙ 

(1)

W11

 √ P (t) exp 1 − ct √  −P (t) exp 1 − ct    √  =  − P (t) exp 1 − ct ˙ ,   0 0 

(2)

W11

 √ P (−t) exp − 1 − ct √  −P (−t) exp − 1 − ct    √  =  − P (−t) exp − 1 − ct ˙ ,   0 0 

(3)

W11

 Q (t) r˙ (t)  Q (t) r˙ (t)     =  (Q (t) r˙ (t))˙ ,  Q (t) r˙ (t)  

(4)

W11

 r˙ (t)  r˙ (t)     =  r¨ (t)  .  r˙ (t)  

(5)

W11

(Q (t) r˙ (t))˙ r¨ (t) h i (1) (2) (3) (4) (5) W12 (t, t0 ) = det W12 , W12 , W12 , W12 , W12 ,

July 19, 2007

10:48


SmoothNonsmooth


where

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙    √  =  R (−t) exp − √1 − 3ct  ,  −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙ 

(1)

W12



(2) W12

  =   

  (3) W12 =   

 √ P (t) exp 1 − ct √ P (t) exp 1 − ct ˙  √ −P (t) exp 1 − ct  ,  0 0

 √ P (−t) exp − 1 − ct √ P (−t) exp − 1 − ct ˙  √ −P (−t) exp − 1 − ct  ,  0 0

 Q (t) r˙ (t)  (Q (t) r˙ (t))˙    =  Q (t) r˙ (t)  ,  Q (t) r˙ (t)  (Q (t) r˙ (t))˙ 

(4)

W12

 r˙ (t)  r¨ (t)     =  r˙ (t)  .  r˙ (t)  r¨ (t) 

(5)

W12

h i (1) (2) (3) (4) (5) W13 (t, t0 ) = det W13 , W13 , W13 , W13 , W13 , where 

(1) W13

  =  

 √ R (−t) exp − 1 − 3ct √ R (−t) exp − 1 − 3ct ˙  √ R (−t) exp − 1 − 3ct  , √ R (−t) exp − 1 − 3ct ˙ √ −R (−t) exp − 1 − 3ct

 √ P (t) exp 1 − ct √  P (t) exp 1 − ct ˙    √  =  −P (t) exp √1 − ct  ,  − P (t) exp 1 − ct ˙ 0 

(2)

W13

201

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SmoothNonsmooth


 √ P (−t) exp − 1 − ct √  P (−t) exp − 1 − ct ˙    √  =  −P (−t) exp −√1 − ct  ,  − P (−t) exp − 1 − ct ˙ 0 

(3)

W13

 Q (t) r˙ (t)  (Q (t) r˙ (t))˙    =  Q (t) r˙ (t)  ,  (Q (t) r˙ (t))˙ Q (t) r˙ (t) 

(4)

W13

 r˙ (t)  r¨ (t)     =  r˙ (t)  .  r¨ (t)  r˙ (t) 

(5)

W13

Expansion of W11 (t, t0 ) with respect to the fifth row gives √ W11 (t, t0 ) = − R (−t) exp − 1 + 3at ˙W111 (t, t0 )

− (Q (t) r˙ (t))˙W112 (t, t0 ) + r¨ (t) W113 (t, t0 ) ,

where h i (1) (2) (3) (4) W111 (t, t0 ) = det W111 , W111 , W111 , W111 , where

 √ P (t) exp 1 − ct √  −P (t) exp 1 − ct  (1)  √ W111 =   − P (t) exp 1 − ct ˙ , 0  √  P (−t) exp − 1 − ct √  −P (−t) exp − 1 − ct  (2)  √ W111 =   − P (−t) exp − 1 − ct ˙ , 0     Q (t) r˙ (t) r˙ (t)  Q (t) r˙ (t)   r˙ (t)  (3) (4)    W111 =   (Q (t) r˙ (t))˙ , W111 =  r¨ (t)  . Q (t) r˙ (t) r˙ (t) h i (1) (2) (3) (4) W112 (t, t0 ) = det W112 , W112 , W112 , W112 , 

where (1)

W112

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct   √ =  R (−t) exp − 1 − 3ct ˙ , √ −R (−t) exp − 1 − 3ct 

(8.19)

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SmoothNonsmooth


(2)

W112

(3)

W112

203

 √ P (t) exp 1 − ct √  −P (t) exp 1 − ct   √ =  − P (t) exp 1 − ct ˙ , 0 

 √ P (−t) exp − 1 − ct √  −P (−t) exp − 1 − ct   √ =  − P (−t) exp − 1 − ct ˙ , 0 

(4)

W112



 r˙ (t)  r˙ (t)   =  r¨ (t)  . r˙ (t)

h i (1) (2) (3) (4) W113 (t, t0 ) = det W113 , W113 , W113 , W113 , where (1)

W113

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct   √ =  R (−t) exp − 1 − 3ct ˙ , √ −R (−t) exp − 1 − 3ct

(2)

W113

(3)

W113



 √ P (t) exp 1 − ct √  −P (t) exp 1 − ct   √ =  − P (t) exp 1 − ct ˙ , 0 


(4)

W113



 Q (t) r˙ (t)  Q (t) r˙ (t)   =  (Q (t) r˙ (t))˙ . Q (t) r˙ (t)

Expansion of W111 (t, t0 ) with respect to the fourth row yields W111 (t, t0 ) = −Q (t) r˙ (t) W1111 (t, t0 ) + r˙ (t) W1112 (t, t0 ) , where h i (1) (2) (3) W1111 (t, t0 ) = det W1111 , W1111 , W1111 , where (1)

W1111

 √ P (t) exp 1 − ct √ =  −P (t) exp 1 − ct  , √ − P (t) exp 1 − ct ˙ 

(8.20)

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SmoothNonsmooth


(2)

W1111

 √ P (−t) exp − 1 − ct √ =  −P (−t) exp − 1 − ct  , √ − P (−t) exp − 1 − ct ˙ 

(3)

W1111



 r˙ (t) =  r˙ (t)  . r¨ (t)

h i (1) (2) (3) W1112 (t, t0 ) = det W1112 , W1112 , W1112 , where (1)

W1112

(2)

W1112



(3)

W1112



 Q (t) r˙ (t) =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙

Expansion of W1111 (t, t0 ) with respect to the third column gives W1111 (t, t0 ) = 2A1 r˙ (t) where

√ √ P (t) exp 1 − ct P (−t) exp − √1 − ct = const. √ A1 = P (t) exp 1 − ct ˙ P (−t) exp − 1 − ct ˙

Expansion of W1112 (t, t0 ) with respect to the third column gives W1112 (t, t0 ) = 2A1 Q (t) r˙ (t) Substitution of the above formulas to (8.20) yields W111 (t, t0 ) = 0.

Expansion of W112 (t, t0 ) with respect to the fourth row gives √ W112 (t, t0 ) = R (−t) exp − 1 − 3ct W1121 (t, t0 ) + r˙ (t) W1122 (t, t0 ) , (8.21) where W1121 (t, t0 ) = W1111 (t, t0 ) = 2A1 r˙ (t) , h i (1) (2) (3) W1122 (t, t0 ) = det W1122 , W1122 , W1122 , where (1)

W1122

 √ R (−t) exp − 1 − 3ct √ =  R (−t) exp − 1 − 3ct  , √ R (−t) exp − 1 − 3ct ˙ 

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(2)

W1122

(3) W1122

SmoothNonsmooth

205


 √ P (−t) exp − 1 − ct √ =  −P (−t) exp − 1 − ct  . √ − P (−t) exp − 1 − ct ˙ 

Developing of W1122 (t, t0 ) with respect to the first column we have √ W1122 (t, t0 ) = 2A1 R (−t) exp − 1 − 3ct . Substituting the above formulas to (8.21) we get √ W112 (t, t0 ) = 4A1 r˙ (t) R (−t) exp − 1 − 3ct . Developing of W113 (t, t0 ) with respect to the fourth row we have √ W113 (t, t0 ) = R (−t) exp − 1 − 3ct W1131 (t, t0 )+Q (t) r˙ (t) W1132 (t, t0 ) , (8.22) where W2131 (t, t0 ) = W1112 (t, t0 ) = 2A1 Q (t) r˙ (t) , √ W1132 (t, t0 ) = W1122 (t, t0 ) = 2A1 R (−t) exp − 1 − 3ct . Substitution of the above formulas to (8.22) yields √ W113 (t, t0 ) = 4A1 Q (t) r˙ (t) R (−t) exp − 1 − 3ct . Substitution of the above formulas to (8.19) gives √ W11 (t, t0 ) = 4A1 r˙ (t) R (−t) exp − 1 − 3ct (¨ r (t) Q (t) − (Q (t) r˙ (t))˙) . Expansion of W12 (t, t0 ) with respect to the fifth row yields √ W12 (t, t0 ) = − R (−t) exp − 1 − 3ct ˙W121 (t, t0 ) + − (Q (t) r˙ (t))˙W122 (t, t0 ) + r¨ (t) W123 (t, t0 ) , where h i (1) (2) (3) (4) W121 (t, t0 ) = det W121 , W121 , W121 , W121 , where (1)

W121

 √ P (t) exp 1 − ct √  P (t) exp 1 − ct ˙  √ =  −P (t) exp 1 − ct  , 0 

(8.23)

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SmoothNonsmooth


(2)

W121

(3)

 √ P (−t) exp − 1 − ct √  P (−t) exp − 1 − ct ˙  √ =  −P (−t) exp − 1 − ct  , 0 



W121

 Q (t) r˙ (t)  (Q (t) r˙ (t))˙  =  Q (t) r˙ (t)  , Q (t) r˙ (t)

(4)

W121



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r˙ (t)


W122

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙  √ =  R (−t) exp − 1 − 3ct  , √ −R (−t) exp − 1 − 3ct 

(2)

W122

(3)

W122



(4)

W122



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r˙ (t)


W123

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙  √ =  R (−t) exp − 1 − 3ct  , √ −R (−t) exp − 1 − 3ct

(2)

W123




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SmoothNonsmooth


(3)

W123


(4)

W123

207



 Q (t) r˙ (t)  (Q (t) r˙ (t))˙  =  Q (t) r˙ (t)  . Q (t) r˙ (t)

Expansion of W121 (t, t0 ) with respect to the fourth row gives W121 (t, t0 ) = −Q (t) r˙ (t) W1211 (t, t0 ) + r˙ (t) W1212 (t, t0 ) ,

(8.24)

where h i (1) (2) (3) W1211 (t, t0 ) = det W1211 , W1211 , W1211 , , and (1)

W1211

  √ √  P (t) exp 1 − ct P (−t) exp − 1 − ct √ √ (2) =  P (t) exp 1 − ct ˙ , W1211 =  P (−t) exp − 1 − ct ˙ , √ √ −P (t) exp 1 − ct −P (−t) exp − 1 − ct 

(3)

W1211



 r˙ (t) =  r¨ (t)  , r˙ (t)


W1212

(2)

W1212

 √ P (t) exp 1 − ct √ =  P (t) exp 1 − ct ˙ , √ −P (t) exp 1 − ct 

 √ P (−t) exp − 1 − ct √ =  P (−t) exp − 1 − ct ˙ , √ −P (−t) exp − 1 − ct 

(3)

W1212



 Q (t) r˙ (t) =  (Q (t) r˙ (t))˙ . Q (t) r˙ (t)

Expansion of W1211 (t, t0 ) with respect to the third column yields W1211 (t, t0 ) = 2A1 r˙ (t) , whereas the same procedure applied to W1212 (t, t0 ) with respect to the first row gives W1212 (t, t0 ) = 2A1 Q (t) r˙ (t) . Substituting the above formulas to (8.24) we obtain W121 (t, t0 ) = 0.

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10:48


SmoothNonsmooth


Expansion of W122 (t, t0 ) with respect to the fourth row gives √ W122 (t, t0 ) = R (−t) exp − 1 − 3ct W1221 (t, t0 ) + r˙ (t) W1222 (t, t0 ) , (8.25) where h i (1) (2) (3) W1221 (t, t0 ) = det W1221 , W1221 , W1221 , and (1)

W1221

(2)

W1221


 √ P (−t) exp − 1 − ct √ =  P (−t) exp − 1 − ct ˙ , √ −P (−t) exp − 1 − ct 

(3)

W1221



 r˙ (t) =  r¨ (t)  . r˙ (t)


W1222

 √ R (−t) exp − 1 − 3ct √ =  R (−t) exp − 1 − 3ct ˙ , √ R (−t) exp − 1 − 3ct 

(2)

W1222

(3)

W1222


 √ P (−t) exp − 1 − ct √ =  P (−t) exp − 1 − ct ˙ . √ −P (−t) exp − 1 − ct 

Expansion of W1221 (t, t0 ) with respect to the third column yields W1221 (t, t0 ) = 2A1 r˙ (t) , whereas development of W1222 (t, t0 ) with respect to the first column gives √ W1222 (t, t0 ) = 2A1 R (−t) exp − 1 − 3ct . Substituting the above formulas to (8.25) gives √ W122 (t, t0 ) = 4A1 r˙ (t) R (−t) exp − 1 − 3ct ,

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SmoothNonsmooth

209

development of W123 (t, t0 ) with respect to the fourth row yields √ W123 (t, t0 ) = R (−t) exp − 1 − 3ct W1231 (t, t0 )+Q (t) r˙ (t) W1232 (t, t0 ) , (8.26) where W1231 (t, t0 ) = W1212 (t, t0 ) , W1232 (t, t0 ) = W1222 (t, t0 ) . Substitution of the above formulas to (8.26) yields √ W123 (t, t0 ) = 4A1 Q (t) r˙ (t) R (−t) exp − 1 − 3ct , whereas their substitution to (8.23) gives √ W12 (t, t0 ) = 4A1 r˙ (t) (¨ r (t) Q (t) − (Q (t) r˙ (t))˙) R (−t) exp − 1 − 3ct . Expansion of W13 (t, t0 ) with respect to the fifth row gives √ W13 (t, t0 ) = −R (−t) exp − 1 − 3ct W131 (t, t0 ) −Q (t) r˙ (t) W132 (t, t0 ) + r˙ (t) W133 (t, t0 ) ,

(8.27)

where h i (1) (2) (3) (4) W131 (t, t0 ) = det W131 , W131 , W131 , W131 , and (1)

W131

  √ √  P (t) exp 1 − ct P (−t) exp − 1 − ct √ √  P (t) exp 1 − ct ˙   P (−t) exp − 1 − ct ˙  (2)    √ √ =  −P (t) exp 1 − ct  , W131 =  −P (−t) exp − 1 − ct  , √ √ − P (t) exp 1 − ct ˙ − P (−t) exp − 1 − ct ˙ 

(3)

W131



   Q (t) r˙ (t) r˙ (t)  (Q (t) r˙ (t))˙  r¨ (t)  (4)    =  Q (t) r˙ (t)  , W131 =  r˙ (t)  . (Q (t) r˙ (t))˙ r¨ (t)

h i (1) (2) (3) (4) W132 (t, t0 ) = det W132 , W132 , W132 , W132 , and (1)

W132



R (−t) exp  R (−t) exp =  R (−t) exp R (−t) exp

  √ √  − 1 − 3ct P (t) exp 1 − ct √ √   − 1 − 3ct ˙  , W (2) =  P (t) exp √1 − ct ˙  , √ 132  −P (t) exp 1 − ct  − 1 − 3ct  √ √ − 1 − 3ct ˙ − P (t) exp 1 − ct ˙

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SmoothNonsmooth


(3)

W132

 √   P (−t) exp − 1 − ct r˙ (t) √  r¨ (t)   P (−t) exp − 1 − ct ˙  (4)    , W132 √ = =  r˙ (t)  .  −P (−t) exp − 1 − ct  √ − P (−t) exp − 1 − ct ˙ r¨ (t) 

On the other hand h i (1) (2) (3) (4) W133 (t, t0 ) = det W133 , W133 , W133 , W133 , where (1)

W133



R (−t) exp  R (−t) exp =  R (−t) exp R (−t) exp

(3)

W133

  √ √  − 1 − 3ct P (t) exp 1 − ct √ √   − 1 − 3ct ˙  , W (2) =  P (t) exp √1 − ct ˙  , √ 133   − 1 − 3ct −P (t) exp 1 − ct  √ √ − 1 − 3ct ˙ − P (t) exp 1 − ct ˙

 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  P (−t) exp − 1 − ct ˙   (Q (t) r˙ (t))˙ (4)    √ =  −P (−t) exp − 1 − ct  , W133 =  Q (t) r˙ (t)  . √ − P (−t) exp − 1 − ct ˙ (Q (t) r˙ (t))˙ 

Expansion of W131 (t, t0 ) with respect to the fourth column gives W131 (t, t0 ) = −r˙ (t) W1311 (t, t0 ) + r¨ (t) W1312 (t, t0 ) − r˙ (t) W1313 (t, t0 ) + r¨ (t) W1314 (t, t0 ) , (8.28) where h i (1) (2) (3) W1311 (t, t0 ) = det W1311 , W1311 , W1311 , and (1)

W1311

(2)

W1311

 √ P (t) exp 1 − ct ˙ √ =  −P (t) exp 1 − ct  , √ − P (t) exp 1 − ct ˙ 

 √ P (−t) exp − 1 − ct ˙ √ =  −P (−t) exp − 1 − ct  , √ − P (−t) exp − 1 − ct ˙ 

(3)

W1311



 (Q (t) r˙ (t))˙ =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙

July 19, 2007

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SmoothNonsmooth

211

i h (1) (2) (3) W1312 (t, t0 ) = det W1312 , W1312 , W1312 , where (1)

W1312

(2)

W1312



(3)

W1312



 Q (t) r˙ (t) =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙

We have also h i (1) (2) (3) W1313 (t, t0 ) = det W1313 , W1313 , W1313 , where (1)

W1313

(2)

W1313

 √ P (t) exp 1 − ct √ =  P (t) exp 1 − ct ˙  , √ − P (t) exp 1 − ct ˙ 

 √ P (−t) exp − 1 − ct √ =  P (−t) exp − 1 − ct ˙  , √ − P (−t) exp − 1 − ct ˙ 

(3)

W1313



 Q (t) r˙ (t) =  (Q (t) r˙ (t))˙ . (Q (t) r˙ (t))˙

h i (1) (2) (3) W1314 (t, t0 ) = det W1314 , W1314 , W1314 , and (1)

W1314


(3)

W1314



 Q (t) r˙ (t) =  (Q (t) r˙ (t))˙ . Q (t) r˙ (t)

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SmoothNonsmooth


Expansion of W1311 (t, t0 ), W1312 (t, t0 ), W1313 (t, t0 ), W1314 (t, t0 ) yields, respectively W1311 (t, t0 ) = 2A1 (Q (t) r˙ (t))˙, W1312 (t, t0 ) = 2A1 Q (t) r˙ (t) , W1313 (t, t0 ) = 2A1 (Q (t) r˙ (t))˙, W1314 (t, t0 ) = 2A1 Q (t) r˙ (t) . Substitution of the above formulas to (8.28) yields W131 (t, t0 ) = 4A1 r˙ (t) (¨ r (t) Q (t) − (Q (t) r˙ (t))˙) . Expansion of W132 (t, t0 ) with respect to the first column gives √ − 3ct (W1321 (t, t0 ) + W1323 (t, t0 )) + W132 (t, t0 ) = R (−t) exp − 1 √ − R (−t) exp − 1 − 3ct ˙(W1322 (t, t0 ) + W1324 (t, t0 )) , (8.29) where h i (1) (2) (3) W1321 (t, t0 ) = det W1321 , W1321 , W1321 , and (1)

W1321

(2)

W1321

 √ P (t) exp 1 − ct ˙ √ =  −P (t) exp 1 − ct  , √ − P (t) exp 1 − ct ˙ 


(3)

W1321



 r¨ (t) =  r˙ (t)  . r¨ (t)

On the other hand h i (1) (2) (3) W1322 (t, t0 ) = det W1322 , W1322 , W1322 , where (1)

W1322


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(2)

W1322

SmoothNonsmooth

213


(3)

W1322



 r˙ (t) =  r˙ (t)  . r¨ (t)


W1323

(2)

W1323

 √ P (t) exp 1 − ct √ =  P (t) exp 1 + at ˙  , √ − P (t) exp 1 − ct ˙ 


(3) W1323



 r˙ (t) =  r¨ (t)  , r¨ (t)


W1324


(3)

W1324



 r˙ (t) =  r¨ (t)  . r˙ (t)

Expansion of W1321 (t, t0 ),W1322 (t, t0 ), W1323 (t, t0 ), W1324 (t, t0 ) with respect to the third column gives, respectively W1321 (t, t0 ) = 2A1 r¨ (t) , W1322 (t, t0 ) = 2A1 r˙ (t) ,

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SmoothNonsmooth


W1323 (t, t0 ) = 2A1 r¨ (t) , W1324 (t, t0 ) = 2A1 r˙ (t) . Substitution of the above formulas to (8.29) yields W132 (t, t0 ) = 4A1 A2 , where

√ R (−t) exp − 1 − 3ct r ˙ (t) = const. √ A2 = R (−t) exp − 1 − 3ct ˙ r¨ (t)

Expansion of W133 (t, t0 ) with respect to the first column yields √ W133 (t, t0 ) = R (−t) exp − 1 − 3ct (W1331 (t, t0 ) + W1333 (t, t0 )) + √ − R (−t) exp − 1 − 3ct ˙(W1332 (t, t0 ) + W1334 (t, t0 )) , (8.30) where i h (1) (2) (3) W1331 (t, t0 ) = det W1331 , W1331 , W1331 , and (1)

W1331

(2)

, W1331

 √ P (t) exp 1 − ct ˙ √ =  −P (t) exp 1 − ct  √ − P (t) exp 1 − ct ˙ 


(3)

W1331



 (Q (t) r˙ (t))˙ =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙


W1332

(2)

, W1332

 √ P (t) exp 1 − ct √ =  −P (t) exp 1 − ct  √ − P (t) exp 1 − ct ˙ 


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(3)

W1332

SmoothNonsmooth

215



 Q (t) r˙ (t) =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙

We have h i (1) (2) (3) W1333 (t, t0 ) = det W1333 , W1333 , W1333 , where (1)

W1333

(2)

W1333

 √ P (t) exp 1 − ct √ =  P (t) exp 1 − ct ˙  , √ − P (t) exp 1 − ct ˙ 


(3)

W1333



 Q (t) r˙ (t) =  (Q (t) r˙ (t))˙ . (Q (t) r˙ (t))˙

h i (1) (2) (3) W1334 (t, t0 ) = det W1334 , W1334 , W1334 , and (1)

W1334


(3) W1334



 Q (t) r˙ (t) =  (Q (t) r˙ (t))˙ . Q (t) r˙ (t)

Expansion of W1331 (t, t0 ), W1332 (t, t0 ), W1333 (t, t0 ), W1334 (t, t0 ) with respect to the third column yields W1331 (t, t0 ) = 2A1 (Q (t) r˙ (t))˙, W1332 (t, t0 ) = 2A1 Q (t) r˙ (t) , W1333 (t, t0 ) = 2A1 (Q (t) r˙ (t))˙, W1334 (t, t0 ) = 2A1 Q (t) r˙ (t) .

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SmoothNonsmooth


Substitution of the above formulas to (8.30) gives W133 (t, t0 ) = 4A1 A3 , where

√ R (−t) exp − 1 − 3ct Q (t) r˙ (t) √ A3 = = const. R (−t) exp − 1 − 3ct ˙ (Q (t) r˙ (t))˙

Substitution of the above formulas to (8.27) yields

√ W13 (t, t0 ) = 8A1 r˙ (t) ((Q (t) r˙ (t))˙− r¨ (t) Q (t)) R (−t) exp − 1 − 3ct , whereas substitution of the above formulas to (8.18) gives √ K1 (t, t0 ) = −4A1 A4 R (−t) exp − 1 − 3ct × γ cos ω (t + t0 ) − δ1 x˙ 10 − T (x˙ 10 − v∗) − T (x˙ 20 − v∗ ) −δ2 x˙ 20 + 2T (x˙ 30 − v∗ ) , and Q (t) r˙ (t) r˙ (t) = const. A4 = (Q (t) r˙ (t))˙ r¨ (t)

Since in the considered case x10 = x20 = x30 , hence √ K1 (t, t0 ) = −4A1 A4 R (−t) exp − 1 − 3ct

× (γ cos ω (t + t0 ) − (δ1 + δ2 ) x˙ 10 ) .

(8.31)

Substitution of the second column of the fundamental solutions matrix (8.17) by vector h (q01 , t) yields the following determinant i h K2 (t, t0 ) = det K2(1) , K2(2) , K2(3) , K2(4) , K2(5) , K2(6) , where

 √ R (t) exp 1 − 3ct  R (t) exp √1 − 3ct ˙    √    R (t) exp √ 1 − 3ct  = ,  R (t) exp 1 − 3ct ˙   √   −R (t) exp 1 − 3ct  √ − R (t) exp 1 − 3ct ˙ 

(1)

K2



(2)

K2

0



 γ cos ωt − δ x˙ − T (x˙ − v )  1 10 10 ∗     0   = ,   −T (x˙ 20 − v∗ ) − δ2 x˙ 20     0 −T (x˙ 30 − v∗ )

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SmoothNonsmooth

217

 √ P (t) exp 1 − ct √  P (t) exp 1 − ct ˙    √    −P (t) exp √1 − ct  = ,  − P (t) exp 1 − ct ˙     0 0 

(3)

K2

 √ P (−t) exp − 1 − ct √  P (−t) exp − 1 − ct ˙    √    −P (−t) exp −√1 − ct  = ,  − P (−t) exp − 1 − ct ˙     0 0 

(4)

K2

   Q (t) r˙ (t) r˙ (t)  (Q (t) r˙ (t))˙  r¨ (t)           Q (t) r˙ (t)   r˙ (t)  (6) =  , K2 =  .  r¨ (t)   (Q (t) r˙ (t))˙      r˙ (t)   Q (t) r˙ (t)  (Q (t) r˙ (t))˙ r¨ (t) 

(5)

K2

Expansion with respect to the second column gives K2 (t, t0 ) = (γ cos ω (t + t0 ) − δ1 x˙ 10 − T (x˙ 10 − v∗ )) W21 (t, t0 ) + (8.32) − (T (x˙ 20 − v∗ ) + δ2 x˙ 20 ) W22 (t, t0 ) − T (x˙ 30 − v∗ ) W23 (t, t0 ) , where i h (1) (2) (3) (4) (5) W21 (t, t0 ) = det W21 , W21 , W21 , W21 , W21 , and

  √ √  R (t) exp 1 − 3ct P (t) exp 1 − ct √ √  R (t) exp 1 − 3ct   −P (t) exp 1 − ct      √ √ (2)    =  R (t) exp √1 − 3ct ˙  , W21 =  − P (t) exp 1 − ct ˙ ,  −R (t) exp 1 − 3ct    0 √ − R (t) exp 1 − 3ct ˙ 0 

(1)

W21

 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  −P (−t) exp − 1 − ct   Q (t) r˙ (t)      √  , W (4) =  (Q (t) r˙ (t))˙ , = − P (−t) exp − 1 − ct ˙ 21      Q (t) r˙ (t)    0 0 (Q (t) r˙ (t))˙ 

(3)

W21

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SmoothNonsmooth


 r˙ (t)  r˙ (t)     =  r¨ (t)  .  r˙ (t)  

(5)

W21

r¨ (t)

We have h i (1) (2) (3) (4) (5) W22 (t, t0 ) = det W22 , W22 , W22 , W22 , W22 , where

  √ √  R (t) exp 1 − 3ct P (t) exp 1 − ct √ √  R (t) exp 1 − 3ct ˙   P (t) exp 1 − ct ˙     √ √ (1) (2)    W22 =   R (t) exp √1 − 3ct  , W22 =  −P (t) exp 1 − ct  ,  −R (t) exp 1 − 3ct    0 √ − R (t) exp 1 − 3ct ˙ 0  √    P (−t) exp − 1 − ct Q (t) r˙ (t) √  (Q (t) r˙ (t))˙  P (−t) exp − 1 − ct ˙     √ (3)  , W (4) =  Q (t) r˙ (t)  , W22 =  −P (−t) exp − 1 − ct 22        Q (t) r˙ (t)  0 

0

(Q (t) r˙ (t))˙

r˙ (t)  r¨ (t)     =  r˙ (t)  .  r˙ (t)  

(5)

W22



r¨ (t)

h

i (1) (2) (3) (4) (5) W23 (t, t0 ) = det W23 , W23 , W23 , W23 , W23 , where

  √ √  R (t) exp 1 − 3ct P (t) exp 1 − ct √ √  R (t) exp 1 − 3ct ˙  P (t) exp 1 − ct ˙      √ √ (1) (2)    W23 =   R (t) exp √ 1 − 3ct  , W23 =  −P (t) exp √1 − ct  ,  R (t) exp 1 − 3ct ˙  − P (t) exp 1 − ct ˙ √ −R (t) exp 1 − 3ct 0  √    P (−t) exp − 1 − ct Q (t) r˙ (t) √  P (−t) exp − 1 − ct ˙   (Q (t) r˙ (t))˙     √ (3)  , W (4) =  Q (t) r˙ (t)  , W23 =  −P (−t) exp − 1 − ct 23     √  − P (−t) exp − 1 − ct ˙  (Q (t) r˙ (t))˙ 

0

Q (t) r˙ (t)

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SmoothNonsmooth


219

 r˙ (t)  r¨ (t)     =  r˙ (t)  .  r¨ (t)  

(5)

W23

r˙ (t)

Expansion of W21 (t, t0 ) with respect to the fifth row gives √ W21 (t, t0 ) = − R (t) exp 1 − 3ct ˙W211 (t, t0 ) − (Q (t) r˙ (t))˙W212 (t, t0 ) + r¨ (t) W213 (t, t0 ) ,

(8.33)

where W211 (t, t0 ) = W111 (t, t0 ) = 0, h i (1) (2) (3) (4) W212 (t, t0 ) = det W212 , W212 , W212 , W212 , and (1)

W212

  √ √  R (t) exp 1 − 3ct P (t) exp 1 − ct √ √  R (t) exp 1 − 3ct   −P (t) exp 1 − ct  (2)   , W212  √ √ = =  R (t) exp 1 − 3ct ˙  − P (t) exp 1 − ct ˙ , √ −R (t) exp 1 − 3ct 0 

(3)

W212

where (1)

W213

 √   P (−t) exp − 1 − ct r˙ (t) √  −P (−t) exp − 1 − ct   r˙ (t)  (4)    √ =  − P (−t) exp − 1 − ct ˙ , W212 =  r¨ (t)  . 0 r˙ (t) h i (1) (2) (3) (4) W213 (t, t0 ) = det W213 , W213 , W213 , W213 , 

  √ √  R (t) exp 1 − 3ct P (t) exp 1 − ct √ √  R (t) exp 1 − 3ct   −P (t) exp 1 − ct  (2)    √ √ =  R (t) exp 1 − 3ct ˙ , W213 =  − P (t) exp 1 − ct ˙ , √ −R (t) exp 1 − 3ct 0 

(3)

W213

 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  −P (−t) exp − 1 − ct   Q (t) r˙ (t)  (4)    √ =  − P (−t) exp − 1 − ct ˙ , W213 =  (Q (t) r˙ (t))˙ . 0 Q (t) r˙ (t) 

Expansion of W212 (t, t0 ) with respect to the fourth row gives √ W212 (t, t0 ) = R (t) exp 1 − 3ct W2121 (t, t0 )+r˙ (t) W2122 (t, t0 ) , (8.34)

July 19, 2007

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SmoothNonsmooth


where W2121 (t, t0 ) = W1111 (t, t0 ) = 2A1 r˙ (t) , h i (1) (2) (3) W2122 (t, t0 ) = det W2122 , W2122 , W2122 , and (1)

W2122

  √ √  R (t) exp 1 − 3ct P (t) exp 1 − ct √ √ (2) =  R (t) exp 1 − 3ct  , W2122 =  −P (t) exp 1 − ct  , √ √ R (t) exp 1 − 3ct ˙ − P (t) exp 1 − ct ˙ 

(3)

W2122

 √ P (−t) exp − 1 − ct √ =  −P (−t) exp − 1 − ct  . √ − P (−t) exp − 1 − ct ˙ 

Substitution of W212 (t, t0 ) with respect to the first column gives √ W2122 (t, t0 ) = 2A1 R (t) exp 1 − 3ct . Substitution of the above formulas to (8.34) gives √ W212 (t, t0 ) = 4A1 r˙ (t) R (t) exp 1 − 3ct . Expansion of W213 (t, t0 ) with respect to the fourth row gives √ W213 (t, t0 ) = R (t) exp 1 − 3ct W2131 (t, t0 ) + Q (t) r˙ (t) W2132 (t, t0 ) , (8.35) where W2131 (t, t0 ) = W1112 (t, t0 ) = 2A1 Q (t) r˙ (t) , W2132 (t, t0 ) = W2122 (t, t0 ) = 2A1 R (t) exp

√

1 − 3ct .

Substitution of the above formulas to (8.35) yields W213 (t, t0 ) = 4A1 Q (t) r˙ (t) R (t) exp

√

1 − 3ct ,

whereas substitution of the above formulas to (8.33) allows us to obtain √ W21 (t, t0 ) = 4A1 A4 R (t) exp 1 − 3ct . Developing of W22 (t, t0 ) with respect to the fifth row gives √ W22 (t, t0 ) = − R (t) exp 1 − 3ct ˙W221 (t, t0 ) − (Q (t) r˙ (t))˙W222 (t, t0 ) + r¨ (t) W223 (t, t0 ) , where W221 (t, t0 ) = W121 (t, t0 ) = 0,

(8.36)

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SmoothNonsmooth


221

i h (1) (2) (3) (4) W222 (t, t0 ) = det W222 , W222 , W222 , W222 , where (1)

W222

  √ √  R (t) exp 1 − 3ct P (t) exp 1 − ct √ √  R (t) exp 1 − 3ct ˙  P (t) exp 1 − ct ˙ (2)    √ √ =  R (t) exp 1 − 3ct  , W222 =  −P (t) exp 1 − ct  , √ −R (t) exp 1 − 3ct 0 

(3)

W222


(4)

W222



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r˙ (t)


W223

  √ √  P (t) exp 1 − ct R (t) exp 1 − 3ct √ √  P (t) exp 1 − ct ˙  R (t) exp 1 − 3ct ˙ (2)   , W223  √ √ = =  R (t) exp 1 − 3ct   −P (t) exp 1 − ct  , √ −R (t) exp 1 − 3ct 0 

(3)

W223


(4)

W223



 Q (t) r˙ (t)  (Q (t) r˙ (t))˙  =  Q (t) r˙ (t)  . Q (t) r˙ (t)

Substitution of W222 (t, t0 ) with respect to the fourth row gives √ W222 (t, t0 ) = R (t) exp 1 − 3ct W2221 (t, t0 )+r˙ (t) W2222 (t, t0 ) , (8.37) where W2221 (t, t0 ) = W1211 (t, t0 ) = 2A1 r˙ (t) , h i (1) (2) (3) W2222 (t, t0 ) = det W2222 , W2222 , W2222 , and (1)

W2222

  √ √  R (t) exp 1 − 3ct P (t) exp 1 − ct √ √ (2) =  R (t) exp 1 − 3ct ˙ , W2222 =  P (t) exp 1 − ct ˙ , √ √ R (t) exp 1 − 3ct −P (t) exp 1 − ct 

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SmoothNonsmooth


(3)

W2222


Expansion of W2222 (t, t0 ) with respect to the first row gives √ W2222 (t, t0 ) = 2A1 R (t) exp 1 − 3ct . Substitution of the above formulas to (8.37) yields √ W222 (t, t0 ) = 4A1 r˙ (t) R (t) exp 1 − 3ct . Expansion of W223 (t, t0 ) with respect to the fourth row gives √ W223 (t, t0 ) = R (t) exp 1 − 3ct W2231 (t, t0 ) + Q (t) r˙ (t) W2232 (t, t0 ) , (8.38) where W2231 (t, t0 ) = W1212 (t, t0 ) = 2A1 Q (t) r˙ (t) , h i (1) (2) (3) W2232 (t, t0 ) = det W2232 , W2232 , W2232 , and (1)

W2232

  √ √  R (t) exp 1 − 3ct P (t) exp 1 − ct √ √ (2) =  R (t) exp 1 − 3ct ˙ , W2232 =  P (t) exp 1 − ct ˙ , √ √ R (t) exp 1 − 3ct −P (t) exp 1 − ct 

(3)

W2232


Expansion of W2232 (t, t0 ) with respect to the first column gives √ W2232 (t, t0 ) = 2A1 R (t) exp 1 − 3ct , whereas substitution of the above formulas to (8.38) gives √ W223 (t, t0 ) = 4A1 Q (t) r˙ (t) R (t) exp 1 − 3ct . Substitution of the above expressions to (8.36) yields √ W22 (t, t0 ) = 4A1 A4 R (t) exp 1 − 3ct . Expansion of W23 (t, t0 ) with respect to the fifth row allows us to get √ W23 (t, t0 ) = −R (t) exp 1 − 3ct W231 (t, t0 ) − Q (t) r˙ (t) W232 (t, t0 ) + r˙ (t) W233 (t, t0 ) ,

(8.39)

July 19, 2007

10:48


SmoothNonsmooth


223

where W231 (t, t0 ) = W131 (t, t0 ) = 4A1 r˙ (t) (¨ r (t) Q (t) − (Q (t) r˙ (t))˙) , i h (1) (2) (3) (4) W232 (t, t0 ) = det W232 , W232 , W232 , W232 , and (1)

W232

  √ √  R (t) exp 1 − 3ct P (t) exp 1 − ct √ √  R (t) exp 1 − 3ct ˙  P (t) exp 1 − ct ˙  (2)    √ √ =  R (t) exp 1 − 3ct  , W232 =  −P (t) exp 1 − ct  , √ √ − P (t) exp 1 − ct ˙ R (t) exp 1 − 3ct ˙ 

(3)

W232

 √ P (−t) exp − 1 − ct √  P (−t) exp − 1 − ct ˙   √ =  −P (−t) exp − 1 − ct  , √ − P (−t) exp − 1 − ct ˙ 

(4)

W232



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r¨ (t)

i h (1) (2) (3) (4) W233 (t, t0 ) = det W233 , W233 , W233 , W233 , where (1)

W233

  √ √  R (t) exp 1 − 3ct P (t) exp 1 − ct √ √  R (t) exp 1 − 3ct ˙  P (t) exp 1 − ct ˙  (2)   , W233  √ √ = =  R (t) exp 1 − 3ct   −P (t) exp 1 − ct  , √ √ R (t) exp 1 − 3ct ˙ − P (t) exp 1 − ct ˙ 

(3)

W233

 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  P (−t) exp − 1 − ct ˙   (Q (t) r˙ (t))˙ (4)    √ =  −P (−t) exp − 1 − ct  , W233 =  Q (t) r˙ (t)  . √ − P (−t) exp − 1 − ct ˙ (Q (t) r˙ (t))˙ 

Expansion of W232 (t, t0 ) with respect to the first column yields √ W232 (t, t0 ) = R (t) exp 1 − 3ct (W2321 (t, t0 ) + W2323 (t, t0 )) + √ − R (t) exp 1 − 3ct ˙(W2322 (t, t0 ) + W2324 (t, t0 )) , (8.40) where W2321 (t, t0 ) = W1321 (t, t0 ) = 2A1 r¨ (t) , W2322 (t, t0 ) = W1322 (t, t0 ) = 2A1 r˙ (t) , W2323 (t, t0 ) = W1323 (t, t0 ) = 2A1 r¨ (t) ,

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W2324 (t, t0 ) = W1324 (t, t0 ) = 2A1 r˙ (t) . Substitution of the above formulas to (8.40) gives √ √ W232 (t, t0 ) = 4A1 r¨ (t) R (t) exp 1 − 3ct − r˙ (t) R (t) exp 1 − 3ct ˙ . Substitution of W233 (t, t0 ) with respect to the first column gives √ W233 (t, t0 ) = R (t) exp 1 − 3ct (W2331 (t, t0 ) + W2333 (t, t0 )) + √ − R (t) exp 1 − 3ct ˙(W2332 (t, t0 ) + W2334 (t, t0 )) , (8.41) where W2331 (t, t0 ) = W1331 (t, t0 ) = 2A1 (Q (t) r˙ (t))˙, W2332 (t, t0 ) = W1332 (t, t0 ) = 2A1 Q (t) r˙ (t) , W2333 (t, t0 ) = W1333 (t, t0 ) = 2A1 (Q (t) r˙ (t))˙, W2334 (t, t0 ) = W1334 (t, t0 ) = 2A1 Q (t) r˙ (t) . Substitution of the above formulas to (8.41) allows us to get √ W233 (t, t0 ) = 4A1 (Q (t) r˙ (t))˙R (t) exp 1 − 3ct −Q (t) r˙ (t) R (t) exp

√

1 − 3ct ˙.

Substitution of the above formulas to (8.39) gives √ W23 (t, t0 ) = −8A1 A4 R (t) exp 1 − 3ct , and similarly, substitution of the above formulas to (8.32) allows us to obtain √ K2 (t, t0 ) = 4A1 A4 R (t) exp 1 − 3ct × × γ cos ω (t + t0 ) − δ1 x˙ 10 − T (x˙ 10 − v∗) − T (x˙ 20 − v∗ ) −δ2 x˙ 20 + 2T (x˙ 30 − v∗ ) , where Q (t) r˙ (t) r˙ (t) = const. A4 = (Q (t) r˙ (t))˙ r¨ (t)

Since in the considered case x10 = x20 = x30 , one gets √ K2 (t, t0 ) = 4A1 A4 R (t) exp 1 − 3ct (γ cos ω (t + t0 ) − (δ1 + δ2 ) x˙ 10 ) . (8.42)

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225

Substitution of the third column of the fundamental solutions matrix (8.17) by vector h (q01 , t) yields a matrix and the following determinant h i K3 (t, t0 ) = det K3(1) , K3(2) , K3(3) , K3(4) , K3(5) , K3(6) , where


(1)

K3

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙    √    R (−t) exp −√ 1 − 3ct  = ,  R (−t) exp − 1 − 3ct ˙   √   −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙ 

(2)

K3



(3)

K3



0

 γ cos ωt − δ x˙ − T (x˙ − v )  1 10 10 ∗     0   = ,   −T (x˙ 20 − v∗ ) − δ2 x˙ 20     0 −T (x˙ 30 − v∗ )  √ P (−t) exp − 1 − ct √  P (−t) exp − 1 − ct ˙    √    −P (−t) exp −√1 − ct  = ,  − P (−t) exp − 1 − ct ˙     0 0 

(4)

K3

 Q (t) r˙ (t)  (Q (t) r˙ (t))˙      Q (t) r˙ (t)  = ,  (Q (t) r˙ (t))˙    Q (t) r˙ (t)  (Q (t) r˙ (t))˙





(5)

K3

(6)

K3

 r˙ (t)  r¨ (t)       r˙ (t)  = .  r¨ (t)     r˙ (t)  r¨ (t)

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SmoothNonsmooth


Expansion with respect to the third column gives K3 (t, t0 ) = − (γ cos ω (t + t0 ) − δ1 x˙ 10 − T (x˙ 10 − v∗ )) W31 (t, t0 ) + + (T (x˙ 20 − v∗ ) + δ2 x˙ 20 ) W32 (t, t0 ) + T (x˙ 30 − v∗ ) W33 (t, t0 ) , (8.43) where h i (1) (2) (3) (4) (5) W31 (t, t0 ) = det W31 , W31 , W31 , W31 , W31 , and

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct    √ , = R (t) exp 1 − 3ct ˙    −R (t) exp √1 − 3ct  √ − R (t) exp 1 − 3ct ˙ 

(1)

W31


(2)

W31

 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  −P (−t) exp − 1 − ct   Q (t) r˙ (t)      √ (4)    =  − P (−t) exp − 1 − ct ˙ , W31 =   (Q (t) r˙ (t))˙ ,    0 Q (t) r˙ (t)  0 (Q (t) r˙ (t))˙ 

(3)

W31

 r˙ (t)  r˙ (t)     =  r¨ (t)  .  r˙ (t)  

(5)

W31

r¨ (t)

h

i (1) (2) (3) (4) (5) W32 (t, t0 ) = det W32 , W32 , W32 , W32 , W32 , where

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙    √  =  R (t) exp √1 − 3ct  ,  −R (t) exp 1 − 3ct  √ − R (t) exp 1 − 3ct ˙ 

(1)

W32

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227


(2)

W32



(3) W32

  =  

 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  (Q (t) r˙ (t))˙ P (−t) exp − 1 − ct ˙    √ (4)   −P (−t) exp − 1 − ct  , W32 =   Q (t) r˙ (t)  ,   0 Q (t) r˙ (t)  0 (Q (t) r˙ (t))˙  r˙ (t)  r¨ (t)     =  r˙ (t)  .  r˙ (t)  

(5)

W32

r¨ (t)

On the other hand i h (4) (5) (3) (1) (2) W33 (t, t0 ) = det W33 , W33 , W33 , W33 , W33 , where 

(1) W33

  =  

  √ √  R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct √ √  R (−t) exp − 1 − 3ct ˙ R (t) exp 1 − 3ct ˙    √ √ (2)   R (t) exp 1 − 3ct   , W33 =  R (−t) exp −√ 1 − 3ct  , √   R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ √ √ −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct  √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  P (−t) exp − 1 − ct ˙   (Q (t) r˙ (t))˙     √ (4)   =  −P (−t) exp − 1 − ct  , W33 =  Q (t) r˙ (t)   ,  − P (−t) exp −√1 − ct ˙  (Q (t) r˙ (t))˙ 0 Q (t) r˙ (t) 

(3)

W33

 r˙ (t)  r¨ (t)     =  r˙ (t)  .  r¨ (t)  

(5)

W33

r˙ (t)

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Expansion of W31 (t, t0 ) with respect to the third column gives √ W31 (t, t0 ) = 2P (−t) exp − 1 − ct W311 (t, t0 ) ,

(8.44)

where h i (1) (2) (3) (4) W311 (t, t0 ) = det W311 , W311 , W311 , W311 , where (1)

W311

(2)

W311

(3)

W311

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙   √ =  −R (t) exp 1 − 3ct  , √ − R (t) exp 1 − 3ct ˙ 

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙   √ =  −R (−t) exp − 1 − 3ct  , √ − R (−t) exp − 1 − 3ct ˙ 




(4)

W311



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r¨ (t)

Expansion of W311 (t, t0 ) with respect to the first row gives √ − 3ct (W3111 (t, t0 ) − W3113 (t, t0 )) + W311 (t, t0 ) = R (t) exp 1 √ − R (t) exp 1 − 3ct ˙(W3112 (t, t0 ) − W3114 (t, t0 )) , (8.45) where √ R (−t) exp − 1 − 3ct ˙ (Q (t) r˙ (t))˙ r¨ (t) √ W3111 (t, t0 ) = −R (−t) exp − 1 − 3ct Q (t) r˙ (t) r˙ (t) , √ − R (−t) exp − 1 − 3ct ˙ (Q (t) r˙ (t))˙ r¨ (t) √ R (−t) exp − 1 − 3ct Q (t) r˙ (t) r˙ (t) √ W3112 (t, t0 ) = −R (−t) exp − 1 − 3ct Q (t) r˙ (t) r˙ (t) , − R (−t) exp −√1 − 3ct ˙ (Q (t) r˙ (t))˙ r¨ (t)

√ R (−t) exp − 1 − 3ct Q (t) r˙ (t) r˙ (t) √ W3113 (t, t0 ) = R (−t) exp − 1 − 3ct ˙ (Q (t) r˙ (t))˙ r¨ (t) , − R (−t) exp −√1 − 3ct ˙ (Q (t) r˙ (t))˙ r¨ (t)

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229

√ R (−t) exp − 1 − 3ct Q (t) r˙ (t) r˙ (t) √ W3114 (t, t0 ) = R (−t) exp − 1 − 3ct ˙ (Q (t) r˙ (t))˙ r¨ (t) . −R (−t) exp −√1 − 3ct Q (t) r˙ (t) r˙ (t)

Expansion of W3111 (t, t0 ), W3112 (t, t0 ), W3113 (t, t0 ), W3114 (t, t0 ) with respect to the first column gives, respectively √ W3111 (t, t0 ) = 2A4 R (−t) exp − 1 − 3ct ˙, √ W3112 (t, t0 ) = 2A4 R (−t) exp − 1 − 3ct , √ W3113 (t, t0 ) = −2A4 R (−t) exp − 1 − 3ct ˙, √ W3114 (t, t0 ) = −2A4 R (−t) exp − 1 − 3ct . Substitution of the above formulas to (8.45) gives W311 (t, t0 ) = 4A4 A5 , where

√ √ R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct √ √ A5 = = const. R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙

Substitution of the above formulas to (8.44) allows us to obtain √ W31 (t, t0 ) = 8A4 A5 P (−t) exp − 1 − ct .

Expansion of W32 (t, t0 ) with respect to the fifth row gives √ W32 (t, t0 ) = P (−t) exp − 1 − ct (W321 (t, t0 ) − W322 (t, t0 )) , (8.46) where W321 (t, t0 ) = −W311 (t, t0 ) = −4A4 A5 , W322 (t, t0 ) = W311 (t, t0 ) = 4A4 A5 . Substitution of the above formulas to (8.46) gives √ W32 (t, t0 ) = −8A4 A5 P (−t) exp − 1 − ct . Expansion of W33 (t, t0 ) with respect to the third column yields W33 (t, t0 ) = 0.

(8.47)

Substitution of the above equations to (8.43) allows us to find that K3 (t, t0 ) = √ = −8A4 A5 P (−t) exp − 1 − ct × × (γ cos ω (t + t0 ) − δ1 x˙ 10 − T (x˙ 10 − v∗ ) + T (x˙ 20 − v∗ ) + δ2 x˙ 20 ) .

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SmoothNonsmooth


Since in the considered case x10 = x20 = x30 , one gets √ K3 (t, t0 ) = −8A4 A5 P (−t) exp − 1 − ct (γ cos ω (t + t0 ) − (δ1 − δ2 ) x˙ 10 ) . (8.48) Substitution of the fourth column of the fundamental solutions matrix (8.17) by vector h (q01 , t) gives the following determinant h i K4 (t, t0 ) = det K4(1) , K4(2) , K4(3) , K4(4) , K4(5) , K4(6) , where

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙    √    R (t) exp √ 1 − 3ct  (1) K4 =  ,  R (t) exp 1 − 3ct ˙  √    −R (t) exp 1 − 3ct  √ − R (t) exp 1 − 3ct ˙   √ R (−t) exp − 1 − 3ct  R (−t) exp −√1 − 3ct ˙    √    R (−t) exp −√ 1 − 3ct  (2) K4 =  ,  R (−t) exp − 1 − 3ct ˙  √    −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙   √ P (t) exp 1 − ct √  P (t) exp 1 − ct ˙    √    −P (t) exp √1 − ct  (3) K4 =  ,  − P (t) exp 1 − ct ˙     0 0   0  γ cos ωt − δ x˙ − T (x˙ − v )  1 10 10 ∗     0   (4) K4 =  ,   −T (x˙ 20 − v∗ ) − δ2 x˙ 20     0 −T (x˙ 30 − v∗ )     Q (t) r˙ (t) r˙ (t)  (Q (t) r˙ (t))˙  r¨ (t)           Q (t) r˙ (t)   r˙ (t)  (5) (6) K4 =   , K4 =  .  (Q (t) r˙ (t))˙  r¨ (t)       Q (t) r˙ (t)   r˙ (t)  (Q (t) r˙ (t))˙ r¨ (t) 

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SmoothNonsmooth

231

Expansion regarding its fourth column gives K4 (t, t0 ) = (γ cos ω (t + t0 ) − δ1 x˙ 10 − T (x˙ 10 − v∗ )) W41 (t, t0 ) + (8.49) − (T (x˙ 20 − v∗ ) + δ2 x˙ 20 ) W42 (t, t0 ) − T (x˙ 30 − v∗ ) W43 (t, t0 ) , where h i (1) (2) (3) (4) (5) W41 (t, t0 ) = det W41 , W41 , W41 , W41 , W41 , , and

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct    √  =  R (t) exp 1 − 3ct ˙  ,  −R (t) exp √1 − 3ct  √ − R (t) exp 1 − 3ct ˙ 

(1)

W41


(2)

W41

 √   P (t) exp 1 − ct Q (t) r˙ (t) √  −P (t) exp 1 − ct   Q (t) r˙ (t)      √ (4)    =  − P (t) exp 1 − ct ˙ , W41 =   (Q (t) r˙ (t))˙ ,    0 Q (t) r˙ (t)  0 (Q (t) r˙ (t))˙ 

(3)

W41

 r˙ (t)  r˙ (t)     =  r¨ (t)  .  r˙ (t)  r¨ (t) 

(5)

W41

h i (1) (2) (3) (4) (5) W42 (t, t0 ) = det W42 , W42 , W42 , W42 , W42 , , where


(1)

W42

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SmoothNonsmooth



(2)

W42



(3) W42

  =  

 √   P (t) exp 1 − ct Q (t) r˙ (t) √  (Q (t) r˙ (t))˙ P (t) exp 1 − ct ˙    √ (4)   −P (t) exp 1 − ct  , W42 =   Q (t) r˙ (t)  ,   0 Q (t) r˙ (t)  0 (Q (t) r˙ (t))˙  r˙ (t)  r¨ (t)     =  r˙ (t)  .  r˙ (t)  

(5)

W42

r¨ (t)

We have also i h (4) (5) (3) (1) (2) W43 (t, t0 ) = det W43 , W43 , W43 , W43 , W43 , where 

(1) W43

  =   

(2) W43

(3)

 √ R (−t) exp − 1 − 3ct √ R (−t) exp − 1 − 3ct ˙  √ R (−t) exp − 1 − 3ct  , √ R (−t) exp − 1 − 3ct ˙ √ −R (−t) exp − 1 − 3ct

 √   P (t) exp 1 − ct Q (t) r˙ (t) √  P (t) exp 1 − ct ˙   (Q (t) r˙ (t))˙     √  , W (4) =  Q (t) r˙ (t)  , = −P (t) exp 1 − ct 43     √  − P (t) exp 1 − ct ˙  (Q (t) r˙ (t))˙ 0 Q (t) r˙ (t) 

W43

  =  

 √ R (t) exp 1 − 3ct √ R (t) exp 1 − 3ct ˙  √ R (t) exp 1 − 3ct  , √ R (t) exp 1 − 3ct ˙ √ −R (t) exp 1 − 3ct

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SmoothNonsmooth


233

 r˙ (t)  r¨ (t)     =  r˙ (t)  .  r¨ (t)  

(5)

W43

r˙ (t)

Expansion of W41 (t, t0 ) regarding the third column gives √ W41 (t, t0 ) = 2P (t) exp 1 − ct W411 (t, t0 ) ,

(8.50)

where W411 (t, t0 ) = W311 (t, t0 ) = 4A4 A5 . Substitution of the above equations to (8.50) yields √ W41 (t, t0 ) = 8A4 A5 P (t) exp 1 − ct . Expansion of W42 (t, t0 ) with respect to the third column gives √ W42 (t, t0 ) = P (t) exp 1 − ct (W421 (t, t0 ) − W422 (t, t0 )) ,

(8.51)

where W421 (t, t0 ) = −W311 (t, t0 ) = −4A4 A5 , W422 (t, t0 ) = W311 (t, t0 ) = 4A4 A5 . Substitution of the above formulas to (8.51) allows us to get √ W42 (t, t0 ) = −8A4 A5 P (t) exp 1 − ct . Expansion of W43 (t, t0 ) with respect to the third column gives W43 (t, t0 ) = 0.

(8.52)

Substitution of the above relations to (8.49) yields K4 (t, t0 ) √ = 8A4 A5 P (t) exp 1− ct γ cos ω (t + t0 ) − δ1 x˙ 10 − T (x˙ 10 − v∗ ) +T (x˙ 20 − v∗ ) + δ2 x˙ 20 . Since in our case x10 = x20 = x30 , hence √ K4 (t, t0 ) = 8A4 A5 P (t) exp 1 − ct (γ cos ω (t + t0 ) − (δ1 − δ2 ) x˙ 10 ) . (8.53) Substituting the fifth column of the fundamental solutions matrix (8.17) by vector h (q01 , t), we obtain the following determinant h i K5 (t, t0 ) = det K5(1) , K5(2) , K5(3) , K5(4) , K5(5) , K5(6) ,

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SmoothNonsmooth


where

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙    √    R (t) exp √ 1 − 3ct  = ,  R (t) exp 1 − 3ct ˙  √    −R (t) exp 1 − 3ct  √ − R (t) exp 1 − 3ct ˙ 

(1)

K5

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙    √    R (−t) exp −√ 1 − 3ct  = ,  R (−t) exp − 1 − 3ct ˙   √   −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙ 

(2)

K5


(3)

K5

 √ P (−t) exp − 1 − ct  P (−t) exp −√1 − ct ˙    √    −P (−t) exp −√1 − ct  = ,  − P (−t) exp − 1 − ct ˙     0 0 

(4)

K5



(5)

K5

0



 γ cos ωt − δ x˙ − T (x˙ − v )  1 10 10 ∗     0   = ,   −T (x˙ 20 − v∗ ) − δ2 x˙ 20     0 −T (x˙ 30 − v∗ )



(6)

K5

 r˙ (t)  r¨ (t)       r˙ (t)  = .  r¨ (t)     r˙ (t)  r¨ (t)

Expansion regarding the fifth column gives K5 (t, t0 ) = − (γ cos ω (t + t0 ) − δ1 x˙ 10 − T (x˙ 10 − v∗ )) W51 (t, t0 ) + + (T (x˙ 20 − v∗ ) + δ2 x˙ 20 ) W52 (t, t0 ) + T (x˙ 30 − v∗ ) W53 (t, t0 ) , (8.54)

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SmoothNonsmooth

235

where the following notation is applied h i (1) (2) (3) (4) (5) W51 (t, t0 ) = det W51 , W51 , W51 , W51 , W51 , where

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct    √ (1)  W51 =  R (t) exp 1 − 3ct ˙  ,  −R (t) exp √1 − 3ct  √ − R (t) exp 1 − 3ct ˙  √  R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct    √ (2)  W51 =  R (−t) exp − 1 − 3ct ˙  ,  −R (−t) exp −√1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙  √  P (t) exp 1 − ct √  −P (t) exp 1 − ct    √ (3)  W51 =   − P (t) exp 1 − ct ˙ ,   0 0  √    P (−t) exp − 1 − ct r˙ (t) √  −P (−t) exp − 1 − ct   r˙ (t)      √ (4) (5)    W51 =  − P (−t) exp − 1 − ct ˙ , W51 =   r¨ (t)  .    0 r˙ (t)  0 r¨ (t) h i (1) (2) (3) (4) (5) W52 (t, t0 ) = det W52 , W52 , W52 , W52 , W52 , 

where

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙    √ (1)  W52 =   R (t) exp √1 − 3ct  ,  −R (t) exp 1 − 3ct  √ − R (t) exp 1 − 3ct ˙   √ √   P (t) exp 1 − ct R (−t) exp − 1 − 3ct √ √  R (−t) exp − 1 − 3ct ˙   P (t) exp 1 − ct ˙     √ √  , W (3) =  −P (t) exp 1 − ct  , = R (−t) exp − 1 − 3ct 52     √  −R (−t) exp − 1 − 3ct    0 √ − R (−t) exp − 1 − 3ct ˙ 0 

(2)

W52

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10:48


SmoothNonsmooth




  (4) W52 =   

 √   P (−t) exp − 1 − ct r˙ (t) √  r¨ (t)  P (−t) exp − 1 − ct ˙    √ (5)  r˙ (t)  . , W = −P (−t) exp − 1 − ct  52      r˙ (t)  0 0 r¨ (t)

On the other hand h i (1) (2) (3) (4) (5) W53 (t, t0 ) = det W53 , W53 , W53 , W53 , W53 , where  √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙   √ (1)  W53 =   R (t) exp √ 1 − 3ct  ,  R (t) exp 1 − 3ct ˙ √ −R (t) exp 1 − 3ct  √  R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙   √ (2)  W53 =   R (−t) exp −√ 1 − 3ct  ,  R (−t) exp − 1 − 3ct ˙ √ −R (−t) exp − 1 − 3ct  √  P (t) exp 1 − ct √  P (t) exp 1 − ct ˙    √ (3)  W53 =   −P (t) exp √1 − ct  ,  − P (t) exp 1 − ct ˙ 0  √    r˙ (t) P (−t) exp − 1 − ct √  P (−t) exp − 1 − ct ˙   r¨ (t)      √  , W (5) =  r˙ (t)  . = −P (−t) exp − 1 − ct 53      − P (−t) exp −√1 − ct˙  r¨ (t)  0 r˙ (t) 

(4)

W53

Expansion of W51 (t, t0 ) with respect to the third column gives √ W51 (t, t0 ) = P (t) exp 1 − ct (W511 (t, t0 ) + W512 (t, t0 )) √ − P (t) exp 1 + at ˙W513 (t, t0 ) ,

where

h i (1) (2) (3) (4) W511 (t, t0 ) = det W511 , W511 , W511 , W511 ,

(8.55)

July 19, 2007

10:48



and

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙  (1)  √ W511 =   −R (t) exp 1 − 3ct  , √ − R (t) exp 1 − 3ct ˙  √  R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙  (2)  √ W511 =   −R (−t) exp − 1 − 3ct  , √ − R (−t) exp − 1 − 3ct ˙  √    −P (−t) exp − 1 − ct r˙ (t) √  − P (−t) exp − 1 − ct ˙    , W (4) =  r¨ (t)  . = 511    r˙ (t)  0 0 r¨ (t) 

(3)

W511 We have

h i (1) (2) (3) (4) W512 (t, t0 ) = det W512 , W512 , W512 , W512 , where

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙  (1)  √ W512 =   −R (t) exp 1 − 3ct  , √ − R (t) exp 1 − 3ct ˙  √  R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙  (2)  √ W512 =   −R (−t) exp − 1 − 3ct  , √ − R (−t) exp − 1 − 3ct ˙  √    P (−t) exp − 1 − ct r˙ (t) √  − P (−t) exp − 1 − ct ˙    , W (4) =  r¨ (t)  . = 512    0 r˙ (t)  0 r¨ (t) h i (1) (2) (3) (4) W513 (t, t0 ) = det W513 , W513 , W513 , W513 , 

(3)

W512

and (1)

W513

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct   √ =  −R (t) exp 1 − 3ct  , √ − R (t) exp 1 − 3ct ˙ 

SmoothNonsmooth

237

July 19, 2007

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SmoothNonsmooth


(2)

W513

(3)

W513

 √ R (−t) exp − 1 − 3ct √   R (−t) exp − 1 − 3ct , √ =  −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙ 

 √ P (−t) exp − 1 − ct √  −P (−t) exp − 1 − ct  , =   0 0 

(4)

W513



 r˙ (t)  r˙ (t)   =  r˙ (t)  . r¨ (t)

Expansion of W511 (t, t0 ) with respect to the third column gives W511 (t, t0 ) = −P (−t) √ √ × exp − 1 − ct W5111 (t, t0 ) + P (−t) exp − 1 − ct ˙W5112 (t, t0 ) , (8.56) where W5111 (t, t0 )

√ √ R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ r¨ (t) √ √ = −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct r˙ (t) , − R (t) exp √1 − 3ct ˙ − R (−t) exp −√1 − 3ct ˙ r¨ (t)

W5112 (t, t0 ) √ √ R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct r˙ (t) √ √ = −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct r˙ (t) . − R (t) exp √1 − 3ct˙ − R (−t) exp −√1 − 3ct˙ r¨ (t)

Expansion of W5111 (t, t0 ) and W5112 (t, t0 ) with respect to the third column yields, W5111 (t, t0 ) = 2A5 r¨ (t) , W5112 (t, t0 ) = 2A5 r˙ (t) , respectively. Substitution of the above formulas to (8.56) yields √ W511 (t, t0 ) = 2A5 r˙ (t) P (−t) exp − 1 − ct ˙ √ − r¨ (t) P (−t) exp − 1 − ct . Expansion of W512 (t, t0 ) with respect to the third column gives √ W512 (t, t0 ) = P (−t) exp − 1 − ct W5121 (t, t0 ) √ + P (−t) exp − 1 − ct ˙W5122 (t, t0 ) ,

(8.57)

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SmoothNonsmooth

239

where W5121 (t, t0 ) √ √ R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ r¨ (t) √ √ = −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct r˙ (t) , − R (t) exp √1 − 3ct ˙ − R (−t) exp −√1 − 3ct ˙ r¨ (t) W5122 (t, t0 ) √ √ R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct r˙ (t) √ √ = −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct r˙ (t) . − R (t) exp √1 − 3ct ˙ − R (−t) exp −√1 − 3ct ˙ r¨ (t)

Note that

W5121 (t, t0 ) = W5111 (t, t0 ) = 2A5 r¨ (t) , W5122 (t, t0 ) = W5112 (t, t0 ) = 2A5 r˙ (t) , and substituting the above formulas to (8.57) we get √ W512 (t, t0 ) = 2A5 r¨ (t) P (−t) exp − 1 − ct √ + r˙ (t) P (−t) exp − 1 − ct ˙ . Expansion of W513 (t, t0 ) with respect to the third column gives √ W513 (t, t0 ) = 2P (−t) exp − 1 − ct W5131 (t, t0 ) ,

(8.58)

where W5131 (t, t0 ) √ √ R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct r˙ (t) √ √ = −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct r˙ (t) . − R (t) exp √1 − 3ct ˙ − R (−t) exp −√1 − 3ct ˙ r¨ (t)

Observe that

W5132 (t, t0 ) = W5112 (t, t0 ) = 2A5 r˙ (t) . Substituting the above formulas to (8.58) one gets

√ W513 (t, t0 ) = 4A5 r˙ (t) P (−t) exp − 1 − ct ,

whereas substitution of the above equations to (8.55) yields W51 (t, t0 ) = 4A1 A5 r˙ (t) . Expansion of W52 (t, t0 ) with respect to the third column yields √ W52 (t, t0 ) = P (t) exp 1 − ct (W521 (t, t0 ) − W523 (t, t0 )) √ = − P (t) exp 1 − ct ˙W522 (t, t0 ) ,

(8.59)

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SmoothNonsmooth



W521

(2)

W521

(3)

W521

 √ R (t) exp 1 − 3ct ˙ √  R (t) exp 1 − 3ct   √ =  −R (t) exp 1 − 3ct  , √ − R (t) exp 1 − 3ct ˙ 

 √ R (−t) exp − 1 − 3ct ˙ √  R (−t) exp − 1 − 3ct  , √ =  −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙ 

 √ P (−t) exp − 1 − ct ˙ √  −P (−t) exp − 1 − ct  , =   0 0 

(4)

W521



 r¨ (t)  r˙ (t)   =  r˙ (t)  . r¨ (t)


W522

(2)

W522

(3)

W522


 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct   √ =  −R (−t) exp − 1 − 3ct  , √ − R (−t) exp − 1 − 3ct ˙ 

 √ P (−t) exp − 1 − ct √  −P (−t) exp − 1 − ct  , =   0 0 

(4)

W522



 r˙ (t)  r˙ (t)   =  r˙ (t)  . r¨ (t)

h i (1) (2) (3) (4) W523 (t, t0 ) = det W523 , W523 , W523 , W523 ,

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SmoothNonsmooth


and (1)

W523

(2)

W523

(3)

W523

241



 √ P (−t) exp − 1 − ct √  P (−t) exp − 1 − ct ˙ , =   0 0 

(4)

W523



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r¨ (t)

Expansion of W521 (t, t0 ) with respect to the third column gives √ W521 (t, t0 ) = P (−t) exp − 1 − ct ˙ √ ×W5211 (t, t0 ) + P (−t) exp − 1 − ct W5212 (t, t0 ) ,

(8.60)

where W5211 (t, t0 ) √ √ R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct r˙ (t) √ √ = −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct r˙ (t) , − R (t) exp √1 − 3ct˙ − R (−t) exp −√1 − 3ct˙ r¨ (t) W5212 (t, t0 ) √ √ R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ r¨ (t) √ √ = −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct r˙ (t) . − R (t) exp √1 − 3ct ˙ − R (−t) exp −√1 − 3ct ˙ r¨ (t)

Expansion of W5211 (t, t0 ), W5212 (t, t0 ) with respect to the third column gives W5211 (t, t0 ) = W5112 (t, t0 ) = 2A5 r˙ (t) , W5212 (t, t0 ) = W5111 (t, t0 ) = 2A5 r¨ (t) . Substitution of the above equations to (8.60) gives

√ W521 (t, t0 ) = 2A5 r˙ (t) P (−t) exp − 1 − ct ˙ √ + r¨ (t) P (−t) exp − 1 − ct .

July 19, 2007

10:48


SmoothNonsmooth


Expansion of W522 (t, t0 ) with respect to the third column yields √ W522 (t, t0 ) = 2P (−t) exp − 1 − ct W5221 (t, t0 ) ,

(8.61)

where W5221 (t, t0 ) = W5112 (t, t0 ) = 2A5 r˙ (t) . Substitution of the above equations to (8.61) yields √ W522 (t, t0 ) = 4A5 r˙ (t) P (−t) exp − 1 − ct . Expansion of W523 (t, t0 ) with respect to the third column gives √ W523 (t, t0 ) = P (−t) exp− 1 − ct W5231 (t, t0 ) √ − P (−t) exp − 1 − ct ˙W5232 (t, t0 ) ,

(8.62)

where W5231 (t, t0 ) = W5111 (t, t0 ) = 2A5 r¨ (t) , W5232 (t, t0 ) = W5112 (t, t0 ) = 2A5 r˙ (t) . Substitution of the above equations to (8.62) gives √ W523 (t, t0 ) = 2A5 r¨ (t) P (−t) exp − 1 − ct √ − r˙ (t) P (−t) exp − 1 − ct ˙ , whereas their substitution to (8.59) allows us to find that W52 (t, t0 ) = 4A1 A5 r˙ (t) . Expansion of W53 (t, t0 ) with respect to the third column gives √ W53 (t, t0 ) = P (t) exp 1− ct (W531 (t, t0 ) − W533 (t, t0 )) + √ − P (t) exp 1 − ct ˙(W532 (t, t0 ) − W534 (t, t0 )) ,

(8.63)


W531

  √ √  R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ √ √  R (t) exp 1 − 3ct   R (−t) exp − 1 − 3ct  (2)    √ √ =  R (t) exp 1 − 3ct ˙ , W531 =  R (−t) exp − 1 − 3ct ˙ , √ √ −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct 

July 19, 2007

10:48


SmoothNonsmooth


(3)

W531

 √ P (−t) exp − 1 − ct ˙ √  −P (−t) exp − 1 − ct   √ =  − P (−t) exp − 1 − ct ˙ , 0 

(4)

W531

243



 r¨ (t)  r˙ (t)   =  r¨ (t)  . r˙ (t)

On the other hand h i (1) (2) (3) (4) W532 (t, t0 ) = det W532 , W532 , W532 , W532 , where (1)

W532

  √ √  R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct √ √  R (t) exp 1 − 3ct   R (−t) exp − 1 − 3ct  (2)    √ √ =  R (t) exp 1 − 3ct ˙ , W532 =  R (−t) exp − 1 − 3ct ˙ , √ √ −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct 

(3)

W532


(4)

W532



 r˙ (t)  r˙ (t)   =  r¨ (t)  . r˙ (t)


W533

  √ √  R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct √ √  R (t) exp 1 − 3ct ˙  R (−t) exp − 1 − 3ct ˙ (2)   , W533  √ √ = =  R (t) exp 1 − 3ct ˙  R (−t) exp − 1 − 3ct ˙ , √ √ −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct 

(3)

W533

 √ P (−t) exp − 1 − ct √  P (−t) exp − 1 − ct ˙   √ =  − P (−t) exp − 1 − ct ˙ , 0 

(4)

W533



 r˙ (t)  r¨ (t)   =  r¨ (t)  . r˙ (t)


W534

  √ √  R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct √ √  R (t) exp 1 − 3ct ˙  R (−t) exp − 1 − 3ct ˙ (2)    √ √ =  R (t) exp 1 − 3ct  , W534 =  R (−t) exp − 1 − 3ct  , √ √ −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct 

July 19, 2007

10:48


SmoothNonsmooth


(3)

W534


(4)

W534



 r˙ (t)  r¨ (t)   =  r˙ (t)  . r˙ (t)

Expansion of W531 (t, t0 ) with respect to the third column gives √ W531 (t, t0 ) = 2 P (−t) exp − 1 − ct ˙W5311 (t, t0 ) ,

(8.64)

where

√ √ R (t) exp 1 − 3ct R (−t) exp − √1 − 3ct r˙ (t) √ W5311 (t, t0 ) = R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ r¨ (t) . −R (t) exp √1 − 3ct −R (−t) exp −√1 − 3ct r˙ (t)

Observe that

W5311 (t, t0 ) = W5112 (t, t0 ) = 2A5 r˙ (t) . Substitution of the above equations to (8.64) yields √ W531 (t, t0 ) = 4A5 r˙ (t) P (−t) exp − 1 − ct ˙. Expansion of W532 (t, t0 ) with respect to the third column yields √ W532 (t, t0 ) = 2P (−t) exp − 1 − ct W5321 (t, t0 ) ,

(8.65)

where


Note that

W5321 (t, t0 ) = W5311 (t, t0 ) = 2A5 r˙ (t) . Substitution of the above equations to (8.65) gives √ W532 (t, t0 ) = 4A5 r˙ (t) P (−t) exp − 1 − ct . Expansion of W533 (t, t0 ) with respect to the third column yields √ W533 (t, t0 ) = −2 P (−t) exp − 1 − ct ˙W5331 (t, t0 ) , where

(8.66)


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SmoothNonsmooth


245

Observe that W5331 (t, t0 ) = W5311 (t, t0 ) = 2A5 r˙ (t) . Substitution of the above equations to (8.66) gives

√ W533 (t, t0 ) = −4A5 r˙ (t) P (−t) exp − 1 − ct ˙.

Expansion of W534 (t, t0 ) with respect to the third column gives √ W534 (t, t0 ) = −2 P (−t) exp − 1 − ct ˙W5341 (t, t0 ) ,

(8.67)

where

√ √ R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct r ˙ (t) √ √ W5341 (t, t0 ) = R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ r¨ (t) . −R (t) exp √1 − 3ct −R (−t) exp −√1 − 3ct r˙ (t)

Note that

W5341 (t, t0 ) = W5311 (t, t0 ) = 2A5 r˙ (t) . Substitution of the above relations to (8.67) gives

√ W534 (t, t0 ) = −4A5 r˙ (t) P (−t) exp − 1 − ct ˙,

whereas their substitution to (8.63) gives W53 (t, t0 ) = 8A1 A5 r˙ (t) . Upon substitution of the above relations to (8.54) we get K5 (t, t0 ) = 4A1 A5 r˙ (t) − γ cos ω (t + t0 ) + δ1 x˙ 10 + T (x˙ 10 − v∗ ) + T (x˙ 20 − v∗ ) + δ2 x˙ 20 + 2T (x˙ 30 − v∗ ) .

Since in the considered case x10 = x20 = x30 , one gets

K5 (t, t0 ) = 4A1 A5 r˙ (t) (−γ cos ω (t + t0 ) + (δ1 + δ2 ) x˙ 10 + 4T (x˙ 10 − v∗ )) . (8.68) Substitution of the sixth column of the fundamental solutions matrix (8.17) by vector h (q01 , t) yields the following determinant i h K6 (t, t0 ) = det K6(1) , K6(2) , K6(3) , K6(4) , K6(5) , K6(6) , where


(1)

K6

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SmoothNonsmooth


 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙    √   R (−t) exp − 1 − 3ct  (2)  √ K6 =  ,  R (−t) exp − 1 − 3ct ˙  √    −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙   √ P (t) exp 1 − ct √  P (t) exp 1 − ct ˙    √    −P (t) exp √1 − ct  (3) K6 =  ,  − P (t) exp 1 − ct ˙     0 0   √ P (−t) exp − 1 − ct √  P (−t) exp − 1 − ct ˙    √    −P (−t) exp −√1 − ct  (4) K6 =  ,  − P (−t) exp − 1 − ct ˙     0 0     Q (t) r˙ (t) 0  γ cos ωt − δ x˙ − T (x˙ − v )   (Q (t) r˙ (t))˙ 1 10 10 ∗         0    Q (t) r˙ (t)  (5) (6) K6 =   , K6 =  .    (Q (t) r˙ (t))˙ −δ2 x˙ 20 − T (x˙ 20 − v∗ )      Q (t) r˙ (t)    0 (Q (t) r˙ (t))˙ −T (x˙ 30 − v∗ ) Expansion with respect to the sixth column yields 

K6 (t, t0 ) = (γ cos ωt − δ1 x˙ 10 − T (x˙ 10 − v∗ )) W51 (t, t0 ) − (δ2 x˙ 20 + T (x˙ 20 − v∗ )) W52 (t, t0 ) − T (x˙ 30 − v∗ ) W53 (t, t0 ) , (8.69) where i h (5) (2) (3) (4) (1) W61 (t, t0 ) = det W61 , W61 , W61 , W61 , W61 , and

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct    √ , = R (t) exp 1 − 3ct ˙   √  −R (t) exp 1 − 3ct  √ − R (t) exp 1 − 3ct ˙ 

(1)

W61

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SmoothNonsmooth

247

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct    √ , = R (−t) exp − 1 − 3ct ˙    −R (−t) exp −√1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙ 

(2)

W61

  √ √  P (t) exp 1 − ct P (−t) exp − 1 − ct √ √  −P (t) exp 1 − ct   −P (−t) exp − 1 − ct      √ √ (4)    =  − P (t) exp 1 − ct ˙ , W61 =  − P (−t) exp − 1 − ct ˙ ,     0 0 0 0 

(3)

W61

 Q (t) r˙ (t)  Q (t) r˙ (t)     =  (Q (t) r˙ (t))˙ .  Q (t) r˙ (t)  (Q (t) r˙ (t))˙ 

(5)

W61

We have i h (3) (4) (5) (2) (1) W62 (t, t0 ) = det W62 , W62 , W62 , W62 , W62 , where


(1)

W62


(2)

W62



  (3) W62 =   

  √ √  P (t) exp 1 − ct P (−t) exp − 1 − ct √ √  P (−t) exp − 1 − ct ˙ P (t) exp 1 − ct ˙    √ √ (4)  −P (−t) exp − 1 − ct  , , W = −P (t) exp 1 − ct  62       0 0 0 0

July 19, 2007

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SmoothNonsmooth


 Q (t) r˙ (t)  (Q (t) r˙ (t))˙    =  Q (t) r˙ (t)  .  Q (t) r˙ (t)  

(5)

W62

(Q (t) r˙ (t))˙

h i (1) (2) (3) (4) (5) W63 (t, t0 ) = det W63 , W63 , W63 , W63 , W63 , and 

(1) W63

  =  

  √ √  R (−t) exp − 1 − 3ct R (t) exp 1 − 3ct √ √  R (−t) exp − 1 − 3ct ˙ R (t) exp 1 − 3ct ˙    √ √ (2)   R (t) exp 1 − 3ct   , W63 =  R (−t) exp −√ 1 − 3ct  , √   R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ √ √ −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct

  √ √  P (t) exp 1 − ct P (−t) exp − 1 − ct √ √  P (t) exp 1 − ct ˙   P (−t) exp − 1 − ct ˙      √ √  , W (4) =  −P (−t) exp − 1 − ct  , = −P (t) exp 1 − ct 63      − P (t) exp √1 − ct˙  − P (−t) exp −√1 − ct˙ 0 0 

(3)

W63

 Q (t) r˙ (t)  (Q (t) r˙ (t))˙    =  Q (t) r˙ (t)  .  (Q (t) r˙ (t))˙ 

(5)

W63

Q (t) r˙ (t)

Expansion of W61 (t, t0 ) with respect to the third column yields √ W61 (t, t0 ) = P (t) exp 1 − ct (W611 (t, t0 ) + W612 (t, t0 )) √ − P (t) exp 1 − ct ˙W613 (t, t0 ) where h i (1) (2) (3) (4) W611 (t, t0 ) = det W611 , W611 , W611 , W611 , and (1)

W611


(8.70)

July 19, 2007

10:48



(2)

W611

(3)

W611


 √   −P (−t) exp − 1 − ct Q (t) r˙ (t) √  − P (−t) exp − 1 − ct ˙    , W (4) =  (Q (t) r˙ (t))˙ . = 611    0 Q (t) r˙ (t)  0 (Q (t) r˙ (t))˙ h i (1) (2) (3) (4) W612 (t, t0 ) = W612 , W612 , W612 , W612 , 

where (1)

W612

(2)

W612

(3)

W612



 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  − P (−t) exp − 1 − ct ˙    , W (4) =  (Q (t) r˙ (t))˙ . = 612    Q (t) r˙ (t)  0 0 (Q (t) r˙ (t))˙ 

We have also h i (1) (2) (3) (4) W613 (t, t0 ) = det W613 , W613 , W613 , W613 , where (1)

W613

(2)

W613



SmoothNonsmooth

249

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SmoothNonsmooth


(3)

W613

 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  −P (−t) exp − 1 − ct     , W (4) =  Q (t) r˙ (t)  . = 613   Q (t) r˙ (t)   0 0 (Q (t) r˙ (t))˙ 

Expansion of W511 (t, t0 ) with respect to the third column yields √ W611 (t, t0 ) = −P (−t) exp − 1 − ct W6111 (t, t0 ) √ + P (−t) exp − 1 − ct ˙W6112 (t, t0 ) , where h i (1) (2) (3) W6111 (t, t0 ) = det W6111 , W6111 , W6111 , and (1)

W6111

(2)

W6111

 √ R (t) exp 1 − 3ct ˙ √ =  −R (t) exp 1 − 3ct  , √ − R (t) exp 1 − 3ct ˙ 

 √ R (−t) exp − 1 − 3ct ˙ √ =  −R (−t) exp − 1 − 3ct  , √ − R (−t) exp − 1 − 3ct ˙ 

(3)

W6111



 (Q (t) r˙ (t))˙ =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙


W6112

(2)

W6112

 √ R (t) exp 1 − 3ct √ =  −R (t) exp 1 − 3ct  , √ − R (t) exp 1 − 3ct ˙ 

 √ R (−t) exp − 1 − 3ct √ =  −R (−t) exp − 1 − 3ct  , √ − R (−t) exp − 1 − 3ct ˙ 

(3)

W6112



 Q (t) r˙ (t) =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙

(8.71)

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SmoothNonsmooth

251

Expansion of W6111 (t, t0 ) with respect to the third column gives W6111 (t, t0 ) = 2A5 (Q (t) r˙ (t))˙. Expansion of W6112 (t, t0 ) with respect to the third column yields W6112 (t, t0 ) = 2A5 Q (t) r˙ (t) . Substitution of the above equations to (8.71) gives √ W611 (t, t0 ) = 2A5 Q (t) r˙ (t) P (−t) exp − 1 − ct ˙ √ − (Q (t) r˙ (t))˙P (−t) exp − 1 − ct . Expansion of W612 (t, t0 ) with respect to the third column gives √ W612 (t, t0 ) = P (−t) exp − 1 − ct W6121 (t, t0 ) √ + P (−t) exp − 1 − ct ˙W6122 (t, t0 ) , where h i (1) (2) (3) W6121 (t, t0 ) = det W6121 , W6121 , W6121 , where (1)

W6121

(2)

W6121



(3)

W6121



 (Q (t) r˙ (t))˙ =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙

i h (1) (2) (3) W6122 (t, t0 ) = det W6122 , W6122 , W6122 , where (1)

W6122

(2)

W6122



(8.72)

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SmoothNonsmooth


(3)

W6122



 Q (t) r˙ (t) =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙

Note that W6121 (t, t0 ) = W6111 (t, t0 ) = 2A5 (Q (t) r˙ (t))˙, W6122 (t, t0 ) = W6112 (t, t0 ) = 2A5 Q (t) r˙ (t) . Substitution of the above equations to (8.72) gives √ W612 (t, t0 ) = 2A5 (Q (t) r˙ (t))˙P (−t) exp − 1 − ct √ + Q (t) r˙ (t) P (−t) exp − 1 − ct ˙ . Expansion of W613 (t, t0 ) with respect to the third column gives √ W613 (t, t0 ) = 2P (−t) exp − 1 − ct W6131 (t, t0 ) , where h i (1) (2) (3) W6131 (t, t0 ) = det W6131 , W6131 , W6131 , and (1)

W6131

(2)

W6131



(3)

W6131



 Q (t) r˙ (t) =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙

Note that W6132 (t, t0 ) = W6112 (t, t0 ) = 2A5 Q (t) r˙ (t) . Substitution of the above formulas to (8.73) allows us to get √ W613 (t, t0 ) = 4A5 Q (t) r˙ (t) P (−t) exp − 1 − ct ,

(8.73)

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SmoothNonsmooth

253

whereas substituting them into (8.70) we obtain W61 (t, t0 ) = 4A1 A5 Q (t) r˙ (t) . Expansion of W62 (t, t0 ) with respect to the third column gives √ W62 (t, t0 ) = P (t) exp 1− ct (W621 (t, t0 ) − W623 (t, t0 )) √ − P (t) exp 1 − ct ˙W622 (t, t0 ) , where h i (1) (2) (3) (4) W621 (t, t0 ) = det W621 , W621 , W621 , W621 , and (1)

W621

(2)

W621

(3)

W621

 √ R (t) exp 1 − 3ct ˙ √  R (t) exp 1 − 3ct   √ =  −R (t) exp 1 − 3ct  , √ − R (t) exp 1 − 3ct ˙ 

 √ R (−t) exp − 1 − 3ct ˙ √  R (−t) exp − 1 − 3ct   √ =  −R (−t) exp − 1 − 3ct  , √ − R (−t) exp − 1 − 3ct ˙ 

 √   (Q (t) r˙ (t))˙ P (−t) exp − 1 − ct ˙ √  −P (−t) exp − 1 − ct     , W (4) =  Q (t) r˙ (t)  . = 621    0 Q (t) r˙ (t)  0 (Q (t) r˙ (t))˙ 

We have h i (1) (2) (3) (4) W622 (t, t0 ) = det W622 , W622 , W622 , W622 , where (1)

W622

(2)

W622



(8.74)

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SmoothNonsmooth


(3)

W622

 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  −P (−t) exp − 1 − ct     , W (4) =  Q (t) r˙ (t)  . = 622   Q (t) r˙ (t)   0 0 (Q (t) r˙ (t))˙ h i (1) (2) (3) (4) W623 (t, t0 ) = det W623 , W623 , W623 , W623 , 

and

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙  (1)  √ W623 =   −R (t) exp 1 − 3ct  , √ − R (t) exp 1 − 3ct ˙  √  R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙  (2)  √ W623 =   −R (−t) exp − 1 − 3ct  , √ − R (−t) exp − 1 − 3ct ˙  √   P (−t) exp − 1 − ct Q (t) r˙ (t) √   P (−t) exp − 1 − ct ˙  , W (4) =  (Q (t) r˙ (t))˙ . 623   0 Q (t) r˙ (t)  0 (Q (t) r˙ (t))˙ 



 (3) W623 =  

Expansion of W621 (t, t0 ) with respect to the third column yields √ W621 (t, t0 ) = P (−t) exp − 1 − ct ˙W6211 (t, t0 ) √ +P (−t) exp − 1 − ct W6212 (t, t0 ) ,

where:

h i (1) (2) (3) W6211 (t, t0 ) = det W6211 , W6211 , W6211 , and

 √ R (t) exp 1 − 3ct √ (1) W6211 =  −R (t) exp 1 − 3ct  , √ − R (t) exp 1 − 3ct ˙  √  R (−t) exp − 1 − 3ct √ (2) W6211 =  −R (−t) exp − 1 − 3ct  , √ − R (−t) exp − 1 − 3ct ˙ 

(3)

W6211



 Q (t) r˙ (t) =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙

(8.75)

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SmoothNonsmooth

255

On the other hand h i (1) (2) (3) W6212 (t, t0 ) = det W6212 , W6212 , W6212 , where (1)

W6212

(2)

W6212



(3) W6212



 (Q (t) r˙ (t))˙ =  Q (t) r˙ (t)  . (Q (t) r˙ (t))˙

Expansion of W6211 (t, t0 )and W6212 (t, t0 ) with respect to the third column gives W6211 (t, t0 ) = W6112 (t, t0 ) = 2A5 Q (t) r˙ (t) , W6212 (t, t0 ) = W6111 (t, t0 ) = 2A5 (Q (t) r˙ (t))˙. Substitution of the above equations into (8.75) yields √ W621 (t, t0 ) = 2A5 Q (t) r˙ (t) P (−t) exp − 1 − ct ˙ √ + (Q (t) r˙ (t))˙P (−t) exp − 1 − ct . Expansion of W622 (t, t0 ) with respect to the third column gives √ W622 (t, t0 ) = 2P (−t) exp − 1 − ct W6221 (t, t0 ) ,

(8.76)

where W6221 (t, t0 ) = W6112 (t, t0 ) = 2A5 Q (t) r˙ (t) . Substitution of the above relations to (8.76) allows us to write √ W622 (t, t0 ) = 4A5 Q (t) r˙ (t) P (−t) exp − 1 − ct . Substitution of W623 (t, t0 ) with respect to the third column gives √ W623 (t, t0 ) = P (−t) exp − 1 − ct W6231 (t, t0 ) √ − P (−t) exp − 1 − ct ˙W6232 (t, t0 ) ,

(8.77)

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SmoothNonsmooth


where W6231 (t, t0 ) = W6111 (t, t0 ) = 2A5 (Q (t) r˙ (t))˙, W6232 (t, t0 ) = W6112 (t, t0 ) = 2A5 Q (t) r˙ (t) . Substitution of the above equations to (8.77) gives √ W623 (t, t0 ) = 2A5 (Q (t) r˙ (t))˙P (−t) exp − 1 − ct √ − Q (t) r˙ (t) P (−t) exp − 1 − ct ˙ , whereas their substitution into (8.69) yields W62 (t, t0 ) = 4A1 A5 Q (t) r˙ (t) . Expansion of W63 (t, t0 ) with respect to the third column gives √ W63 (t, t0 ) = P (t) exp 1 − ct (W631 (t, t0 ) − W633 (t, t0 )) √ − P (t) exp 1 − ct ˙(W632 (t, t0 ) − W634 (t, t0 )) ,

(8.78)


W631

  √ √  R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ √ √  R (t) exp 1 − 3ct   R (−t) exp − 1 − 3ct  (2)   , W631  √ √ = =  R (t) exp 1 − 3ct ˙  R (−t) exp − 1 − 3ct ˙ , √ √ −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct 

(3)

W631

 √   (Q (t) r˙ (t))˙ P (−t) exp − 1 − ct ˙ √  −P (−t) exp − 1 − ct   Q (t) r˙ (t)  (4)    √ =  − P (−t) exp − 1 − ct ˙ , W631 =  (Q (t) r˙ (t))˙ . 0 Q (t) r˙ (t) 

We have h i (1) (2) (3) (4) W632 (t, t0 ) = det W632 , W632 , W632 , W632 , where (1)

W632

  √ √  R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct √ √  R (t) exp 1 − 3ct   R (−t) exp − 1 − 3ct  (2)    √ √ =  R (t) exp 1 − 3ct ˙ , W632 =  R (−t) exp − 1 − 3ct ˙ , √ √ −R (t) exp 1 − 3ct −R (−t) exp − 1 − 3ct 

July 19, 2007

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(3)

W632

 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  −P (−t) exp − 1 − ct   Q (t) r˙ (t)  (4)    , W632 √ = =  (Q (t) r˙ (t))˙ .  − P (−t) exp − 1 − ct ˙ 0 Q (t) r˙ (t) h i (1) (2) (3) (4) W633 (t, t0 ) = det W633 , W633 , W633 , W633 , 

where (1)

W633

(2)

W633

(3)

W633

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙  √ =  R (t) exp 1 − 3ct ˙ , √ −R (t) exp 1 − 3ct 

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙  √ =  R (−t) exp − 1 − 3ct ˙ , √ −R (−t) exp − 1 − 3ct 

 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  P (−t) exp − 1 − ct ˙   (Q (t) r˙ (t))˙ (4)    √ =  − P (−t) exp − 1 − ct ˙ , W633 =  (Q (t) r˙ (t))˙ . 0 Q (t) r˙ (t) 

We have also h i (1) (2) (3) (4) W634 (t, t0 ) = det W634 , W634 , W634 , W634 , where (1)

W634

(2)

W634

(3)

W634

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙  √ =  R (t) exp 1 − 3ct  , √ −R (t) exp 1 − 3ct 

 √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙  √ =  R (−t) exp − 1 − 3ct  , √ −R (−t) exp − 1 − 3ct 

 √   P (−t) exp − 1 − ct Q (t) r˙ (t) √  P (−t) exp − 1 − ct ˙  (Q (t) r˙ (t))˙ (4)    √ =  −P (−t) exp − 1 − ct  , W634 =  Q (t) r˙ (t)  . 0 Q (t) r˙ (t) 

SmoothNonsmooth

257

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Expansion of W631 (t, t0 ) with respect to the third column gives √ W631 (t, t0 ) = 2 P (−t) exp − 1 − ct ˙W6311 (t, t0 ) ,

(8.79)

where W6311 (t, t0 ) √ √ R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct Q (t) r ˙ (t) √ √ = R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ (Q (t) r˙ (t))˙ . −R (t) exp √1 − 3ct −R (−t) exp −√1 − 3ct Q (t) r˙ (t)

Note that

W6311 (t, t0 ) = W6112 (t, t0 ) = 2A5 Q (t) r˙ (t) . Substitution of the above equations to (8.79) yields √ W631 (t, t0 ) = 4A5 Q (t) r˙ (t) P (−t) exp − 1 − ct ˙. Expansion of W632 (t, t0 ) with respect to the third column gives √ W632 (t, t0 ) = 2P (−t) exp − 1 − ct W6321 (t, t0 ) ,

(8.80)

where W6321 (t, t0 ) √ √ R (t) exp 1 − 3ct R (−t) exp − √1 − 3ct Q (t) r˙ (t) √ = R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ (Q (t) r˙ (t))˙ . −R (t) exp √1 − 3ct −R (−t) exp −√1 − 3ct Q (t) r˙ (t)

Note that

W6321 (t, t0 ) = W6311 (t, t0 ) = 2A5 Q (t) r˙ (t) . Substitution of the above equations to (8.80) yields √ W632 (t, t0 ) = 4A5 Q (t) r˙ (t) P (−t) exp − 1 − ct . Expansion of W633 (t, t0 ) with respect to the third column gives √ W633 (t, t0 ) = −2 P (−t) exp − 1 − ct ˙W6331 (t, t0 ) ,

(8.81)

where W6331 (t, t0 ) √ √ R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct Q (t) r˙ (t) √ √ = R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ (Q (t) r˙ (t))˙ . −R (t) exp √1 − 3ct −R (−t) exp −√1 − 3ct Q (t) r˙ (t)

Note that

W6331 (t, t0 ) = W6311 (t, t0 ) = 2A5 Q (t) r˙ (t) .

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259

Substitution of the above equations to (8.81) gives √ W633 (t, t0 ) = −4A5 Q (t) r˙ (t) P (−t) exp − 1 − ct ˙. Expansion of W634 (t, t0 ) with respect to the third column yields √ W634 (t, t0 ) = −2 P (−t) exp − 1 − ct ˙W6341 (t, t0 ) ,

(8.82)

where W6341 (t, t0 ) √ √ R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct Q (t) r ˙ (t) √ √ = R (t) exp 1 − 3ct ˙ R (−t) exp − 1 − 3ct ˙ (Q (t) r˙ (t))˙ . −R (t) exp √1 − 3ct −R (−t) exp −√1 − 3ct Q (t) r˙ (t)

Note that

W6341 (t, t0 ) = W6311 (t, t0 ) = 2A5 Q (t) r˙ (t) . Substitution of the above relations to (8.82) gives √ W634 (t, t0 ) = −4A5 Q (t) r˙ (t) P (−t) exp − 1 − ct ˙. On the other hand, substitution of the above equations to (8.78) gives W63 (t, t0 ) = 8A1 A5 Q (t) r˙ (t) , whereas their substitution to (8.69) yields K6 (t, t0 ) = 4A1 A5 Q (t) r˙ (t) γ cos ω (t + t0) − δ1 x˙ 10 − T (x˙ 10 − v∗ ) −T (x˙ 20 − v∗ ) − δ2 x˙ 20 − 2T (x˙ 30 − v∗ ) . Since in the considered case x10 = x20 = x30 , one gets K6 (t, t0 ) = 4A1 A5 Q (t) r˙ (t) (γ cos ω (t + t0 ) − (δ1 + δ2 ) x˙ 10 − 4T (x˙ 10 − v∗ )) . (8.83) The perturbation vector along the homoclinic orbit (8.6) for sign “-” is given by   0  γ cos ωt + δ x˙ − T (−x˙ − v )  1 10 10 ∗     0   h (q02 , t) =  .   −T (−x˙ 20 − v∗ ) + δ2 x˙ 20     0 −T (−x˙ 30 − v∗ ) Substituting the first column of (8.17) by vector h (q02 , t) we obtain the following determinant h i (1) (2) (3) (4) (5) (6) K1− (t, t0 ) = det K1− , K1− , K1− , K1− , K1− , K1− ,

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where 

(1)

K1−

 0  γ cos ωt + δ x˙ − T (−x˙ − v )  1 10 10 ∗     0   = ,   −T (−x˙ 20 − v∗ ) + δ2 x˙ 20     0 −T (−x˙ 30 − v∗ )  √ R (−t) exp − 1 − 3ct  R (−t) exp −√1 − 3ct ˙    √    R (−t) exp −√ 1 − 3ct  = ,  R (−t) exp − 1 − 3ct ˙  √    −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙ 

(2)

K1−

   √ √ P (t) exp 1 − ct P (−t) exp − 1 − ct √ √  P (t) exp 1 − ct ˙   P (−t) exp − 1 − ct ˙      √ √     −P (−t) exp − 1 − ct   −P (t) exp √1 − ct  (4)  √ =  , K1− =  ,  − P (t) exp 1 − ct ˙  − P (−t) exp − 1 − ct ˙         0 0 0 0 

(3)

K1−

   r˙ (t) Q (t) r˙ (t)  r¨ (t)   (Q (t) r˙ (t))˙          Q (t) r˙ (t)   r˙ (t)  (6) =  , K1− =  .  (Q (t) r˙ (t))˙  r¨ (t)       Q (t) r˙ (t)   r˙ (t)  (Q (t) r˙ (t))˙ r¨ (t) 

(5)

K1−

Expansion with respect to the first column gives K1− (t, t0 ) = − (γ cos ω (t + t0 ) + δ1 x˙ 10 − T (−x˙ 10 − v∗ )) W11− (t, t0 ) + (T (−x˙ 20 − v∗ ) − δ2 x˙ 20 ) W12− (t, t0 ) + T (−x˙ 30 − v∗ ) W13− (t, t0 ) , (8.84) where √ W11− (t, t0 ) = W11 (t, t0 ) = 4A1 r˙ (t) R (−t) exp − 1 − 3ct × (¨ r (t) Q (t) − (Q (t) r˙ (t))˙) ,

W12− (t, t0 ) = W12 (t, t0 ) = 4A1 r˙ (t) (¨ r (t) Q (t) − (Q (t) r˙ (t))˙) √ × R (−t) exp − 1 − 3ct ,

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261

W13− (t, t0 ) = W13 (t, t0 ) = 8A1 r˙ (t) ((Q (t) r˙ (t))˙− r¨ (t) Q (t)) √ × R (−t) exp − 1 − 3ct . Substitution of the above relations to (8.84) yields √ K1− (t, t0 ) = −4A1 A4 R (−t) exp − 1 − 3ct γ cos ω (t + t0 ) + δ1 x˙ 10

− T (−x˙ 10 − v∗ ) − T (−x˙ 20 − v∗ ) + δ2 x˙ 20 + 2T (−x˙ 30 − v∗ ) . Since in our case x10 = x20 = x30 , hence √ K1− (t, t0 ) = −4A1 A4 R (−t) exp − 1 − 3ct (8.85) × γ cos ω (t + t0 ) − (δ1 + δ2 ) x˙ 10 . Substitution of the second column of the fundamental solutions matrix (8.17) by vector h (q02 , t) yields a matrix whose determinant is h i (1) (2) (3) (4) (5) (6) K2− (t, t0 ) = det K2− , K2− , K2− , K2− , K2− , K2− , where

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙    √    R (t) exp √ 1 − 3ct  (1) K2− =  ,  R (t) exp 1 − 3ct ˙  √    −R (t) exp 1 − 3ct  √ − R (t) exp 1 − 3ct ˙   0  γ cos ωt + δ x˙ − T (−x˙ − v )  1 10 10 ∗     0   (2) K2− =  ,   −T (−x˙ 20 − v∗ ) + δ2 x˙ 20     0 −T (−x˙ 30 − v∗ )     √ √ P (t) exp 1 − ct P (−t) exp − 1 − ct  P (t) exp √1 − ct ˙   P (−t) exp −√1 − ct ˙      √ √      −P (t) exp √1 − ct   −P (−t) exp −√1 − ct  (4) =  , K2− =  ,  − P (t) exp 1 − ct ˙  − P (−t) exp − 1 − ct ˙         0 0 0 0     Q (t) r˙ (t) r˙ (t)  (Q (t) r˙ (t))˙  r¨ (t)           Q (t) r˙ (t)   r˙ (t)  (5) (6) K2− =   , K2− =  .  (Q (t) r˙ (t))˙  r¨ (t)       Q (t) r˙ (t)   r˙ (t)  (Q (t) r˙ (t))˙ r¨ (t) 

(3)

K2−

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Expansion with respect to the second column gives K2− (t, t0 ) = (γ cos ω (t + t0 ) + δ1 x˙ 10 − T (−x˙ 10 − v∗ )) W21− (t, t0 ) − (T (−x˙ 20 − v∗ ) − δ2 x˙ 20 ) W22− (t, t0 ) − T (−x˙ 30 − v∗ ) W23− (t, t0 ) , (8.86) where √ W21− (t, t0 ) = W21 (t, t0 ) = 4A1 A4 R (t) exp 1 − 3ct , W22− (t, t0 ) = W22 (t, t0 ) = 4A1 A4 R (t) exp W23− (t, t0 ) = W23 (t, t0 ) = −8A1 A4 R (t) exp

√

1 − 3ct ,

√

1 − 3ct ,

Substitution of the above formulas into (8.86) yields √ K2− (t, t0 ) = 4A1 A4 R (t) exp 1 − 3ct γ cos ω (t + t0 ) + δ1 x˙ 10 − T (−x˙ 10 − v∗ ) − T (−x˙ 20 − v∗ ) + δ2 x˙ 20 + 2T (−x˙ 30 − v∗ ) . Since in the considered case x10 = x20 = x30 , then √ K2− (t, t0 ) = 4A1 A4 R (t) exp 1 − 3ct (γ cos ω (t + t0 ) + (δ1 + δ2 ) x˙ 10 ) . (8.87) Substitution of the third column of fundamental solutions matrix (8.17) by vector h (q02 , t) gives a matrix whose determinant is as follows h i (1) (2) (3) (4) (5) (6) K3− (t, t0 ) = det K3− , K3− , K3− , K3− , K3− , K3− , where  √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙    √    R (t) exp √ 1 − 3ct  = ,  R (t) exp 1 − 3ct ˙  √    −R (t) exp 1 − 3ct  √ − R (t) exp 1 − 3ct ˙ 

(1)

K3−

 √ R (−t) exp − 1 − 3ct  R (−t) exp −√1 − 3ct ˙    √    R (−t) exp −√ 1 − 3ct  = ,  R (−t) exp − 1 − 3ct ˙  √    −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙ 

(2)

K3−

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SmoothNonsmooth

263



(3)

K3−

 0  γ cos ωt + δ x˙ − T (−x˙ − v )  1 10 10 ∗     0   = ,   −T (−x˙ 20 − v∗ ) + δ2 x˙ 20     0 −T (−x˙ 30 − v∗ )  √ P (−t) exp − 1 − ct  P (−t) exp −√1 − ct ˙    √    −P (−t) exp −√1 − ct  = ,  − P (−t) exp − 1 − ct ˙     0 0 

(4)

K3−

   Q (t) r˙ (t) r˙ (t)  (Q (t) r˙ (t))˙  r¨ (t)           Q (t) r˙ (t)   r˙ (t)  (6) =  , K3− =  .  r¨ (t)   (Q (t) r˙ (t))˙      r˙ (t)   Q (t) r˙ (t)  (Q (t) r˙ (t))˙ r¨ (t) 

(5)

K3−

Expansion with respect to the third column gives K3− (t, t0 ) = − (γ cos ω (t + t0 ) + δ1 x˙ 10 − T (−x˙ 10 − v∗ )) W31− (t, t0 ) + + (T (−x˙ 20 − v∗ ) − δ2 x˙ 20 ) W32− (t, t0 ) + T (−x˙ 30 − v∗ ) W33− (t, t0 ) , (8.88) where √ W31− (t, t0 ) = W31 (t, t0 ) = 8A4 A5 P (−t) exp − 1 − ct , √ W32− (t, t0 ) = W32 (t, t0 ) = −8A4 A5 P (−t) exp − 1 − ct , W33− (t, t0 ) = 0. Substitution of the above relations to (8.88) yields √ K3− (t, t0 ) = −8A4 A5 P (−t) exp − 1 − ct × = × (γ cos ω (t + t0 ) + δ1 x˙ 10 − T (−x˙ 10 − v∗ ) + T (−x˙ 20 − v∗ ) − δ2 x˙ 20 ) . Since in the considered case x10 = x20 = x30 , one gets √ K3− (t, t0 ) = −8A4 A5 P (−t) exp − 1 − ct

× (γ cos ω (t + t0 ) + (δ1 − δ2 ) x˙ 10 ) .

(8.89)

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Substitution of the fourth column of the fundamental solutions matrix (8.17) by vector h (q02 , t) gives the following determinant h i (1) (2) (3) (4) (5) (6) K4− (t, t0 ) = det K4− , K4− , K4− , K4− , K4− , K4− , where

   √ √ R (t) exp 1 − 3ct R (−t) exp − 1 − 3ct  R (t) exp √1 − 3ct ˙   R (−t) exp −√1 − 3ct ˙      √ √      R (t) exp √ 1 − 3ct   R (−t) exp −√ 1 − 3ct  (2) =  , K4− =  ,  R (t) exp 1 − 3ct ˙   R (−t) exp − 1 − 3ct ˙  √ √      −R (t) exp 1 − 3ct   −R (−t) exp − 1 − 3ct  √ √ − R (t) exp 1 − 3ct ˙ − R (−t) exp − 1 − 3ct ˙ 

(1)

K4−


(3)

K4−



(4)

K4−

 0  γ cos ωt + δ x˙ − T (−x˙ − v )  1 10 10 ∗     0   = ,   −T (−x˙ 20 − v∗ ) + δ2 x˙ 20     0 −T (−x˙ 30 − v∗ ) 

(5)

K4−

   Q (t) r˙ (t) r˙ (t)  (Q (t) r˙ (t))˙  r¨ (t)           Q (t) r˙ (t)   r˙ (t)  (6) =  , K4− =  .  (Q (t) r˙ (t))˙  r¨ (t)       Q (t) r˙ (t)   r˙ (t)  (Q (t) r˙ (t))˙ r¨ (t)

Expansion with respect to the fourth column yields K4− (t, t0 ) = (γ cos ω (t + t0 ) + δ1 x˙ 10 − T (−x˙ 10 − v∗ )) W41− (t, t0 ) − (T (−x˙ 20 − v∗ ) − δ2 x˙ 20 ) W42− (t, t0 ) − T (−x˙ 30 − v∗ ) W43 (t, t0 ) , (8.90) where the following notation is applied √ W41− (t, t0 ) = W41 (t, t0 ) = 8A4 A5 P (t) exp 1 − ct ,

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W42− (t, t0 ) = W42 (t, t0 ) = −8A4 A5 P (t) exp

265

√

1 − ct ,

W43− (t, t0 ) = 0. Substitution of the above relations to (8.90) gives √ K4− (t, t0 ) = 8A4 A5 P (t) exp 1 − ct × (γ cos ω (t + t0 ) + δ1 x˙ 10 − T (−x˙ 10 − v∗ ) + T (−x˙ 20 − v∗ ) − δ2 x˙ 20 ) . Taking into account that in our case x10 = x20 = x30 , one gets √ K4− (t, t0 ) = 8A4 A5 P (t) exp 1 − ct (γ cos ω (t + t0 ) + (δ1 − δ2 ) x˙ 10 ) . (8.91) Substitution of the fifth column of fundamental solutions matrix (8.17) by vector h (q02 , t) allows us to get the following determinant h i (1) (2) (3) (4) (5) (6) K5− (t, t0 ) = det K5− , K5− , K5− , K5− , K5− , K5− , where

 √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙    √    R (t) exp √ 1 − 3ct  (1) K5− =  ,  R (t) exp 1 − 3ct ˙  √    −R (t) exp 1 − 3ct  √ − R (t) exp 1 − 3ct ˙   √ R (−t) exp − 1 − 3ct √  R (−t) exp − 1 − 3ct ˙    √    R (−t) exp −√ 1 − 3ct  (2) K5− =  ,  R (−t) exp − 1 − 3ct ˙  √    −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙     √ √ P (t) exp 1 − ct P (−t) exp − 1 − ct  P (t) exp √1 − ct ˙   P (−t) exp −√1 − ct ˙      √ √      −P (t) exp √1 − ct   −P (−t) exp −√1 − ct  (4) =  , K5− =  ,  − P (t) exp 1 − ct ˙  − P (−t) exp − 1 − ct ˙         0 0 0 0     r˙ (t) 0  γ cos ωt + δ x˙ − T (−x˙ − v )   r¨ (t)  1 10 10 ∗         0    r˙ (t)  (5) (6) K5− =   , K5− =  .    r¨ (t)  −T (−x˙ 20 − v∗ ) + δ2 x˙ 20        r˙ (t)  0 −T (−x˙ 30 − v∗ ) r¨ (t) 

(3)

K5−

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Expansion with respect to the fifth column yields K5− (t, t0 ) = − (γ cos ω (t + t0 ) + δ1 x˙ 10 − T (−x˙ 10 − v∗ )) W51− (t, t0 ) + + (T (−x˙ 20 − v∗ ) − δ2 x˙ 20 ) W52− (t, t0 ) + T (−x˙ 30 − v∗ ) W53− (t, t0 ) , (8.92) where W51− (t, t0 ) = W51 (t, t0 ) = 4A1 A5 r˙ (t) , W52− (t, t0 ) = W52 (t, t0 ) = 4A1 A5 r˙ (t) , W53− (t, t0 ) = W53 (t, t0 ) = 8A1 A5 r˙ (t) . Substituting the above equations to (8.92) we have K5− (t, t0 ) = 4A1 A5 r˙ (t) − γ cos ω (t + t0) − δ1 x˙ 10 + T (−x˙ 10 − v∗ ) +T (−x˙ 20 − v∗ ) − δ2 x˙ 20 + 2T (−x˙ 30 − v∗ ) . Since x10 = x20 = x30 , hence K5− (t, t0 ) = 4A1 A5 r˙ (t) (4T (−x˙ 10 − v∗ ) − γ cos ω (t + t0 ) − (δ1 + δ2 ) x˙ 10 ) . (8.93) Substitution of the sixth column of (8.17) by vector h (q02 , t) gives the following determinant h i (1) (2) (3) (4) (5) (6) K6− (t, t0 ) = det K6− , K6− , K6− , K6− , K6− , K6− , where  √ R (t) exp 1 − 3ct √  R (t) exp 1 − 3ct ˙    √    R (t) exp √ 1 − 3ct  = ,  R (t) exp 1 − 3ct ˙  √    −R (t) exp 1 − 3ct  √ − R (t) exp 1 − 3ct ˙ 

(1)

K6−

 √ R (−t) exp − 1 − 3ct  R (−t) exp −√1 − 3ct ˙    √    R (−t) exp −√ 1 − 3ct  = ,  R (−t) exp − 1 − 3ct ˙  √    −R (−t) exp − 1 − 3ct  √ − R (−t) exp − 1 − 3ct ˙ 

(2)

K6−

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   √ √ P (t) exp 1 − ct P (−t) exp − 1 − ct √ √  P (t) exp 1 − ct ˙   P (−t) exp − 1 − ct ˙      √ √     −P (t) exp 1 − ct −P (−t) exp − 1 − ct    (3) (4)  , K6−  √ √ K6− =  = ,  − P (t) exp 1 − ct ˙  − P (−t) exp − 1 − ct ˙         0 0 0 0     0 Q (t) r˙ (t)  (Q (t) r˙ (t))˙  γ cos ωt + δ x˙ − T (−x˙ − v )  1 10 10 ∗         0   Q (t) r˙ (t)   (5) (6) K6− =  .  , K6− =   (Q (t) r˙ (t))˙   −T (−x˙ 20 − v∗ ) + δ2 x˙ 20        Q (t) r˙ (t)  0 −T (−x˙ 30 − v∗ ) (Q (t) r˙ (t))˙ 

Expansion with respect to the sixth column yields K6− (t, t0 ) = (γ cos ω (t + t0 ) + δ1 x˙ 10 − T (−x˙ 10 − v∗ )) W51− (t, t0 ) − (T (−x˙ 20 − v∗ ) − δ2 x˙ 20 ) W52− (t, t0 ) − T (−x˙ 30 − v∗ ) W53− (t, t0 ) , (8.94) where the following notation is applied W61− (t, t0 ) = W61 (t, t0 ) = 4A1 A5 Q (t) r˙ (t) , W62− (t, t0 ) = W62 (t, t0 ) = 4A1 A5 Q (t) r˙ (t) , W63− (t, t0 ) = W63 (t, t0 ) = 8A1 A5 Q (t) r˙ (t) . Substitution of the above formulas into (8.94) gives K6− (t, t0 ) = 4A1 A5 Q (t) r˙ (t) γ cos ω (t + t0 ) + δ1 x˙ 10 − T (−x˙ 10 − v∗ ) − T (−x˙ 20 − v∗ ) + δ2 x˙ 20 − 2T (−x˙ 30 − v∗ ) .

Since for the considered case x10 = x20 = x30 , one gets K6− (t, t0 ) = 4A1 A5 Q (t) r˙ (t)

× (γ cos ω (t + t0 ) + (δ1 + δ2 ) x˙ 10 − 4T (−x˙ 10 − v∗ )) .

(8.95)

The Melnikov function is defined by (8.31) and is as follows M1 (t0 ) = −M11 (t0 ) + M12 ,

(8.96)

where M11 (t0 ) = 4A1 A4 γ

Z∞

√ R (−t) exp − 1 − 3ct cos ω (t + t0 ) dt = 4A1 A4

0

×γ

Z∞ 0

−c −

√

√ 1 − 3c + tanh t tanh t exp − 1 − 3ct cos ω (t + t0 ) dt

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and M12 = 4A1 A4 (δ1 + δ2 )

Z∞

√ R (−t) exp − 1 − 3ct x˙ 10 (t) dt

0

= −4A1 A4 (δ1 + δ2 ) Z∞ √ √ × −c − 1 − 3c + tanh t tanh t exp − 1 − 3ct sech t tanh tdt. 0

Formula M11 (t0 ) is defined as follows M11 (t0 ) = M111 (t0 ) − M112 (t0 ) ,

(8.97)

where M111 (t0 ) = 4A1 A4 γ cos ωt0 Z∞ √ √ × −c − 1 − 3c + tanh t tanh t exp − 1 − 3ct cos ωtdt 0

and M112 (t0 ) = 4A1 A4 γ sin ωt0 Z∞ √ √ × −c − 1 − 3c + tanh t tanh t exp − 1 − 3ct sin ωtdt. 0

Upon integration we have √ 4 ω 2 + 1 − 3c (1 + c) M111 (t0 ) = A1 A4 γ cos ωt0 3c − 1 − ω 2 √ 1 √ 1 + ωiψ 1 − 3c − ωi − ωiψ 2 + 1 − 3c − ωi 4 4 √ √ 1 1 −ωiψ 1 − 3c + ωi + ωiψ 2 + 1 − 3c + ωi , 4 4 √

1 − 3c − (1 + c) M112 (t0 ) = A1 A4 γ sin ωt0 1 − 3c + ω 2 √ 1 √ 1 + ωψ 1 − 3c − ωi − ωψ 2 + 1 − 3c − ωi 4 4 √ √ 1 1 +ωψ 1 − 3c + ωi − ωψ 2 + 1 − 3c + ωi , 4 4 4ω

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269

where ψ (z) denotes the so called digamma function introduced in chapter 5. Substitution the above equations into (8.97) gives M11 (t0 ) = A1 A4 γ A˜1 cos ωt0 − A˜2 sin ωt0 = A˜ sin (ωt0 − κ) , where

√ 2 ω + 1 − 3c (1 + c) 4 A˜1 = 3c − 1 − ω 2 √ 1 √ 1 + ωi ψ 1 − 3c − ωi − ψ 2 + 1 − 3c − ωi 4 4 √ 1 √ 1 −ψ 1 − 3c + ωi + ψ 2 + 1 − 3c + ωi , 4 4 √ 4ω 1 − 3c − (1 + c) ˜ A2 = 1 − 3c + ω 2 √ 1 1 √ 1 − 3c − ωi − ψ 2 + 1 − 3c − ωi +ω ψ 4 4 √ √ 1 1 1 − 3c + ωi − ψ 2 + 1 − 3c + ωi , +ψ 4 4 q . and A˜ = A1 A4 γ A˜21 + A˜22 , tan κ = A˜1 A˜2 . Computing the original in relation M12 (t0 ) one gets √ 4 2 M12 = A1 A4 (δ1 + δ2 ) 3 √ √ √ 1 1 × 1 + 6c + 3c 1 − 3c ψ 1 + 1 − 3c − ψ 3 + 1 − 3c . 4 4 Substitution of the above relations to (8.96) one gets √ 4 2 ˜ M1 (t0 ) = A sin (ωt0 − κ) − A1 A4 (δ1 + δ2 ) 3 √ √ √ 1 1 × 1 + 6c + 3c 1 − 3c ψ 1 + 1 − 3c − ψ 3 + 1 − 3c . 4 4 (8.98) The criterion of chaos is as follows √ 4 2 ˜ A> A1 A4 (δ1 + δ2 ) 3 √ √ √ 1 1 . × 1 + 6c + 3c 1 − 3c ψ 1 + 1 − 3c − ψ 3 + 1 − 3c 4 4 (8.99)

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According to (8.42), the Melnikov function is M2 (t0 ) = M21 (t0 ) − M22

(8.100)

where Z0

M21 (t0 ) = 4A1 A4 γ

R (t) exp

−∞

= 4A1 A4 γ

Z0

−c +

−∞

√

√

1 − 3ct cos ω (t + t0 ) dt

√ 1 − 3c + tanh t tanh t exp 1 − 3ct

× cos ω (t + t0 ) dt and M22 = 4A1 A4 (δ1 + δ2 )

Z0

R (t) exp

−∞

√

1 − 3ct x˙ 10 (t) dt

= −4A1 A4 (δ1 + δ2 ) ×

Z0

−c +

−∞

√

√ 1 − 3c − tanh t tanh t exp 1 − 3ct sech t tanh tdt.

Relation M21 (t0 ) is as follows M21 (t0 ) = M211 (t0 ) − M212 (t0 ) ,

(8.101)

where M211 (t0 ) = 4A1 A4 γ cos ωt0 Z0

×

−c +

−∞

√

√ 1 − 3c − tanh t tanh t exp 1 − 3ct cos ωtdt,

and M212 (t0 ) = 4A1 A4 γ sin ωt0 ×

Z0

−∞

−c +

√

√ 1 − 3c − tanh t tanh t exp 1 − 3ct sin ωtdt.

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271

Upon integration we have √ 4 ω 2 + 1 − 3c (1 + c) M211 (t0 ) = M111 (t0 ) = A1 A4 γ cos ωt0 3c − 1 − ω 2 √ 1 1 √ +ωiψ 1 − 3c − ωi − ωiψ 2 + 1 − 3c − ωi 4 4 √ √ 1 1 −ωiψ 1 − 3c + ωi + ωiψ 2 + 1 − 3c + ωi , 4 4 √

1 − 3c − (1 + c) M212 (t0 ) = M112 (t0 ) = A1 A4 γ cos ωt0 1 − 3c + ω 2 √ 1 √ 1 +ωψ 1 − 3c − ωi − ωψ 2 + 1 − 3c − ωi 4 4 √ 1 √ 1 +ωψ 1 − 3c + ωi − ωψ 2 + 1 − 3c + ωi , 4 4 4ω

Substitution of the above relations to (8.97) gives M21 (t0 ) = M11 (t0 ) = A1 A4 γ A˜1 cos ωt0 − A˜2 sin ωt0 = A˜ sin (ωt0 − κ) , and the original function in expression M22 (t0 ) is √ 4 2 M22 = −M12 = − A1 A4 (δ1 + δ2 ) 3 √ √ √ 1 1 × 1 + 6c + 3c 1 − 3c ψ 1 + 1 − 3c − ψ 3 + 1 − 3c . 4 4 Substitution of the above relations to (8.100) gives √ 4 2 M2 (t0 ) = A˜ sin (ωt0 − κ) + A1 A4 (δ1 + δ2 ) 3 √ √ √ 1 1 × 1 + 6c + 3c 1 − 3c ψ 1 + 1 − 3c − ψ 3 + 1 − 3c . 4 4 The chaos criterion is as follows √ 4 2 ˜ A> A1 A4 (δ1 + δ2 ) 3 √ √ √ 1 1 . × 1 + 6c + 3c 1 − 3c ψ 1 + 1 − 3c − ψ 3 + 1 − 3c 4 4 (8.102) According to (8.48), the Melnikov function is M3 (t0 ) = −M31 (t0 ) + M32 ,

(8.103)

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where M31 (t0 ) = 8A4 A5 γ

Z∞

√ P (−t) exp − 1 − ct cos ω (t + t0 ) dt

0

= 8A4 A5 γ

Z∞

− 13 c −

√

√ 1 − c + tanh t tanh t exp − 1 − ct

0

× cos ω (t + t0 ) dt, and M32 = 8A4 A5 (δ1 − δ2 )

Z∞

√ P (−t) exp − 1 − ct x˙ 10 (t) dt

0

= −8A4 A5 (δ1 − δ2 ) Z∞ √ √ × − 13 c − 1 − c + tanh t tanh t exp − 1 − ct sech t tanh tdt. 0

The relation for M31 (t0 ) is defined in the following way M31 (t0 ) = M311 (t0 ) − M312 (t0 ) ,

(8.104)

where M311 (t0 ) = 8A4 A5 γ cos ωt0 Z∞ √ √ × − 31 c − 1 − c + tanh t tanh t exp − 1 − ct cos ωtdt, 0

and M312 (t0 ) = 8A4 A5 γ sin ωt0 Z∞ √ √ × − 31 c − 1 − c + tanh t tanh t exp − 1 − ct sin ωtdt. 0

Upon integration one gets √ 8 1 − c (3 + c) + 4ω 2 2 M311 (t0 ) = A4 A5 γ cos ωt0 ( . 3 3 (c − 1 − ω 2 ) √ 1 √ 1 + 3ωiψ 1 − c − ωi − 3ωiψ 2 + 1 − c − ωi 4 4 √ √ 1 1 − 3ωiψ 1 − c + ωi + 3ωiψ 2 + 1 − c + ωi ), 4 4

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273

√ 8ω (3 + c) − 3 1 − c M312 (t0 ) = A4 A5 γ sin ωt0 3 (c − 1 − ω 2 ) √ 1 1 √ 1 − c − ωi − 2ωψ 2 + 1 − c − ωi + 2ωψ 4 4 √ 1 √ 1 +2ωψ 1 − c + ωi − 2ωψ 2 + 1 − c + ωi . 4 4 Substitution of the above relations to (8.104) gives the following result M31 (t0 ) = A4 A5 γ A˜1 cos ωt0 − A˜2 sin ωt0 = A˜ sin (ωt0 − κ) , where √

1 − c (3 + c) + 4ω 2 3 (c − 1 − ω 2 ) √ 1 √ 1 + 3ωiψ 1 − c − ωi − 3ωiψ 2 + 1 − c − ωi 4 4 √ 1 1 √ 1 − c + ωi + 3ωiψ 2 + 1 − c + ωi , −3ωiψ 4 4

2 A˜1 = 3

8

√ 8ω (3 + c) − 3 1 − c ˜ A2 = 3 (c − 1 − ω 2 ) √ 1 √ 1 + 2ωψ 1 − c − ωi − 2ωψ 2 + 1 − c − ωi 4 4 √ 1 √ 1 +2ωψ 1 − c + ωi − 2ωψ 2 + 1 − c + ωi , 4 4 q . ˜ and A = A4 A5 γ A˜21 + A˜22 , tan κ = A˜1 A˜2 . Computing the original function in equation M32 (t0 ) one gets √ 8 2 M32 = A4 A5 (δ1 − δ2 ) 3 √ √ √ 1 1 × 1 + 2c + c 1 − c ψ 1+ 1−c −ψ 3+ 1−c . 4 4 Substituting the above relations to (8.103) one has √ 8 2 ˜ M3 (t0 ) = −A sin (ωt0 − κ) + A4 A5 (δ1 − δ2 ) 3 √ √ √ 1 1 × 1 + 2c + c 1 − c ψ 1+ 1−c −ψ 3+ 1−c . 4 4

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SmoothNonsmooth


The chaos √ criterion is as follows 8 2 A˜ > A4 A5 (δ1 − δ2 ) 3 √ √ √ 1 1 . 1+ 1−c −ψ 3+ 1−c × 1 + 2c + c 1 − c ψ 4 4 (8.105) According to (8.53), the Melnikov function is M4 (t0 ) = M41 (t0 ) − M42 ,

(8.106)

where M41 (t0 ) = 8A4 A5 γ

Z0

P (t) exp

√

−∞

= 8A4 A5 γ

Z0

√

− 13 c +

−∞

1 − ct cos ω (t + t0 ) dt

√ 1 − c − tanh t tanh t exp 1 − ct

× cos ω (t + t0 ) dt, and M42 = 8A4 A5 (δ1 − δ2 )

Z0

P (t) exp

√

−∞

= −8A4 A5 (δ1 − δ2 ) × exp

√

Z0

− 13 c +

−∞

1 − ct x˙ 10 (t) dt

√

1 − c − tanh t tanh t

1 − ct sech t tanh tdt.

Expression M41 (t0 ) is defined as follows

M41 (t0 ) = M411 (t0 ) − M412 (t0 ) ,

(8.107)

where M411 (t0 ) = 8A4 A5 γ cos ωt0 × exp and

√

− 13 c +

√

−∞


1 − ct cos ωtdt

M412 (t0 ) = 8A4 A5 γ sin ωt0 × exp

Z0

√

Z0

− 13 c +

−∞

1 − ct sin ωtdt.

√


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275

Upon integration we have √ 8 1 − c (3 + c) + 4ω 2 2 M411 (t0 ) = M311 (t0 ) = A4 A5 γ cos ωt0 3 3 (c − 1 − ω 2 ) √ 1 √ 1 +3ωiψ 1 − c − ωi − 3ωiψ 2 + 1 − c − ωi 4 4 √ √ 1 1 −3ωiψ 1 − c + ωi + 3ωiψ 2 + 1 − c + ωi , 4 4 √ 8ω (3 + c) − 3 1 − c M412 (t0 ) = M312 (t0 ) = A4 A5 γ sin ωt0 3 (c − 1 − ω 2 ) √ 1 √ 1 +2ωψ 1 − c − ωi − 2ωψ 2 + 1 − c − ωi 4 4 √ 1 √ 1 +2ωψ 1 − c + ωi − 2ωψ 2 + 1 − c + ωi , 4 4 Substitution of the above formulas into (8.107) gives M41 (t0 ) = M31 (t0 ) = A4 A5 γ A˜1 cos ωt0 − A˜2 sin ωt0 = A˜ sin (ωt0 − κ) . Computation of the original function in M42 (t0 ) gives √ 8 2 M42 = −M32 = − A4 A5 (δ1 − δ2 ) 3 √ √ √ 1 1 × 1 + 2c + c 1 − c ψ 1+ 1−c −ψ 3+ 1−c . 4 4 Substituting the above relations to (8.106) we get √ 8 2 M4 (t0 ) = A˜ sin (ωt0 − κ) − A4 A5 (δ1 − δ2 ) 3 √ √ √ 1 1 × 1 + 2c + c 1 − c ψ 1+ 1−c −ψ 3+ 1−c . 4 4 The chaos criterion is as follows √ 8 2 ˜ A> A4 A5 (δ1 − δ2 ) 3 √ √ √ 1 1 . × 1 + 2c + c 1 − c ψ 1+ 1−c −ψ 3+ 1−c 4 4 (8.108) The Melnikov function (see (8.68)) is M5 (t0 ) = −M51 (t0 ) + M52 + M53 ,

(8.109)

July 19, 2007

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SmoothNonsmooth


where M51 (t0 ) = 4A1 A5 γ

Z∞

r˙ (t) cos ω (t + t0 ) dt

−∞

= −4A1 A5 γ

Z∞

sech t tanh t cos ω (t + t0 ) dt

−∞

and M52 = 4A1 A5 (δ1 + δ2 )

Z∞

x˙ 01 (t) r˙ (t) dt

−∞

Z∞ √ = 4 2A1 A5 (δ1 + δ2 ) sech2 t tanh2 tdt, −∞

M53 = 16A1 A5

Z∞

T (x˙ 01 − v∗ ) r˙ (t) dt

−∞

= −16A1 A5

Z∞

T (x˙ 01 − v∗ ) sech t tanh tdt.

−∞

Expression M51 (t0 ) is defined as follows M51 (t0 ) = −M511 (t0 ) + M512 (t0 ) ,

(8.110)

where M511 = 4A1 A5 γ cos ωt0

Z∞

sech t tanh t cos ωtdt = 0,

−∞

M512 = 4A1 A5 γ sin ωt0

Z∞

sech t tanh t sin ωtdt

−∞

= 4πA1 A5 γω sech

πω

sin ωt0 . 2 Substitution of the above relations into (8.110) allows us to find πω M51 (t0 ) = M512 (t0 ) = 4πA1 A5 γω sech sin ωt0 . 2 Integral M52 is √ 8 2 M52 = A1 A5 (δ1 + δ2 ) . 3

(8.111)

(8.112)

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277

Applying (8.3), relation M53 takes the form M53 = −M531 − M532 + M533 + M534 + M535 ,

(8.113)

where M531 = 16A1 A5 T0

Z∞

sgn (x˙ 01 (t) − v∗ ) sech t tanh tdt,

−∞

M532 = 16A1 A5 v∗ −

βv∗3

Z∞

sech t tanh tdt,

−∞

M533

Z∞ √ 2 = 16 2A1 A5 3βv∗ − α sech2 t tanh2 tdt, −∞

Z∞ √ M534 = 48 2A1 A5 βv∗ sech3 t tanh3 tdt, −∞

M535 Integral M531

Z∞ √ = 16 2A1 A5 β sech4 t tanh4 tdt.

√ for |v∗ | > 2 2 gives M531 = 16A1 A5 T0

−∞

Z∞

sech t tanh tdt = 0.

−∞

In the second case, i.e. for |v∗ |
A1 A4 (δ1 + δ2 ) 3 √ √ √ 1 1 . × 1 + 6c + 3c 1 − 3c ψ 1 + 1 − 3c − ψ 3 + 1 − 3c 4 4 (8.119) According to (8.89), the Melnikov function is defined in the following way M3− (t0 ) = −M31− (t0 ) − M32− ,

(8.120)

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where M31− (t0 ) = M31 (t0 ) = A4 A5 γ A˜1 cos ωt0 − A˜2 sin ωt0 = A˜ sin (ωt0 − κ) , √ 8 2 A4 A5 (δ1 − δ2 ) M32− = M32 = 3 √ √ √ 1 1 × 1 + 2c + c 1 − c ψ 1+ 1−c −ψ 3+ 1−c . 4 4 Substituting the above formulas to (8.120) we get √ 8 2 M3− (t0 ) = −A˜ sin (ωt0 − κ) − A4 A5 (δ1 − δ2 ) 3 √ √ √ 1 1 × 1 + 2c + c 1 − c ψ 1+ 1−c −ψ 3+ 1−c . 4 4 The criterion of chaos is as follows √ 8 2 ˜ A4 A5 (δ1 − δ2 ) A> 3 √ √ √ 1 1 . × 1 + 2c + c 1 − c ψ 1+ 1−c −ψ 3+ 1−c 4 4 (8.121) According to (8.91), the Melnikov function is defined as follows M4− (t0 ) = M41− (t0 ) + M42− ,

(8.122)

where M41− (t0 ) = M41 (t0 ) = A4 A5 γ A˜1 cos ωt0 − A˜2 sin ωt0 = A˜ sin (ωt0 − κ) , and

√ 8 2 M42− = M42 = − A4 A5 (δ1 − δ2 ) 3 √ √ √ 1 1 × 1 + 2c + c 1 − c ψ 1+ 1−c −ψ 3+ 1−c . 4 4

Substituting the above relations to (8.122) we have √ 8 2 ˜ M4− (t0 ) = A sin (ωt0 − κ) − A4 A5 (δ1 − δ2 ) 3 √ √ √ 1 1 × 1 + 2c + c 1 − c ψ 1+ 1−c −ψ 3+ 1−c , 4 4

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281

and the criterion of chaos occurrence is as follows √ 8 2 ˜ A> A4 A5 (δ1 − δ2 ) 3 √ √ √ 1 1 . × 1 + 2c + c 1 − c ψ 1+ 1−c −ψ 3+ 1−c 4 4 (8.123) The next Melnikov function (see (8.93)) is defined in the following way M5− (t0 ) = −M51− (t0 ) − M52− + M53− ,

(8.124)

where M51− (t0 ) = M51 (t0 ) = 4πA1 A5 γω sech

M52− = M52 Z∞

M53− = 16A1 A5

πω 2

sin ωt0 ,

√ 8 2 = A1 A5 (δ1 + δ2 ) , 3 T (−x˙ 01 − v∗ ) r˙ (t) dt

−∞

= −16A1 A5

Z∞

T (−x˙ 01 − v∗ ) sech t tanh tdt.

−∞

Using (8.3), relation M53− takes the form M53− = −M531− − M532− − M533− + M534− − M535− , where M531− = 16A1 A5 T0

Z∞

sgn (−x˙ 01 (t) − v∗ ) sech t tanh tdt,

−∞

M532− = 16A1 A5 v∗ −

βv∗3

Z∞

sech t tanh tdt = 0,

−∞

M533− = M533

Z∞ √ 2 sech2 t tanh2 tdt = 16 2A1 A5 3βv∗ − α

√ 32 2 = A1 A5 3βv∗2 − α , 3

−∞

(8.125)

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Z∞ √ M534− = 48 2A1 A5 βv∗ sech3 t tanh3 tdt = 0, −∞

√

M535− = M535 = 16 2A1 A5 β

Z∞

√ 64 2A1 A5 β sech t tanh tdt = . 35 4

4

−∞

Computing integral M531− for|v∗ | > M531− = 16A1 A5 T0

√ 2 2 one obtains

Z∞

sech t tanh tdt = 0.

−∞

√ In the second case, i.e. for |v∗ | < 2 2 one gets M531− = 16A1 A5 T0   Zt3 Zt4 Z∞ × − sech t tanh tdt + sech t tanh tdt − sech t tanh tdt , −∞

t3

t4

where t3 , t4 are defined by equations (7.1). Upon integration we have t ∞ t3 M531− = 16A1 A5 T0 sech t|−∞ − sech t|t43 + sech t|t4 = 32A1 A5 T0 (sech t3 − sech t4 ) . According to (7.1), one gets M531− = 32A1 A5 T0

r

1 2

−

q

1 4

−

v∗2 2

−

r

1 2

+

q

To sum up this part of investigation one gets  r r q q  v∗2  32A A T 1 1 1 1 − − − + 1 5 0 2 4 2 2 4 − M531− =   0 Substitution of the above formulas to (8.125) gives √ √ 32 2 64 2A1 A5 β M53− = − A1 A5 3βv∗2 − α + − 3 35  r r q q  v∗2  1 1 1 1 32A1 A5 T0 − − − + 2 4 2 2 4 − −   0

1 4

v∗2 2

v∗2 2

!

−

v∗2 2

!

for v∗

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