Nonlinear Science and Complexity
Nonlinear Science and Complexity edited by
Albert C J Luo Southern Illinois University Edwardsville, USA
Liming Dai University ofRegina,
Canada
Hamid R Hamidzadeh Tennessee State University, USA
\[p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI •
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NONLINEAR SCIENCE AND COMPLEXITY 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-13 978-981-270-436-8 ISBN-10 981-270-436-1
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TRANSACTIONS OF NONLINEAR SCIENCE AND COMPLEXITY EDITORS: Albert C.J. Luo Southern Illinois University, Edwardsville, USA
George Zaslavsky New York University, New York, USA
EDITORIAL BOARD Eugene Benilov, University of Limerick, Limerick, Ireland Maurice Courbage, Universite Paris 7, Paris, France Liming Dai, University of Regina, Regina, Saskatchewan, Canada M.H. Elahinia, The University of Toledo, Toledo, USA Maria Luz Gandarias, University of Cadiz, Cadiz, Spain Marian Gidea, Northeastern Illinois University, Chicago, USA James A. Glazier, Indiana University, Bloomington, USA Ling Hong, Xi'an Jiaotong University, Xian, China Nail Ibragimov, Blekinge Institute of Technology, Karlskrona, Sweden Zhongliang Jing, Shanghai Jiaotong University, Shanghai, China V. Kumaran, National Institute of Technology, Tiruchirappalli, India Weihua Li, University of Wollongong, Wollongong, Australia Shijun Liao, Shanghai Jiaotong University, Shanghai, China Jose Antonio Tenreiro Machado, ISEP-Institute of Engineering of Porto, Porto, Portugal Nikolai A. Magnitskii, Russian Academy of Sciences, Moscow, Russia Lev Ostrovsky, Zel Technology/NOAA ETL, Boulder CO, USA Dmitry E. Pelinovsky, McMaster University, Hamilton, Ontario, Canada Sergey Prants, Pacific Oceanological Institute of the Russian Academy of Sciences, Vladivostok, Russia Dirk Roose, Katholieke Universiteit Leuven, Celestijnenleaan, Heverlee-Leuven, Belgium Lev Shemer, Tel Aviv University, Ramat Aviv, Isreal Jian Qiao Sun, University of Delaware, Newark, Delaware, USA Pei Yu, The University of Western Ontario, London, Ontario, Canada M.V. Zakrzhevsky, Riga Technical University, Riga, Latvia
v
Organizing Committee Liming Dai (Canada) Frank Z. Feng (USA) M.H. Elahinia (USA) Nail H. Ibragimov (Sweden) G Nakhaie Jazar (USA) Zhongliang Jing (China) Shijun Liao (China) Albert C.J. Luo (USA) Lev A. Ostrovsky (USA) Sergey Prants (Russia) Victor I. Shrira (UK) Subhash Sinha (USA) Xubin Song (USA) Gazanfer Unal (Turkey) Fei-Yue Wang (USA) Trong Wu (USA) Pei Yu (Canada)
VI
Preface This volume partially contains the papers presented in the 2006 International Conference on Nonlinear Science and Complexity, which held in Beijing, China, August 7-12, 2006. This conference provided a place to exchange recent developments, discoveries and progresses on Nonlinear Science and Complexity. The fundamental and frontier theories and techniques for modern science and technology were presented. In addition, this conference provides a platform to exchange the methodology in applied nonlinear science, nonlinear modeling and intelligent computations. The conference organizers believe this conference to stimulate more research in nonlinear science and complexity. The conference focused on the following topics: • Lie Group Analysis and Applications in Nonlinear Science Complex (Nail H. Ibragimov) • Nonlinear Wave Dynamics and Patterns in Geophysical Flows (Lev A. Ostrovsky and Victor I. Shrira) • Chaotic Dynamics and Transport in Classic and Quantum Systems (Sergey Prants) • Nonlinear Dynamics, Oscillations and Stability (Albert C.J. Luo, Pei Yu, Subhash Sinha ) • Nonlinear Fluid Mechanics (Gazanfer Unal and Shijun Liao) • Dynamics in Continuous Media and Wave Propagations (Liming Dai) • Modeling and Nonlinearity in Sensors, Bio-devices, MEMS and Nano-systems (Frank Z. Feng, G Nakhaie Jazar) • Nonlinear Modeling and Control of Smart Material Systems (M.H. Elahinia, Xubin Song) • Nonlinear Modeling and Control and Intelligent Computing (Zhongliang Jing, Trong Wu) Many papers presented in this conference show excellent achievements in nonlinear science and complexity. The organizers believes this permanent record will make the work last longer and more influence. The conference organizers wish to express their deep appreciation to all the authors and reviewers. Albert C.J. Luo, Liming Dai, H.R. Hamidzadeh
IX
CONTENTS
Preface
ix
Symmetry Reduction for an Inhomogeneous Nonlinear Diffusion Equation M.L. Gandarias and M.S. Bruzon
1
Applying a New Algorithm to Derive Nonclassical Symmetries M.S. Bruzon and M.L. Gandarias
7
Turbulence and Surface Gravity Waves on the Sun N. Mole, R.Erdelyi and A.Kerekes
13
The Linear Stability of Interfacial Solitary Waves in a Two-Layer Fluid Takeshi Kataoka
24
On the Transition to Diffusion Chaos in the Kuramoto-Tsuzuki Equation Nikolai A. Magnitskii
33
Fractional Calculus for Transport in Disordered Semiconductors Vladimir V. Uchaikin and Renat T. Sibatov
43
Resonant Influence of Spatial Oscillations of a Perturbation on Motion of a Nonlinear Oscillator D.V. Makarov and M. Yu. Uleysky
54
Chaotic Transport and Fractals in a Geophysical Jet Current M.V. Budyansky and S.V. Prants
62
Experiments on Pattern Formation in Reacting Systems with Chaotic Advection T.H. Solomon, M.S. Paoletti and C.R. Nugent
72
Properties of Chaotic Advection in a 2-Layer Model of Vortex Flow D.V. Stepanov and K.V. Koshel
82
XI
Xll
Monte Carlo Simulation of Atomic Transport in a Laser Field V. Yu. Argonov and S.V. Prants
89
A Fuzzy Blue Sky Catastrophe J.Q. Sun and Ling Hong
98
Efficient and Reliable Stability Analysis of Solutions of Delay Differential Equations Koen Verheyden, Tatyana Luzyanina and Dirk Roose
109
Grazing Phenomena in a Harmonically Excited Oscillator with Dry-Friction on a Sinusoidally Time-Varying, Traveling Surface Albert C.J. Luo, Brandon Gegg and Steve S. Suh
121
Complex Dynamics in the Trajectory Control of Redundant Manipulators B.M. Fernando Duarte, Maria da Graca Marcos and J.A. Tenreiro Machado
134
Under-Damped Oscillator with Cross-Correlated Colored Noises Input Modulated by Periodic Signal Yanfei Jin and Haiyan Hu
144
A 3D Turning Model for the Interpretation of Machining Stability and Chatter Steve Suh and Achala Y. Dassanayake
149
Nonlinear Dynamics and Optimization of Spur Gears Francesco Pellicano, Giorgio Bonori, Marcello Faggioni and Giorgio Scagliarini
164
Global Bifurcation and Chaotic Behavior Research of a Truncated Conical Shallow Shell Rotating Around a Single Axle Changping Chen, Liming Dai and Simon Y. Sun
180
Xlll
Technology of Magnetic Flux Leakage Signal Detection Based on Scale Transformation Stochastic Resonance Taiyong Wang, Shiguang Hu, Yonggang Leng, Ying Zhang and Li Zhao
187
Periodic Motions and Bifurcations of Vibro-Impact Systems Near a Strong Resonance Point Guanwei Luo, Yanlong Zhang, Jiangang Zhang and Jianhua Xie
193
On the Nonlinear Dynamic Characteristics of Truck Rear Full-Floating Axle T.N. Tongele
204
Nonlinear Vibration Analysis of an Unbalanced Rotor on Rolling Element Bearings Due to Cage Run-Out C. Nataraj and S.P. Harsha
213
Perturbation Analysis of Nip Contact Delay System L. Yuan and V.-M. Jarvenpaa
222
Vibration Signal Analysis and Feature Extraction Based on Wavelet Energy Spectrum Yongqiang Li and Jie Liu
231
Experimental Analysis of Cumulants Scaling Properties in Fully Developed Intermittent Turbulence Francois G. Schmitt
240
Exact Solutions of a Second Grade Fluid in a Porous Medium 5. Islam and C.Y. Zhou
247
Approximate Analytic Solutions of Stagnation-Point Flows in a Porous Medium V. Kumaran and R. Tamizharasi
259
Lump Solutions of 2D Generalized Gardner Equation Y.A. Stepanyants, I.K. Ten and H. Tomita
264
XIV
Wood Fracture Behavior Simulation Using Stochastic FEM Mingbao Li, Jun Cao and Shiqiang Zheng
272
A Novel Numerical Computation Based on FEM for Wood Temperature Distribution Field Liping Sun, Mingbao Li and Shiqiang Zheng
282
Elastic Wave Field in a Porous Medium Fully Saturated with a Newtonian Viscous Fluid Liming Dai and Guoqing Wang
292
Ball Bearing Remnant Life Prediction of Induction Motor - Impact Inspection Approach Lanfu Luo, Liming Dai, Mingzhe Dong and Lisa Fan
301
Nonlinear Airway Smooth Muscle Stiffness Changes Using Artificial Neural Network A.M. Al-Jumaily, L. Chen and Y. Du
308
In-Plane Free Vibrations of Compound High Speed Rotating Disks Hamid R. Hamidzadeh
314
Self-Excited Vibration and Stability Analyses for Axially Traveling Strings under Steady Wind Loadings Yuefang Wang and Lefeng Lu
322
Nonlinear Dynamical Analysis of Micro Self-Acting Gas Journal Bearing Hai Huang, Guang Meng and Long Liu
329
Parametric Modelling of Microresonators Dynamics with Considering Thermal Effects M.A. Tadayon, H. Sayyaadi and G.N. Jazar
338
Design and Analysis of General and Travelling Wave Dielectrophoresis T. Kinkeldei and W.H. Li
346
XV
Self-Propulsion of a Capsule on a Lubricated Surface Driven by Piezoelectric Actuators Z.C. Feng
353
Decomposition-Modeling Smart Struts for Vehicle Suspension Development Xubin Song
365
A Direct Model Reference Adaptive Control System Design and Simulation for the Vibration Suppression of a Piezoelectric Smart Structure Tamara Nestorovic Trajkov, Heinz Koppe and Ulrich Gabbert
375
A New Methodology of Modeling a Novel Large-Scale Magnetorheological Impact Damper Yancheng Li, Jiong Wang and Linfang Qian
382
Nonlinear Characteristics of Magnetorheological Damper under Base Excitation Yancheng Li, Jiong Wang and Linfang Qian
388
Analysis of Distributed Micro-Control Actions on Free Paraboloidal Membrane Shells H.H. Yue, Z.Q. Deng and H.S. Tzou
394
MATLAB Simulation of Semi-Active Skyhook Control of a Quarter Car Incorporating an MR Damper and a Fuzzy Logic Controller W.H. Li, R.S. Ujszaszi, B. Liu, X.Z. Zhang, P.B. Kosaish and X.L. Gong
405
An Effective Permeability Model to Predict Field-Dependent Modulus of Magnetorheological Elastomers X.Z. Zhang, W.H. Li, B. Liu, P.B. Kosasih, X.L. Gong, X.C. Zhang and P.Q. Zhang
412
XVI
The Simulation of Magnetorheological Elastomers Adaptive Tuned Dynamic Vibration Absorber for Automobile Engine Vibration Control X.C. Zhang, X.Z. Zhang, W.H. Li, B. Liu, X.L. Gong and P.Q. Zhang
418
Spatial Signal Characteristics of Shallow Paraboloidal Shell Structronic System H.H. Yue, Z.Q. Deng and H.S. Tzou
425
Investigation of Mechanical Characteristics of EAPap Actuator under Ambient Conditions Lijie Zhao, Yuanxie Li, Heung Soo Kim, Jaehwan Kim and Chulho Yang
434
On the Modeling of Ferromagnetic Shape Memory Alloy Actuators H. Tan and M.H. Elahinia
442
Dynamic Performance Analysis of Nonlinear Tuned Vibration Absorbers Jeong-Hoi Koo, Amit Shukla and Mehdi Ahmadian
454
Application of Magnetorheological Elastomer to Vibration Control H.X. Deng and X.L. Gong
462
Emulation and Analysis of the Response Time of the Magnetrheological Fluid Damper N. Shen and J. Wang
471
On Dynamic Transmitting Property of Circular Plate MR Clutch Chongzhi Guo, Jiangchuan Guo, Yu Guo and Ziyang Ma
476
Web-Based Traffic Noise Control Support System for Sustainable Transportation Lisa Fan, Liming Dai and Anson Li
484
xvii Visual Tracking and Recognition Using Adaptive Probabilistic Appearance Manifold in Particle Filter Yanxia Jiang, Hongren Zhou and Zhongliang Jing
491
Region-Based Infrared and Visible Dynamic Image Fusion Bo Yang, Gang Xiao and Zhongliang Jing
498
Modeling of Complex Systems for Diagnosis Xudong W. Yu
505
Ada Programming for Solving Nonlinear Equations Trong Wu
519
The Study of 3-D Surface Reconstruction in Digital Image Processing Zhong Qu
531
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Symmetry reductions for an inhomogeneus nonlinear diffusion equation M.L. G a n d a r i a s , M.S. Bruzon Departamento de Matematicas, Universidad de Cadiz, POBOX 40 11510 Puerto Real, Cadiz, Spain E-mail
[email protected] Abstract In this work we derive symmetry reductions and exact solutions for an inhomogeneous equation that model fast diffusion. We find the connection between classes of nonclassical symmetries of the equation and of an associated system. These symmetries allow us to increase the number of solutions. Some of theses solutions are unobtainable by classical symmetries. Keywords: Partial differential equations; Nonclassical symmetries; Potential symmetries
1 Introduction The diffusion processes appear in many physics processes such as plasma physics, kinetic theory of gases, solid state, metallurgy and transport in porous medium [1, 12, 15]. In this work we consider a mathematical model for diffusion processes which is the generalised inhomogeneous nonlinear diffusion equation f(x)ut
= [g(x)u"ux]x.
(1)
In [15] P.Rosenau presented a number of remarkable features of the fast diffusion processes: for f(x) = 1, g(x) = 1 and — 2 < n < —1, the family of fast diffusion (1) coexists with a subclass of superfast diffusions where the whole process terminates within a finite time. The special case with n = — 1 emerges in plasma physics and reveals a surprising richness of new physic-mathematical phenomena. In (1) u(x,t) is a function of position x and time t and may represent the temperature, f(x) and g(x) are arbitrary smooth functions of position and may denote the density and the density-dependent part of thermal diffusion, respectively. There is no existing general theory for solving nonlinear partial differential equations and the methods of point transformations are a powerful tool. One of the most useful point transformations are those which form a continuous group. Lie classical symmetries admitted by nonlinear partial differential equations (PDE's) are useful for finding invariant solutions. Motivated by the fact the symmetry reductions for many PDE's are known that are not obtained by using the classical Lie method there have been several generalizations of the classical Lie group method for symmetry reductions.
1
2 Bluman and Cole [3] developed the nonclassical method to study the symmetry reductions of the heat equation. The basic idea of the method is to require that the N order PDE A = A(x,t,u,uW(x,t),...,u(-N)(x,t)>)
=0,
where (x,t) e M2 are the independent variables, u € IR is the dependent variable and u^(x,t) the set of all partial derivatives of I order of u and the invariance surface condition £ux + Tut - <j> = 0,
(2) denote
(3)
which is associated with the vector field v = £(x,t,u)dx
+r(x,t,u)dt
+ <j>{x,t,u)du,
(4)
are both invariant under the transformation with infinitesimal generator (4). Since then, a great number of papers have been devoted to the study of nonclassical symmetries of nonlinear PDE's in both one and several dimensions. In [4, 5] Bluman introduced a method to find a new class of symmetries for a PDE. By writing a given PDE, denoted by R { i , t, u] in a conserved form a related system denoted by S{x, t, u, v} as additional dependent variables is obtained. Any Lie group of point transformations admitted by S{x, t, u, v} induces a symmetry for R{x,t, u); when at least one of the generators of the group depends explicitly of the potential, then the corresponding symmetry is neither a point nor a Lie-Backlund symmetry. These symmetries of R { i , t, u) are called potential symmetries. In [14], C. Sophocleous has classified the nonlocal potential symmetries of (1). He obtained that potential symmetries exists only if the parameter n takes the values —2 or — | and also certain relations must be satisfied by the functions f(x) and g(x). In [10], we have derived nonclassical potential symmetries for the special case of (1), with f(x) = 1 and g(x) = 1 ut = [ u _ 1 u x ] x . (5) In [13] connection between classes of nonclassical symmetries of (5), and of nonclassical symmetries of an associated system as well as some new generators have been found. The aim of this paper is to obtain nonclassical symmetries for (1) and for the associated system given by vx = / ( x ) u , (6) vt = g{x)u~xux, as well as the connection between these symmetries. These symmetries lead to new solutions, some of these solution exhibit an interesting behaviour.
2
Nonclassical symmetries
2.1
Nonclassical symmetries of t h e P D E (1)
To obtain nonclassical symmetries of (1), with n = - 1 , we apply the algorithm described in [6, 7] for calculating the determining equations. We can distinguish two different cases: In the case r ^ 0, without loss of generality, we may set r{x,t,u) = 1. The generators that we obtain can be obtained by Lie classical method consequently the nonclassical method, with T ^ 0 applied to (1) gives only rise to the classical symmetries.
3 In the case r = 0, without loss of generality, we may set £ = 1 and we get that the determining equation for the infinitesimal <j> is « ( n + 2 ) {f94>xx + 2fg'<j>x - f'gx + fg2uu + VgM^ + fg'H>u ~ f'gtu) + fg" ~ f'g'4>) ( +n0utu2 = 0. '' The complexity of this equation is the reason why we cannot solve (7) in general. Thus we proceed, by making an ansatz on the form of <j>(x,t,u), to solve (7) for n = —1. In this way we found, choosing = a(x, t)u2 + /3(x, t)u, after substituting into the determining equation and splitting with respect to u we obtain an overdetermined system for the functions a and /?. So, for equation (1) with n = — 1, we obtain the infinitesimal generator v = dx + (a(x, t)u2 + p(x,
t)u)du,
where / , g, a and /3 satisfy the system fg'a2
- fgaax
+ f2at
f'g'a
- f g"a2 + (fg' + f'g)a/3 + (f'g - 2fg')ax
= 0, + fg(ap,
- ax0 - axx) + f ft = 0, (8)
/ / % - fg0xx + f'gfix = 0, PU'g' - fg") + fiM'g - Vg') + P2f9~ + fgW, - /»..) = o. 2.2
Nonclassical symmetries of the system (6)
We now consider the associated auxiliary system (6) augmented with the invariance surface condition f«x + rvt - ip = 0,
(9)
which is associated with the vector field w = £(z, t, u, v)dx +r(x,t,u,v)dt
+ 4>(x,t,u,v)du
+ tp(x,t,u,v)dv.
(10)
By requiring both (6) and (9) to be invariant under the transformation with infinitesimal generator (10) one obtains an over determined, nonlinear system of equations for the infinitesimals £ ( x , t , u , v ) , T(x,t,u,v), (x,t,u,v), ip(x,t,u,v). When at least one of the generators of the group depend explicitly of the potential, that is if ev+rZ+l?0 (11) then (10) yields a nonlocal symmetry of (1). A nonclassical potential symmetry of (1) is a nonclassical symmetry of the associated potential system (6) that satisfies (11). We are considering r ^ 0, and without loss of generality, we set T = 1. The nonclassical method, with T / 0, applied to (6), give rise to nonlinear determining equations for the infinitesimals. If we require that £ u = ipu = 0, we obtain that 4>=-KvU2+(^v-^-^u and f(x), g(x), £(x,t,v)
+^
(12)
and il>(x, t,v), must satisfy the following equations: gtw -tfv= 0,
(13)
m
4 -fQttx
+ 2fg%x
2
+ fg^v
2
+ 2/' 9 2 £„ - fgfr + fg'? - fg^
2
f 9 H*X + f g^x
2
2
- fg^vv
= 0,
(14)
,2 2
+ ff'g tx - / s « - /VV-? + ff"g Z - f g Z - 2/VV>,* -fgipxx + / # , + f'gipx = o.
,, « (16)
We can distinguish the following cases: If £„ / 0 by solving (13) and substituting into (14), (15), (16) leads to generators for which (11) is satisfied, consequently they are nonclassical potential generators and have been considered in [11]. If £ does not depend on v, by substituting f = £ ( i , t) in (14) and (15) we obtain that
*
=
"(^I~l+ f) + f l (l + l)~fl'+e(a:'t)'
(1?)
By substituting into (12) we get that 0„ = 0. We observe that in this case condition (11) is not satisfied, consequently
v = i{x, t)dx + dt+ ({S(t) -M-£x)u
+ ^'\du + Wv
(18)
is not a nonclassical potential generator.
2.3
Connection between symmetries of the PDE (1) and of the system (6)
If we assume that £ and ip do not depend on v, the system (13-16) becomes
-9tfx-gZt + g'e = 0, f2 g2Sxx + f2g^x
+ ff'92(,x - f2gxt, - fg'M + ff"g2t - }l2g2(. + Pg^Pt = o,
(19)
-fgi>xx + fi»l>x + f'gipx = o. It is easy to check that denoting a = — -£f, /? = ^ systems (8) and (19) coincides. Consequently we can state: w = i{x, t)dx +dt+
4>(x, t)dv - (Jj-Su + ^ \ du
is a generator for system (6) if and only if
v = dx + (--(.u2 + ^-u J du is a generator for equation (1).
3
Some exact solutions
In this section we derive some exact solutions by using some generators: 1- From generator £ = kiy/x, for / = ¥%e *2 exact solution
and g =
ip = 0,
(20)
y , we obtain the similarity solution and the ODE that gives rise to the
5 2*i j
u —
k3 *4 I *4 e
r
*s
-
e
" a
T
(21)
r^.
"a
1
We observe that the solution (21), for x = l M t M _ j blows up at a parabola. 2- From generator a; + kt tor g — 1, f = exp(x) and the surface condition we obtain the similarity solution and the ODE that gives rise to the exact solution
We observe that solution (22) blows up in two straight lines x -1 + *i = 0 and x +1 = 0. 3- Prom generator £ = I,
ip = - 2 * ! tanh[fci(t + / f(x)dx)],
for
