Some problems in the theory of nonlinear oscillations

A tranalaUon of: Nel5,otorye zada!!hi t~orii llelinelnykh kolebanil.. Gosudarstevennoe IZdatel' stvo· Tekhniko:" Teo:reticheskOiLiter~tury;. Moskva. 1956. T;ransla.ted by the Language'I'ranslatien Se~v1ce,Cfevelandf Ohio. •

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BIBLIOTECA FIVI Y ME

IIIIIIIIIIIIII~I~~~~~~I~I~~~~IIIIIIIIIII SOME PROBLEMS IN THE THEORY OF NONLINEAR OSCILLATIONS By I.G. Malkin

BOOK 1

State Publishir.g House of Technicil and Theoretical Literature Moscow, 1956

i

PREFACE At the present time the so called gmethod of small parameter" is widely applied in the theory of nonlinear oscillations. In the present book this method is systematically applied to the solution of a number of problems in the theory of nonlinear oscillations. Herein is set forth the theory of periodic and almost periodic oscillations of quasilinear systems with one and several degrees of freedom, the ,theory of periodic and almost periodic oscillations of systems approximating arbitrary nonlinear system, and the theory of free and forced oscillations ( periodic and almost periodic ) of quasiharmonic systems, i.e. systems described by linear differential equations with periodic coefficients. A good deal of attention is at the same time devoted to the question of the stability of the oscillations. Notwithstanding the rather large size of the book and the wide range of problems which it treats, the book in no wise pretends to an exhaustive presentation of all the problems to which the method of small parameter m~ successfully be applied ( this is reflected in the title of the book). In particular, no consideration has been given to the establishment processes, the slowly varying processes in nonlinear systems and other problems that reduce to finding the solutions of the equations of motion in a finite interval of time. All the problems considered in this book are set forth with sufficient mathematical rigor. In particular, much attention is given to the proofs of the existence of periodic or almost periodic solutions of the equations of the oscillations. The fundamental goal however which the author set himself is the presentation of practical procedures for computing the oscillations. To this question therefore the book devotes its primary attentio~ All the methods discussed are accompanied by examples with detailed computations. In the book is also given the solution ,of a number of concrete physical and technical problems. These too however serve principally for illustrating the computation methods presented. The author has striven to make the book accessible to as wide a circle of readers as possible, which accounts for the somewhat concentrated method of presentation that is used. In particular, the reading of the first chapter requires a very limited amount of mathematical preparation. In order to give this chapter a completed form there is also presented in it an elementary discussion of the

iii

stability problem, the more thorough and detailed presentation of which is found in the third chapter. Similar repetitions are occasionally encountered also in other parts of the book. The author assumes that this shortcoming is compensated by the above indicated considerations of the book· s methodical character. In the writing of th~ book frequent use was made of the authorts earlier published book, "The Methods of Lyapunov and Poincare in the Theory of Nonlinear Oscillations: The author desires to express his thanks to S.N. Shimanov and N.G. Bulgakov for the help he has received in reading the proofs. I. Malkin

CONTENTS BOOK 1

PREFACE CHAPTER I. 1.

Page i

QUASILINEAR OSCILLATIONS WITH ONE DEGREE OF FREEDOM Idea of the Method of Poincare. Small

1

Parameter ...•..•...•.........•....•...•.. 1

2.

Oscillations of a Nonautonomous System Not at Resonance ••••••••••••••••••••••••• 3 3. Oscillations of a Nonautonomous System at Resonance. Conditions for the Existence of a Periodic Solution ••••••••• 9 4. Oscillations of a Nonautonomous System at Resonance. Computation of a Periodic Solution ••••••••••••••••••••••• 16 5. Application to the Theory of a Regenerative Receiver •••••••••••••••••••••••••••••••• 20 6. Application to the Problem of Duffing ••••• 27 7. Resonance of the n-th Kind •••••••••••••••• 34 8. Examples of Resonance of the n-th Kind •••• 37 9. Autonomous Systems. Conditions for the Existence of Periodic Solutions ••••••••• 42 10. Computation .of Periodic Solutions for an Autonomous System ••••••••••••••••••••••• 46 11. Phase Plane for the System Considered in the Preceding Section. Limit Cycles. Self-Oscil1ations •••••••••••• • •• •• •••••• ·52 12. Self-Oscillations of a Tube Generator ••••• 62 13. Problem of the Stability of Periodic Motions ................................. 67 14.· Stability of the Periodic Solutions of Autonomous Systems. Application to the Theory of the Self-Oscillations of a Tube Generator •••••••••••••••••••••••••• 74 15. Stability of the Periodic Motions of Nonautonomous Systems ••••••••••••••••••• 78 16. Stability of the Oscillations Considered in Sections 5 and 6 ••••••••••••••••••••• 85 17. Systems with Nonanalytic Characteristic of the Nonlinearity •••••••••••••••••••••••• 91

Page

[APTER II. 1. 2.

3.

PERIODIC OSCILLATIONS OF QUASILINEAR SYSTEMS WITH MANY DEGP..EES OF FREEDOM 100 Linear Equations with Constant Coefficients •.• " .••. " •........•...•..........• 100 Periodic Solutions of Homogeneous Linear Systems with Constant Coefficients •••••••• 10B Conjugate Systems. Reduction of Linear Equations with Constant Coefficients to Canonic al Form •••••••••••••••••••••••••••• 109

4.

5. 6.

Periodic Solutions of Nonhomogeneous Linear Systems with Constant Coefficients •••••••• 114 Oscillations of Nonautonomous Systems not at Resonance •.•• " •••. ~, ...••..•.••.•.•..••.••• 123 Oscillations of Nonautonomous Systems at Resonance ••••••.•

7.

ot.

.........................

128

Practical Method of Compu~ting the Periodic Solutions of Nonautonomous Systems at Resonance in the Case of Analytic

Equations ••••.•...•..••. " ..•...•.••..•••• 135

B.

Practical Method of Computing the Periodic Solutions of Nonautonomous Systems at Resonance in the Case of Nonanalytic Equations ••••.•••.•...••••...•.••• " •. " ••• 141

9. 10.

Proof of the Convergence of the Successive Approximations ••••.•••.••••••••••.•.••••• 147 Computation of the Periodic Solutions of Nonautonomous Systems at Resonance in the Singular Case ••• " .••.. " .. " .•• "." ....••••• 156

11. 12.

Oscillations of Autonomous Systems ••••••••• 165 Computation of the Periodic Solutions of Autonomous Systems in the Case of Analytic Equations •••••.•••. " .•• " •• " ••••.•••.••••• 172

13.

Self-Oscillations in Two Coupled Circuits ••• 176

IAPTER III. STABILITY OF OSCILLATIONS 1. Statement of the Problem. Equations in

182

Variations ••••..........•.••.•.•.••.••.•. 182

2.

3.

Linear Equations Characteristic Analytic Form of Equations with

with Periodic Coefficients. Equation •••••••••••••••••• 186 the Solutions of Linear Periodic Coefficients ••••• 191 vi

Page

'+. 5. 6.

7.

Proof of the Proposition of the Preceding Section ••••••••••••••.•••••••••••••••••• 19L+

Reduction of Linear Equations with Periodic Coefficients to Equations with Const~~t Coefficients ••••••••••••••••••• 203 Theorem of Lyapunov on the Roots of the Characteristic Equations of Conjugate Systems. Fundamental Equation of the Reduced Sys tem •••••.•••••••••••••••••••• 208 Some General Theorems on Stability of Motion •••••••••••••••••••••••••••••••••• 210

8.

9. 10.

11. 12.

14. •

Theorem of Lyapunov on the Characteristic Equation of Canonical Systems ••••••••••• 215 Theorem of AndroLov and Vitt on the Stability of the Periodic Motions of Autonohlous Systems ••••..•••••••••••••••• 218 Approximate Computation ot the Roots of the Characteristic Equation by the Method of Expanding in Powers of a Parameter •••• 220 A Second Method of Approximate Computation of the Roots of the Characteristic Equation •.•.•...•.••.••••.••••••••• '••• •• 225 Application to the Problem of the Stability of Oscillations of Quasilinear Systems ••.•..•..•.••.•••..•.•••••••••••• 235 Application to the Case of EquB-tions Analytic with Respect to the Parameter ••• 241 Stabi1.i ty of the Self-Oscillations in T'vvo Inductively Coupled Circuits •••••••••••• 246 . 1 Cases •••....••..••••.•••••.•• ~~ ric/, S ome S pecla BOOK 2 IV.

1. 2.

ALMOST PERIODIC OSCILLATIONS OF QUA-BILINEAR SYSTEMS 260 Statement of the Problem. Basic Concepts ••• 260 Almost Periodic Solutions of Nonhomogeneous ~inear

4.

Equations •••••••••.•.•••.••••••••• 263

Certs,..ip. Properties of the Problem of Almost Periodic Oscillations. Small Divisors ••• 269 Almost Periodic Oscil~ations of Nonautqnomous Systems in the Absence of critical Roots of the Fundamental Equation, 273 Transformation of Krylov and Bogolyubov •••• 275

Page 6. Almost Periodic Solutions of Standard Systems ................................. • 278

7. Almost Periodic Oscillations of Nonautonomous Systems for the Case of the Presence of Critical Roots of the Fundamental Equation. Special Case ••••• 282 8. Almost Periodic Oscillations of Nonautonomous Systems for the Case of the Presence of Critical Roots of the Fundamental Equation. General Case ••••• 285 9. Proof of the Convergence of the Successive Approximations ••••••••••••••••••••••••••• 297 10. Practical Methods of Computing the Almost Periodic Solutions Considered in Sec. 8 •• 303 11. Stability Criteria ••••••••••••.••••••••••• 311 12. Application to the Problem of the Forced Oscillations of a Regenerative Receiver •• 3l4 13. Analysis of the Equations Determining the Parameters of the Generating System. Resonance and Nonresonance Frequencies ••• 320 14. Certain Simplifications of the Computation of Almost Periodic Solutions in the Presence of Nonresonance Frequencies ••••• 326 15. Oscillations with Nonresonance Frequencies. Properties of the First Approximation •••• 327 16'. Oscillations with Nonresonance Frequencies. Properties of Exact Solutions •••••••••••• 337 17. Practical Procedures for Obtaining Oscillations with Nonresona.."1ce Frequencies •••• 343 18. :Ei:::camples •••••••••••••••••••••••••••••••••• 347

19. Averaging Principle ••••••••.••••.••••••••• 359 CHAPTER V. QUASIHAlThIONIC SYSTEMS 364A. FREE OSCILLATIONS OF QUASIHARMONIC SYSTEr·:TS 1. Parametric Resonance. Statement of the Pro b 1 em ............. ... "................ 364 2. Regions of Stability and Instability for Equations of the Second O~der •••••••••• 370 3. Practical Method of Determining the Regions of Stability and !nstability For Equations of the Second Order ••••••• 376

Page 4.

5. 6.

Examples of Method of Problems of Canonical

the Application of the the Preceding Section ••••••••• 386 Parametric Resonance for Systems with Many Degrees of Freedom •••••••••••••••••••••••••••••••• 395 Regions of Simple Parametric Resonance for Canonical Systems with Many Degrees of Freedom •••••••••••••••.•••••••••.••• 401

7.

8.

A Second Method of Determining the Regions of Instability of Parametric Resonance for Canonical Systems. Regions of Combined Resonance ••••••••••••••.•••••• 412 Example. Theorem of M.G. Krein •••••••••• 416

B. FORCED OSCILLATIONS OF QUASIF~m~ONIC SYSTEMS 9. Conditions for the Existence of Almost Periodic Solutions of Systems of Linear Equations with Periodic Coefficients •••• 422 10. Conditions of Resonance. Form of the Forced Oscillations ••••••••••••••••.••• 427 11. Dependence of the Forced Oscillations on a Parameter •••••••••••••.•••••••••••••• 431 12. Practical Method of Computing the Forced Oscillations ••..•...•.•.•••.•••.•••••.• 434

13.

Examples of Computation of Forced Oscillations •..•....................•....... 441

CHAPTER VI.

APPROXIMATIONS TO ARBITARY NONLINEAR 447

S~TDS

1. 2.

3. 4.

Periodic Solutions of Nonautonomous Systems in the Case of an Isolated Generating Solution ••••.•••..•••...•••• 447 Periodic Solutions of Nonautonomous Systems in the Case of a Family of Generating Solutions ••.••••..•.•..•.••• 451 Case of AnalytiC Equations ••••..••••••••• 461 Practical Computation of the Periodic Solution ••••.•••.••...•••.•••••••.••••• 466

5. 6.

Stability Criteria of the Periodic Solutions Considered •••.•..•.•.•••.•.•• 470 Nonautonomous System with One Degree of Freedom Approximating a Conservative System ••••••.•.••••••••••••••••..•••••• 474 ix

Page

7.

Stability Criteria of the Periodic Solution Considered in the Preceding Section ••..•..•••.•.•.••.•••••....••••• 483

8.

9. 10. 11. 12.

CHAPTER 1. 2.

3. 4.

5. CHAPTER

Periodic Solutions of Autonomous Systems •• 487 Periodic Solutions of Autonomous Systems. Equations for the Parameters of the Generating Solutions ••••••••••••••••••. 493 Almost Periodic Solutions of Nonautonomous Systems in the Case of an Isolated Generating Solution ••••••..•••••••••••• 497 Almost Periodic Solutions of Nonautonomous Systems in the Case of a Family of Generating Solutions •••••..•••••••••••• 504 Case in which the Number of Parameters of the Generating Solution is Equal to the Order of the System •••...•.•••••.•••••• 514 VII. LYAPUNOV SYSTEMS 519 Statement of the Problem ••••••.••.••••..• 519 Periodic Solutions of Lyapunov System •••• 525 Practical Method of Computing the Periodic Solutions of Lyapunov Systems •••••.•••• 530 Some Properties of the Periodic Solutions of Lyapunov Systems •••••••.••..•...•••. 538 Principal Oscillations of Conservative Sys t ems •••••••••••••••••••••••••••••••• 54--3 VIII.

APPROXIMATIONS TO LYAPUNOV SYSTEl'AS 54--5

Generating Solutions ••.••.•••••.••...•••• 545 The Periodic Solu'tion { Xs (o)} •••••..••.••• 549 3. Periodic Solution for Resonance •••••••••• 550 4. Practical Method of Computing the Resonance Solution •.•...•.••••••...•••• 560 5. The Periodic Solution {x~p)J •••••....•••• 564 6. Stability Criteria ••......••••..•.....••• 568 7. Application to Duffing s Problem •••••••• 571 8. Example of Determination of Subharmonic Oscillations ••••••••..••.•••••.•••.•••• 581 AUTHOR IND.l!!:X. •••••••••••••••••••••••••••••••••••••• 585 SUBJECT INDEX. ••••••••••••••••••••••••••••••••••••• 587 1. 2.

x

CHAPTER I QUASILINEAR OSCILLATIONS WITH ONE DEGREE OF FREEDOM 1. Idea of the Method of Poincare". Small Parameter We shall consider physical systems described by differential equations of the form (s

1, 2, ... , r).

(1.1)

In this chapter we shall assume that if the right hand sides of these equations contain the time t explicitly they are continuous periodic functions of the latter. We shall assume the period of these functions to be equal to 2n, an assumption which can always be satisfied by a suitable choiqe of the unit of time. We shall not exclude from consideration those cases for which the right hand sides of (1.1) do not contain the time explicitly. In these cases we shall say that the system under consideration is AUTONOMOUS. Systems described by equations containing t explicitly we shall call NONAUTONOMOUS The problem consists of obtaining the periodic oscillations of the system. To these oscillations correspond the periodic particular solutions of equations (1.1), i.e. such solutions

of these equations for which fs(t) are periodic functions of t. It is assumed here that the general solution of equations (1.1) is not known. In order to solve the problem posed it is natural to m~~e use of a device, widely applied in practice, which consists of the following. In the majority of practic~1 problems it is possible to separate out from the right hand sides of the investigated differential equations a certain group of terms which can be considered as small in comparison with the reaaining leading terms. Assuming that these terms do not have essential significance for the problem considered (whether it be the problem of periodic solutions or some other 1

problem ) they are rejected and in this way a simpler system of equations is obtained. Having solved the problem for the simplified system we either restrict ourselves to the obtained solution or, taking it as the first approximation, we return to the initial equations and apply to them some special method of successive approximations that is specially worked out for the given problem. I

This is the device that underlies the method of Poincare for obtaining periodic solutions. In order to give the problem an exact mathematical formulation Poincare introduced into consideration the so-called Il small parameterll • He assumed that the right hand sides of (1.1) depend not only on the variables t, xl, ••• ,x r ' but also on a certain parameter ~, and are analytical functions of this parameter for its sufficiently small values. With this assumption, taking ~ to be sufficiently small, equations (1.1) can be represented in the form dxs _ V/lll(1 ) de - . i s , Xl' " ' , Xr

+11

XU) (

s

t,

Xl' ••• , X T

+fl2X~2)(t, Xl' •••• X,)

where the magnitudes The expressions

+ ...

)

+

(s= 1. 2, ... , r),

(1.2)

x~o), X~l), ••• do not depend on~.

represent the set of small terms in question. Rejecting these small terms in equations (1.2) we obtain the simplified system of equations d~ -x

dt 2

+n-B=O

and the initial conditions \ (0) = J,

.

1(0)-=0,

B(O)=O,

.

1J(0)=1,

whence we find . 1= cus lit.

Hence

.

B

lA)=[A]=[B]

and equations

1 .

=~smllt n .

.

[B1==0,

(3.6) on the basis of (3.5) assume the form

The problem thus reduces to the determination of the conditions under which equations (3.7) can be solved for ~l and ~2. It is here a question of a solution of these equations for which the magnitudes ~l and ~2 become zero for ~ = 0, since for ~ = 0 the function (3.5) must reduce to the generating solution xo(t). Equations (3.7), in contrast to the corresponding equations in the nonresonancp case, do not have linear terms with ~l and S2. As a result, the functional determinant of these equations with respect to f3 l and ~2 becomes zero for ~l = 2 = ~ = 0, and thE~ question of the solubility of these equations requires special investigation •

a

.. ..

We can, first of all, reduce these equations by ~. But then the new equations obtained in this way will not be satisfied for ~1 = ~2 = ~ ~ 0 on account of the presence of the free termS CeJ and [eJ in them. Hence, in order that for the solutions ~~(~) of these equations our required .... conditions Pi(O) = 0 be satisfied, it is necessary first of all to require that the above-mentioned free terms reduce to zero. We thus obtain the two following conditions:

.

[C]=O,

(3.8)

[CJ=O.

Let us examine these conditions more closely. For this purpose let us find the coefficient e. To do this we substitute the solution (3.5) in equation (3.1) and equate the coefficients of ~ to the first power. We thus obtain: (3.9)

Moreover, the initial conditions (3.4) give C (0)

.

(3.10)

C (0) = O.

The solution of equation (3.9) for the initial conditions (3.10) is, as is known, of the form

,

C=..!... ( F("C, .To (.),

3

11

;o(~), O)sinn(t-"C)dt.

o

whence

• i'= ~ F(-~, .Til ("C), ;0 ("C), O)t'osn(I-"C)d"C. o

Substituting in (3.10) and takin~ (3.3) into account we find that for the equations (3.7) to have the form of solutions we require it is necessary that the two following conditions be satisfied: 2",

P(M o. No)

=~

)

F("C, J1!ol.:osll't+Nosillll't+p("C),

o

.M oil sin Wt +.IV011 co!' It "C + 'P (,). O} ~ill Il~ d"C 2'1\

Q (Mo. No)

= SF ("

M 0 (:05 1l"C..,.. N. sin Il~+? ("C).

0,

\ r

o - Moll sin

liT

+ A'oll COS m+ 9('t), 0) cns w: d"C = O. J 12

(3.11)

These conditions are not satisfied, generally speaking, for any values of Mo and No' but are equations serving to determine these magnitudes. Consequently there will not to any periodic ~ol~tion of. the generating equation (3.2), correspond. a per10d1C solut10n of the initial equation (3.1). Suc~ solut1ons.can correspond only to those generating solut10ns for Wh1Ch the constants M and N satisfy equationQ (3.11). 0 0 . ~ Let us assume however that the constants M and N in o 0 the ~e~erating solution are actually chosen according to cond1t1ons (3.11). Then equations (3.7) assume the form

cjI: = [~] ~l + [~] ~2 + [~J ~ + ... = 0, 4'~ == [D] ~l

+ [E] ~:! + [F] p. + ... = O.

I

(3.12)

For the functional determinant of these equations with respect to PI and P2 for PI = P2 = ~ = 0 we obtain the expression

I

[D] [E] [D] [i'] .

I

(3.13)

If this expression is different from zero, then on the basis of the theorem on implicit functions equations (3.12), and therefore also equations (3.7), have one and only one solution ~i = ~i(~) for which ~i(O) = 0, and this solution will be analytic with respect to~. Substituting this solution in the function x(t, PI' P2' ~) we obtain a periodic solution of equation (3.1),and this solution is analytic with respect to ~. Thus,for each gen~rating solution for which Mo and N0 satisfy equations (,3.11) and for which the ~agnitude (3.13) is different from zero there exists one and only one periodic solution of the initial equation (3.]) and this solution will be analytic with respect to ~. Let us now consider the computation of the magnitude (3.13). For this we must determine the coefficients D and E in the expansion (3.5). To determine these coefficients we substitute the series (3.5) in equation (3.1) and equate in both sides of this equation the coefficients ~1~ and ~2iJ.·

Let F'lt. (t, PI' ~2' 11) denote the expression to which t!le function F(t, x, ~) :reduces after replacing the

x,

1

magnitude x by the series (3.5). Expanding this expression into a Maclaurin series we can write:

where the parentheses denote that the derivatives are computed for the generating solution, i.e. that after differentiation it is necessary in the derivatives of the function ~ to put ~l = ~2 = ~ = 0, and in the derivatives of the function F to put x = x o , ~ = o. Using (3.14) we readily find that the coefficients D and E satisfy the differential equations

of )A+ (a:)\ ...1= (~F ') cos nt -n (a:) sin nt, ax ox ox tf.2E + n E = (OF) B + (a:) B= ( ~F) ~ sin nt + (0:) eosnt. ox ax ox d2~ +n2D = (

x •

dt

2

dt 2

uX

1&

\

For the initial values of these coefficients we obtain from (3.4)

.

.

D(O)= D(O)= E(O) = E(O)= O.

Whence we find t

D =~ 1&

",

C {( aF ) cos 1l't _ n lo ox t='t

t

D= \~ {( ~F)_ ax

cosn't-n

t-'t

o

(a:) ax,

sin n't} sin n (t - 't)d't, t='t

(~~) Sinn't}cOSIt(t-"C)d't ox t="

and therefore 21'1:

[D)

=_ 21'1:

{r

1 \ aF) _ C051l"; _ n n J \. ax t - t o

iJF)

[D]= \ {( )o \.. ox

cosn't-n t='t

(a:) ax

(iJ:) ax

sin ",";} sin n't cI't. t='t

sin n't} cosn'td't. t=T

Comparing with (3.11) we readily satisfy ourselves that [DJ =

_.! JIP

•

n iJMo '

iJQ

iD] = iJMo •

(3.15)

Similarly we obtain (3.1())

and therefore

Thus, the condition that the magnitude (3.13) does not reduce to zero is ~quivalent to the condition that the functional determinant of equations (3.11) with respect to the unkovms Mo and No does not reduce to zero. Let the equations

m magnitudes xl, ••• ,xm satisfy the (i = 1, ... I In).

m (3.17)

By analogy with the case of a single equation with a single unkown we shall say that the magnitudes ml, ••• x m form a simple solution of equations (3.17) if they satisfy these equations and if at the same time the conditions are satisfied IJ(fh • ··.Im) =F 0 -.. d(xlr •..• xm)

•

The results we have obtained can be formulated in the form of the following fundamental proposition: IN ORDER THAT THERE CORRESPOND TO THE GENERATING SOLUTION (3.3) A SOLUTION OF Th~ INITIAL EQUATION (3.1) IT IS NECESSARY THAT THE CONSTANTS Mo AND No SATISFY EQUATIONS (3.11). TO EACH SIMPLE SOLUTION OF THESE EQUATIONS THERE ACTUALLY CORRESPONDS FOR ~ SUFFICIENTLY S~~LL A PERIODIC SOLUTION OF EQUATION (3.1), AND THIS SOLUTION WILL BE ANALYTIC WITH RESPECT TO ~.

15

From this it is seen that the resonance case differs sharply from the case of nonresonance. In the nonresonance case both the generating equation and the initial equation have one and only one periodic solution. Hence, in the nonresonance case there exists a complete correspondence between the initial and simplified equation. In the resonance case, however, no matter how small the parameter ~ may be, that is, however little the initial and simplified equations differ from each other, there exists a sharp qualitative difference between them. Whereas the generating equation possesses an infinite number of periodic solutions forming a continuous family, the initial equation possesses in general only a finite number of periodic solutions. As we shall see below, we shall always encounter. this circumstance each time that the generating function admits a family of periodic solutions that depend on one or several parameters. 4. Oscillations of a Nonautonomous System at Resonance. Computation of Periodic Solution We shall now show how practically to find the periodic solution of equation (3.1) the existence of which was established in the preceding section. Thus, let us assume that in the generating solution

(3.3) the constants Mo and No satisfy the equations (3.11)

and are simple solutions of tnese equations. Then, as was shown, the required periodic solution of the initial equation (3.1) will exist and will be analytic with respect to ~. We can therefore seek to obtain this solution in the form of the formal series (4. t)

where Xo is the generating solution and Xl' x 2 ' ••• are certain unkown periodic functions of t of period 2n. To determine these functions we substitute the series (4.1) in equation (3.1) and equate the coefficients of like powers of~. We first of all obtain (4.2)

Since n is an integer, equation (4.2), in contrast to the corresponding equation in the nonresonance case, either has no periodic solutions at allor, on the contrary, It:;

all its solutions are periodic of the period 2n. In fact, let co

F(t, x o'

;0'

0)=

a~1

+

~

(a llll cosmt+ bill I sin mt)

(4.3)

m=1

be the Fourier expansion of the right hand side of equation (4.2). To each component of this expansion there corresponds a periodic component in the solution of equation (4.2) of the form ann

('os lilt + b",. sin mt n% -/ll'.:

An exception is formed only by the component containing the

n-th harmonic. To this component in the solution of equation (4.2) there corresponds the periodic term

;n (a"'l Sill III -

bill cos nl).

Hence, in order that equation (4.2) admit a periodic solution it is necessary and sufficient that the equations be satisfied

or, in developed form, 2K

~

o

2K

F(t, ;}.·o'

;'0' O)cosnldt= ~ F(I, xo' ;0' O)sinnldl=O. 0

Equations (4.4) agree with equations (3.11) which according to the choice of MO and N are satisfied. Hence equation (4.2) for xl admits a pa~ticular periodic solution. But then also the general solution of this equation, determined by the formula x --~, ~ alllicosml+b"usinmt

. 1-

2n2

T

..c.J

n~-mz

+ '1/ 1 cos nl + ~T 1 sm nt, •

1

H'

m*n

where M and Nl are arbitrary constants, will likewise be periodi! of period 2x. We now proceed to the determination of x 2 • For this ~agnitude there is obtained an equation of the type (4.2)

17

(4.4)

but with a different expression on the right hand side. This expression will depend on xl and therefore will contain the arbitrary constants Ml and Nl • Like equation (4.2), the equation for x 2 will admit a periodic solution only in the case that the coefficients of cos nt and sin nt in the Fourier expansion of its right hand side are equal to zero. This may be obtained by a suitable choice of the constants Ml and Nl • We shall show how this can be done. The equation for x i ( 1 = 2,3, ••• ) is of the form

(4_5)

where the parentheses have,the same meaning as in (3.14)9 that is, they indicate that the derivatives are computed for the generating solution, and Fl is an integral rational function of x o 'xl , ••• ,x i _ 2 • We explicitly wrote out the terms containing x i _ l since they are required for what follows. Let us assume that all the functions xl, ••• ,xi _ l have already been computed and came out periodic. These functions are of the form

Xj = 'Pj (t)

where

o/j

+M

J

cos nl + "'-j sin lit

(j = 1. 2, ... , i-I),

are certain periodic functions of the time and Mj

and Nj are constants. We shall assume that the constants Ml, ••• ,Mi_2,Nl, ••• ,Ni_2 have already been determined while the constants Mi _ 1 and Ni _ l still remain undetermined. Then, equating to zero the coefficients of cos nt and sin nt in the Fourier expansion of the right hand side of equation (4.5), we obtain the two equations:

2ft

Mi - t

~ {( ~~ )cosnt-n(::) Sinnt}COsntdt+ 2ft

+N

(-I

~ { ( ~~ )

sin

nt + n (::) cos nt} cos nt dt + 2ft

+ ~ Ftcosntdt=O, o

2ft Mit ) { (

o

~~ )

2ft

+Ni- 1 }

{(

cos nt -

~~ )

sin nl

It (

\

a~) sin nt} sin nt dt + ax

(4.6)

+ n (::) cos nt)sinnt dt+ 2ft

) Fi sin nt dt = 0, o

where L'. r i =

F

i

+ ( aF '\ ax )

7i-l

+ ( aF \ a~ ) \

• lii-l

is a known periodic function of the time. Equations (4.6) expressing the necessary and sufficient conditions for the existence of periodic solutions for xi determine the constants Mi _ l and Ni _ l • These equations are linear and their determinant agrees with the magnitude

a(P, Q) a (Mo. No)

•

Since the latter is by assumption different from zero, equations (4.6) are always solvable and have a single solution. Thus, for each set of values of the magnitudes M and No forming a simple solution of equations (3.11) thgre exists only one series (4.1) with periodic coefficients that formally satisfies equation (3.1). This series therefore represents the required periodic solution and converges for sufficiently small ~.

19

5. Application to the Theory of a Regenerative Receiver l

As a first example let us consider the oscillations of a regenerative receiver the circuit diagram of which is shown in fig. 1. Let an external electromotive force E = Psinw t act on its grid circuit. The equation of the oscillatibns of such receiver, if the grid current is neglected, is of the form I

R' L di dt+ l

+c r) 'dt1

l

o

--

u -."l

dia dt =

p'

smwl.

(5:1)

Fig. 1 where i is the current strength in the grid circuit, ia the current strength in the anode circuit. Assuming that the anode current depends only on the grid voltage v, where t

V=~~idl,

(5.2)

o

we consider two cases of the characteristic of the tube: the characteristic of the ~oft operation~ (5.3)

and the characteristic of the

ff hard

i a (v)=8 ov+

operation~

i8 v 2

3

~S4v'.

(5.4)

lSee the work of A.A. ~ndronov and A.A.Vitt cited on p.3 20

Substituting (5.4), (5.3) and (5.2) in (5.1) and denoting the frequency of the circuit by (A.) o (W 02 = l/LC) , we obtain for the case of soft operation the equation (5.5)

and for the case of hard operation the equation dZt'+(!.!.._~llSo_JlSz_2+JIS4 -l)dV+ dl~ L LC LC v LC 0 dt

!!

_

moO -

~. LC sm

t

U)l •

If in these equations we introduce in place of the variable t the variable T with the aid of the substitution T = W It they assume the following form: cl2~,

- + w-u., =

('

~~ + w20 =

(a' + 1'202 -

dv • a -1 '2...2) -v -~' + ,"sm 't

~2

where

w, a.',

are certain constants.

1', 0', ;, ,

Setting, finally we obtain d2v

d-r 2

o'2o.a)~~ -t).. sin 't,

+ w-v = 2...

( 2... ") dr., p. a. - l-v~ d't

• +", sm T',

(5.6) (5.7)

If the magnitude ~ is sufficiently small~ as we shall assume to be the case, equations (5.6) and (5.7) belong precisely to the type we have considered in detail in the preceding sections. The magnitude a entering equations (5.6) and (5.7) can be both positive or negative l (we assume ll>O ).

1--------------The pysical significance

o·f the sign of the magnitude a will be explained below in sec. 12. 21

In the present section we shall consider in detail equation (5.6) on the assumption that cx.>O. In this case, by a suitable choice of the variable x , equation (5.6) may be reduced to the form (5.R)

which is the equation we shall consider. In order to obtain equation (5.8) from equation (5.5) we must ~et:

(5.9)

OSCILLATIONS FAR FROM RESONANCE. Let us first assume that the magnitude~2 in equation (5.8) differs sufficiently from an integer. In this case the generating equation

has the following periodic solution xo =

.

). -11--1 SID 't'. w -

To find the corresponding periodic solution of the complete equation (5.8) we substitute in it x=

For xl

Xo

+ P.,Tl + ...

we then obtain the equation

This equation has the only periodic solution

In a similar manner we can compute also the further approximations, on which however we shall not dwell. OSCILLATIO~S NEAR THE PRINCIPAL RESONANCE. Let us now assume that W differs by a magnitude of the order of smallness of ~ from an integer, namely unity. We shall say that in this case we are dealing with the principal resonance due to the fact that the disturbing force contains only the first harmonic.

Following the rule of the preceding section, we set w2 =

1 + ., ua

\ /-1-"0' ,,=

(5.10)

'I.

after which equation (5.8) assumes the form d 2x

d't

+x -= p. p'osin 't

•

ax+(1 - Z2) xl.

-

(5.11 )

The generating solution in the case considered contains two arbitrary constants and is of the form

where Mo and No are the arbitrary constants.

Putting

we obtain for xl and x2 the following equations:

+ _\ .

d~x! d-;.2

;.1:1 -

d 2x2 d't 2

1 '2. axo + ( - xo) x o'

"0 SIn 't -

+ x:: = -

aX l

• • 2XOXl ZO'

+(1 - x~) .7 1 -

(5.13)

(5.14)

Substituting in (5.13) for x its value (5.12), after simple transformation we obtain 0 t!'d't'l. I.x1

+x1 '=:o.(-aM +T:\r -.IN i1J2_'!N )cos't+ 4 4 + ().o- JI aNo+ ! ilI!+ t A1oN:) sin 't + 1

0

• 0

0

0

0

0-

+! No(N!-3M~)cos3't+! ilI (ll1: O

23

3N~)sin3't.

(5.15)

For this equation to admit a periodic solution it is necessary and sufficient that the equations be satisfied P(M o• N o}= -aNo-/J/o [ Q(Mo• N o}= - aM o

1-

t (M!+ N!)]

+'-0=0. ) (5.1H)

+ No [1- ! (M! + N~)] =0.

These equations determine the constants Mo and No in the generating solution. We shall assume that Mo and No actually satisfy equations (5.16). The general solution of equation (5.15) will then be periodic. This solution will be: XI -= M} cOS"t +NI sin 't+

+ 3~ No (3M! -

N;) cos 3'1:

+ 3~ M" (3N; -

M!) sin 3'1:,

(5.17)

where Ml and Nl are arbitrary constants. These constants are determined from the condition of periodicity of the solution for x 2 ' namely, substituting in (5.14) for the magnitudes Xo and Xl their expressions (5.12) and (5.17) we obtain:

~:i+X2= [( - ! 11/oNo-a)M} + (1 ! M;- ! N; )NI + + 1~8A/o(M:+ N!)' Jcos 'I: + [( - 1 + ! M;+ ! N!) All +

+(! MoNo

a

)NI+1~No(..4I!+N!)2] sin"t+ ..• ,

where the nonwritten down terms do not contain cos ~ and sin ~~ Whence for M1 and Nl we obtain the following two equatl.ons:

( - ! MoNo -

! M! - ! N! ) N + '/ Ja Mo (M: +N!)! = 0. + ( ! MoNo - a ) NI + (

a ) All + ( 1-

I

-I-

(

1+

! M! + ! N!)

iii]

+ 1~No(M!+N!)!=0.

(5.18)

expressing the conditions for the existence of periodic solutions for x 2 - The obtained equations for Ml and Nl , as follows from the general theory, turned out to be linear_ The determinant of these equations identically agrees with the functional determinant 8(P,Q) _ _ 0(.\1 0 , Nt)) -

1_a2+(AI'I..+N2)-~(M'J+N2)S 0

0

16

II

(5.19)

II'

as is likewise in agreement with the general theory_ Hence, if MS and No are a simple solution for equations (5.16), we btain a fully determined solution for Ml and Nl and therefore also for XlRestricting ourselves to the obtained approximations let us examine more closely the equations (5.16) for M and No. Passing to the amplitude and phase in the gengrating solution, which are determined by the relations Mo=Asin?"

No=cos~,

we obtain from (5.16) the following equations for these magnitudes:

14.4:!

tg~=---

a

.'p [ a2+ (1-

"!!)2] = l;.

(5.20) (5.21)

Thus, for determining the magnitude A2 we shall have obtained a cubic equation. Hence, depending on the value of the mistuning a and the amplitude of the external action ~ 0' we have obtained one or three real solutions_ We are interested of course only in the positive solutions. In order to explain better the different cases that may here arise we shall construct the resonance curve, i.e. the curve of A2 as a function of a for fixed value of Ao. Formula (5.21) precisely represents the equations of this curve. In fig. 2 are shown a number ~f curves (5.21) corresponding to different values of~ o. For ~~ = 0 the curve (5.21) consists of the axis A2= 0 and the isolated (double) point (0,4). For I\~ different from zero but sufficiently small the resonance curve consists of two branches: in place of the axis A2 = 0 there 25

is obtained a curve of the form I - I, and in place of the double point (0,4) an oval I' - I' surrounding this point. With gradual increase of l\~ the oval I' - I' and the branch ~ - I approach each other and for a certain value of Ao there is obtained only one branch of the form II - II with the double pOint P. This value of 1\2 is easily found from the condition that equation (5.21)0 for a = must have a double root and therefore the expression (5.19) must also reduce to zero. Solving (5.19) and (5.21) simultaneously with a = we obtain, in addition to the curveA~ with double point (0,4), already mentioned, the curve i\ ~ = 16/27 with double point P( ,4/3) • For A~ > 16/27 there will at first be obtained the curves ~f th2 form III - III and then,for sufficiently large 7\.0 (A 0 )' 32/27), the curves of the type IV - IV intersected by each vertical in only one point. From this it is immediately evident that for A~ a 2 again one solution. For the amplitudes for the curves of the type IV - IV we shall obtain only one solution for any a • The values of a separating the regions for woich equation (5.21) has three solutions from the regions for which this equation has one solution are determined from the condition that for these values the equation has a double root and therefore expression (5.19) reduces to zero. Thus, these values of a are found as abscissas of the points of intersection of the resonance curve with the ellipse

°

°

°

A;

1

(5.22)

Equation (5.22) can also be obtained as the equation of the geometic locus of the points at which the tangents to the reso~ance curves are vertical, i.e. from the condition da/dA = O. If a resonance curve with given value of i\o intersects the ellipse (5.22) equation (5.21) 'will have three real roots for the values of a included between the abscissas of the points of intersection, while for the remaining values of a it will have one solution. If the resonance curve does not intersect the ellipse (5.22) equation (5.21) will have only one solution for all values of a • Both these types of resonance curves are separated by the resonance curve passing through a vertex of the ellipse. From this the result is readily obtained that for this boundary curve l\ ~ = 32/27. Thus, equation (5.21) always has one or three real roots to which correspond one or three periodic solutions of equation (5.11). Whether, for the physical system under consideration, periodic oscillations, corresponding to these periodiC solutions, will arise depends on whether these oscillations are stable or not. The question of the stability of these oscillations will be considered below in sec. 15. 6. Application to the Problem of Duffing Let us now consider the equation 2

d x 1 dt 2 T

k!!

3_ '\

x - lX -

A

.

sm t,

(6.1)

which was first studied by nonrigorous methods by Duffing. It may be considered as the equation of the forced vibrations of a conservative system with one degree of freedom with a nonlinear elastic restoring force. We shall assume that the Lagnitude y is small and shall treat equation (6.1) as quasi~inear. For this purpose we substitute in it y = ~y! after which it assumes the form

(6.2)

27

We shall assume at first that we are dealing with the nonresonance case. The generating equation will then have a single periodic solution Xo

=

A

•

k2-1 SIn t.

To this corresponds a single periodic solution of the equation (6.2) which we shall seek in the form

(6.3) For xl we obtain

the equation

which has the single periodic solution

The computation of the further approximations presents no difficulties. Let us ~ow assume that we are dealing with the resonance case when k is equal to unity or differs little from it. Setting in this case 1 - kl

flat

1. = I'l't

we reduce equation (6.1) to the form (6.4)

The generating equation now has the periodic solution

that depends on the two arbitrary constants Mo and No. We shall seek the periodic solution of equation (6.4) in the form of the series

28

x = Xo + f1Xl +

with periodic coefficients.

...

(6.5)

We have:

For this equation to have a periodic solution it is necessary and sufficient that the equations be satisfied alJJ o

+ 4"3 l'AI.{lll:+N!)=O,

A' +aN.+: l'No{M:+N:)=O.

These equations can be satisfied only on the assumption (6.6)

where A is a root of the cubic equation (6.7)

The magnitude x thus contains only the sine. This is due to the fact that O equation (6.4) does not change when t is replaced by -t and x by -x. As a result, the further approximations will likewise contain only sines. We shall make use of this circumstance for the further computations where it will introduce certain simplifications. With the arbitrary constants determined according to (6.6), we shall now have Xl

y'A' .

.

= Jrsm3t+Nlsmt,

where NI is a new arbitrary constant. This constant is determined from the condition of periodicity of the function x 2 • For this function we have cPZ:I..l dt S T

x! := a (

1'A 3 , " J 32 Sill ·)1

' ) + + N1 sm,

+31'A2 si n3 t(T;1' Sin3t+N.sint).

29

and the condition of periodicity gives:

(a+: l'AI) NI-l~T·JA$=O.

(6.8)

As was to be expected, according to the general theory the equation for Nl was found to be linear. With Nl determined we now have: . _

1

'iAS' 5

:1'2- ;;12 1

810

t+

3 1 'A3+ Jll ,:!,t:!N 3 '!Aa) sm , at +"'asm/. ""' ' .. 1-512 +( -156al 1 "1

where N2 is a new arbitrary constant which is determined from the following approximation. We shall not however concern ourselves with its determination and restrict ourselves to the obtained approximations. Let us consider more closely equation (6.7) which determines A. Returning to the initial notations we can rewrite it in the form: (6.9)

The roots of this equation can be obtained graphically as the abscissas of the points of intersection of the straight line (6.10)

and the cubical parabola

'1- _34 .... , .',1,"1·

(6.11 )

Let us assume for definiteness that y;.O. 2Then, from fig.3 it is seen that when the magnitude k 1 is negative or when it is positive but does not exceed a certain magnitude a , for which the straight line (6.10) is tangent to the cubical parabola (6.9), e~uation (6.9) has only one real negative root. But if k - 1 > ex. equation (6.9) will have three real roots of which one is negative and two positive. We arrive at the same results if we construct the

30

resonance curve tAl = IA(k2 struct separately the parabola

)1 •

For this purpose we con(6.12)

!I

!I-{kl.l)A-l --------~~~--~~~----A

Fig.; t/AI I

A0

I

I

I I

t I I

I

------------~-----L--------~~K'

Fig. 4

and the family of curves (fig. 4) (6.13)

Adding the abscissas of both curves we obtain the resonance curves determined by equation (6.9) for different values of ~. The form of these curves is shown in fig. 5. From a consideration of this figure we at once arrive at the same conclusion with regard to the roots of equation (6.9) to which we arrived on the basis of fig. 3. It is of interest to note that the parabola (6.12) represents the curve of frequency as a function of the amplitude of the free oscillations corresponding to the equation (6.1) for k = 1, i.e. the oscillations determined by the equation fi2:r dl"

+x-jr=O.

(6.14)

In fact, as will be shown in section 3 of chapter VII, the period of the solution of equation (6.14) with the initial conditions x(O) = A, x(O) = 0 is determined by the series

Whence we easily obtain the result that the square of the frequency of these oscillations, with an accuracy up to the second order of magnitude relative to A, is determined by the right hand side of equation (6.12).

IAI

-----.-L----'-------.....Il' Fig. 5 We shall now compare the form of the resonance curves for the nonlinear problem under consideration with the same curves for the case of the corresponding linear problem, i.e. for the case y = O. For the linear problem we have:

32

).

A=

kl-l •

rtence the resonance curves can be obtained by adding the abscissas of the curves (6.13) (the same as for the nonlinear problem) to the abscissas of the straight line k 2 = 1 and are of the form shown in fjg.6. These curves differ from the resonance curves in the nonlinear problem only in the circumstance that instead of the parabolic asymptote (6.12) a rectilinear asymptote is obtained. IAI

------------~------~t~------------~H·

Fig. 6 From this however a very sharp qualitative difference results between the two systems. First of all, ~n the nonlinear system for accurate resonance, i.e. for k = 1, in spite of the absence of damping, there is an enti'rely definite periodic solution with finite amplitude , deter~ined on the basis of (6.9) by the formula

~oreover, as was already noted above, beyond resonance, when the mistuning k 2 = 1 exceeds the magnitude of a, we obtain three different real values for the amplitude ~, to which correspond three periodic solutions of equation (6.1).

The question of the stability of the obtained periodic solutions and the associated question of the character of the development of the oscillations will be considered below in section 16. 1 The amplitude under consideration is of course that of the principal term. 33

7. Resonance of the n-th Kind In nonlinear systems under the action of a disturbing force intense forced oscillations may arise not only when the period of the natural oscillations is near the period T of the disturbing force but also when it is near the magnitude nT, where n is an integer. The period of the forced oscillations will then be equal to nT. For comparison we shall first consider the linear system described by the equation tJls d,1

+ 1.21X + 2~ 0

II;

ds _ ~ • d, - " SID wt.

(7.1)

This equation has one and only one periodic solution determined by the formula

(7.2) To this solution correspond the forced oscillations of the system. Every other solution of the system (7.1) for t ~;o =

! I.~) +No [a+ ! (ill! + N! + ; ).11)]} cast + + { - Alo ra + ~ ( AI: + N! + : A +No ( a + ! ),fj) } sin t + ..., = {A/o(a -

2 )

]

where the terms not written down do not contain cos t and sin t. The condition of periodicity gives:

These two equations, aside from the trivial solution Mo = ~8 = 0, have additional solutions determined by the equatJ. ns

The functional determinant of equations (8.2) for the solutions (8.3) is different from zero. For sufficiently small a , these solutions are real and to them actually correspond periodic solutions of equation (8.1) of period 2n. EXAMPLE 2. RESONANCE OF THE N-TH KIND FOR THE PROBLEM OF DUFFING.I We shall investigate resonance of the n-th I-~ii-~~;~~~~tions in this example were carried out by the stugent V.M. Mendelson.

kind for the equation of Duffing

~~ +k'X-lx3 =).sint. We shall assume therefore that and set

differs little from lin

k

Equation (8.4) then assumes the form tP:t;

d"C1 X = ~ sin no: + f1 (ax+ llz3), (8.6) and the problem reduces to finding the periodic solutions of period 2n of this equation. We shall seek these solutions in the form of the series

(8.7)

where Xo

A... = 1-ra' SID n ..

+ !If0 cos + No sm• JI

't

't

(8.8)

and Mo and No are arbitrary constants. These constants are found from the conditions of the periodicity of the function Xl" For this function we have

39

The conditionsof periodicity of the function xl are found different depending on whether n ~ 3 or n = 3. We shall first assume that n ~ 3. In this case the conditions of periodicity of xl assume the form

The obtained equations, aside from the solution Mo = No = 0, have the further solution M'+Nz= _~ [ •

II

311

a

+ 32 (i_nt)S 1.Al ]

,

where one of the constants, Mo or No' may be chosen arbitrarily. As a result, the functional determinant

a(P. Q)

deMo. N;}

for this solution becomes zero and the method of preceding section cannot be applied.

the

We shall now assume that n: 3. In this case the additional terms with cos T and sin T appear in equation (8.9) and the conditions of periodicity of the function xl assume the form P(AI Ot No> = 1

= Mo [ a + l~ 111.;

Q(M ot N o)= = No [a+l~l)A!+

! 1dM~ +N~> ] -1~ 111.tA/oN. =0,

j

! 11(M!+N:) ] -AT1At(.M:-N!)=0. (S.lO)

These equations, aside from the solution Mo = No = 0, have other solutions. In particular, there is the solution "I -0 V _ -3).111 ±Y -3072a'rt- 63 lflJ if

0 -

, .. 0 -

48Tl

•

(S.11)

In order that this solution be real it is necessary that

the inequality be satisfied

Hence it is necessary that a and y be of signs and that the condition be satisfied Ia1

different

> t~;4 I.: I TIl

or, if with the aid of (8.5) we go over to the initial notations, (8.12)

J. Stoker l , in solving this problem by a less rigorous method, arrived at the inequality

Besides the solution (8.11), equations (8.10) have additional solutions in which both magnitudes, Mo and No, are different from zero. Thus, dividing equations (8.10) one by the other, we have:

after which from these equations we obtain: Mo=

N -

± VaNo .... ± 3)..ITI

± V -3072al"J 96tl

0-

If however account (8.6) does not change ( and T by T + n/7, it is (8.17) in no way differ

tgi No, 1 I" 63 l fA!

(8.13)

is taken of the fact that eouation for n = 7 ) on replacing x~by-x easily concluded that the solutions from the solutions (8.11).

IStoker, J., Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publishers, 1950. 41

o(P, ~r the solutions(8.11) the functional determinant a(Mo. N.) is different from zero, since these solutions are

simple. It is also easy to satisfy oneself of this by direct verification. Hence to the obtained solution Mo and No there actually correspond, for sufficiently small periodic solutions of equation (8.6).

9.

~,

Autonomous Systems.

Conditions for the Existence of Periodic Solutions We now pass on to the consideration of oscillations of autonomous systems. In this chapter we shall restrict ourselves to the consideration of the simple system with one degree of freedom described by the equation 1 (9.t)

f is an analytic function of the variables x and xwhere : dx/dt in a certain region G in which the generating solution under consideration lies, and of the parameter ~

for its sufficiently small values. Equation (9.1) is essentially a special case of the equations considered in the preceding sections. Autonomous systems however have their own characteristics which sharply differentiate them from nonautonomous systems, as a consequence of which equation (9.1) requires special investigation. First of all, in the case of nonautonomous systems the period of the required periodic solution was always entirely definite, equal to or a multiple of the period of the equations themselves. In the case of equation (9.1) however that is now considered, since it does not contain the time explicitly, periodic solutions of arbitrary period are possible which, generally speaking, depend on ~. Moreover, equation (9.1) does not change if t is replaced by t + h. Hence any solution of this equation, including a periodic solution, remains the same if in it t is replaced by t + h. This makes it possible, without disturbing the generality of the considerations, to assume ~~~~_~!_~~~_~~itial instant of time the magn1tude dx/dt 1 Cf. Andronov A.A. and Khaikin S.E., Teoriya kolebanii (Theory of Oscillations), ONTI, 1937.

for the required periodic solution reduces to zero. In fact, let ~ be the period of the required periodic solution. Hence x(O) = x(w) and in the interval (0, w ) there must exist an instant of time tl for which the magnitude dx/dt becomes zero. We can make this instant of time to be the initial instant which, evidently, will only mean that the magnitude t in equation (9.1) is replaced by the magnitude t + t l • Thus, in seeking the periodic solution of equation (9.~ we can always assume that the magnitude dx/dt reduces to zero at t = 0. 1 We now proceed to find the periodic solutions of equation (9.1) following the idea of Poincare. For this purpose we consider first the generating equation (9.2)

The general solution of this equation, for which Xo(O)=O, is of the form Xo

= A/o cos kt,

(9.3)

where Mo is an arbitrary constant. This solution will be periodic of period T = 2n/k. Taking the solution corresponding to some fixed value of Mo as the generating solution, we shall seek to obtain the periodic solution of equation (9.1) which for ~ = 0 reduces to the generating solution. Let this solution, which we shall denote by x(t,S,~) , be determined by the initial conditions x(O.~. l1)=xo(O}+~=Jlo+~'

xo(O,~, fl)=O. ~

(

'~

f':' ,

(9.4) \

where S = p(~) is an unkown function of ~ which becomes zero for ~ = O. Thus, in contrast to autonomous systems, in the case considered the initial conditions of the required periodic solution contain only one unkown function. But on the other hand, as was pointed out above, the period of the required periodic solution for the system under consideration is not known beforehand. This period is equal to lCertain authors, among; them A.A. Andronov and S.E. Khaikin (see also Minorsky N., Introduction to Non-Linear ~echanics, Ann Arbor, Mich., 1947) incorrectly recommend, in case it is not succeeded in finding a solution on the assumption i(O)=O, trying te. find it by assuming x(O)=O.

43

~

where a = a(~) is likewise an unkown function of ~ which, evidently, must reduce to zero for U = O. Both unkown functions ~ and a are found from the conditions of periodicity of the solution x(t, ~, ~). These conditions are of the form .t(T+a, ~,v-) -x(O, .~, p)=x(T+a,~. P)-Mo-~=Ot} r(T + 11,

~,

p) = O.

(9.5)

Let us expand the left hand sides of these equations into a series in a and write out in the first equation the terms not higher than the second order and in the second equation the terms not higher than the first order. Then, taking (9.1) into account, we obtain

.

1

r(T,~, p)+z(T, ~, p)a+

+ ! [- ~'X(T, ~,

f'J+·· .J.'+ ... - M.-P=O, I'

(9.6)

x(T,~, 11 )+[ -k'x(T,~, p)+ ... )a=O.

Further, we have:

and also evidently,

~t1 + kllA =

0,

.4(0)= I,

.

A(O)=O

so that A

= coskl.

Substituting in (9.6) and retaining in the first equation terms up to the second order and in the second equation terms that are linear with respect to a, S, U we obtaiL: pC "T +D -i ~+E T p.+C T a+ ... +

{(2ft:)

(2'1t)

(1

(2n:)

. (2rr)

}

+ ! al ( -- .11,1-1 + ... )= Ot { C( 2; ) + ... } + a ( - J/01-1+ ... ) =- O. 44

1 J

(9.8)

Assuming Mo different from zero we can solve the second equation for ex and this solution will be of the form

_ {_t . (21:) 1 Mok 2C i +---/'

"-fL

where the terms not written down reduce to zero together with 3 and~. Substituting ex in the first of equations (9.8) we obtain for the determination of ~ the following equation: f'

{c (2k~)+ D (~)~+Ql'-+ -.. } =>k2(fork I),

it follows from (11.6) that for sufficiently small ~ a Dositive number a 2 exists such that for the entire region ~ the inequality holds d&

2

de >a .

(11.7)

From this the result immediately follows that when 00 =.-M o ~ o

f'f(MoCOSU, ;:kMoSinu,O)sinu+

s_i_n_u,-, • + k iJ.....if-'-(~_f_o_co_s_u_,_-_kM--=-o ox

0....;.)

cos u } cos u d u =-

2K

=-kMo ~ of(Mocosu,---:kMosinu,O)du_

ox

o 2~

-

AI

r

0 ~

{Of (Jlo cos u, -kMD sin u, 0) iJr

cos u -

o iif(Mo cos u, -kMo sin u, 0). }. d - k . sm u Sin u U =::

or

2K

=::-kM

r of (MDcos

o ~

o

U,

-kMosinu'O)d _ ~f dP(Mo)=O • U .U 0 dll •

ox

I

e

Hence the condition of stability (14.7) can be represented in the following form: dP(llfo) dM

o

> .• 0

(14.8)

As an example let us investigate the stability of the self-oscillations of the tube generator studied in sec.12. Let us consider first the case of a tube with soft operating characteristic. The oscillations in this case are described by the equation (12.1) while the function P(M o ) and the magnitude Mo are determined by formulas (12.7): P (M 0) = _

11:0+ 1~~8 ,

~f2

T

whence dP (Mo) dAle

42

1t 0 = - ' ,

2:%

= -;;- .

The periodic solution will be real only for a> 0 and for it condition (14.8) will be sat fied. Thus, the investigated periodic solution, as was indicated in sec.12, will be stable. Let us now assume that the tube has a hard operating characteristic. The oscillations in this case are described by the equation (12.2) and the function P(M ) and the magnitude Mo are determined by the formulas 0(12.9) and (12.10):

If ~>O there is obtained for Mo only one real value for which, as is easily seen, condition (14.8) is satisfied. If a. < 0 it is necessary, fQr the existence of periodic solutions, that y4 + 32a.62~0. There will here be ontained two positive values for It is easy to see that for the larger value of M~ condition (14.8) is satisfied and therefore the corresponding periodic solution will ~e stable. On the contrary, for the smaller value of Mo the magnitude dFldM o will be negative and therefore the corresponding periodic solution will be unstable.

Mg.

15. Stability of the Periodic Motions of Nonautonomous Systems For all periodic oscillations of the nonautonomous systems considered in this chapter equation (13.1) has the form (15.1)

where the functions f and F have the period 2n with respect to t. The parameter k is here different from an integer if we are dealing with the nonresonance case; k=n, where n is an integer in the case of a simple resonance and k:l/n for the resonance of the n-th kind. The investigated

8

periodic motion has the form (15.2)

where x is the generating solution. The periodW of the solutio£. (15.2) is equal to 21t for the nonresonance case and for simple resonance, andW = 2nn in the case of resonance of the n-th kind. The equation in variations (13.6) in the case considered is of the form (15.3)

We shall seek to obtain the particular solutions Yl(t) and Y2(t) of this equation, satisfying the initial conditions (13.77 , in the form of series in powers of~. We can write: y~o)

Yl 112

For the functions equations

dt'"

+

+ y~lIf1 + y~2)p.2 +

2

0

y

+k y

aF (t. :ro.

2 Ul 1

d'ly~ll + •.,Y2'" -_ de!

ax

iJF (I,

II.

d y;0)

dr ;0,

".~ ~.,

iJx

(15.4)

\ve obtain the differential

1.2 (01 1 -,

II.

}

,

2

Y~O), y~ll, Y~O\ y~ll

d'!!I\O) dt'l.

d2,l/l ll

y~O)

+ y~I)ll + Y~:)I~ + ...

0)

+ k2y(0) =

(0)

Yl

0)

o

2

'"

Y2

,

I

1

0

'

aF (t, xo.

~o.

0) y·(O)

•

+ aF (t, iJ~~•

1

;'0,

iJxo

0)

'

\

'(0)

j'

Y2

(15.5)

and the initial conditions

y~ll (0) = ~~ll (0) = y~OI (0)

0,

• (0) = 1,

y~Ol

79

0, }

y~ll (0) = y~ll (0) = O.

(15.6)

For the coefficients A and B of the characteristic equation we have, on the basis of (13.14), (13.20):

1>

(15.7)

j Having established this, we proceed to the consideration of the stability criteria. From (15.5) and (15.6) we find: k

(0)

Y1 =cos t,

t sm . k t. YzfO) =T

Hence on the basis of (15.7) the first two conditions stability (13.18) assume the form 2 cos k{J) + 2 + Ii ( ... ) + - 2 cos kID + 2 + Ii ( ... ) +

>0, } >0.

(15.8)

of

(15.9)

We shall consider the following three cases: 1. NONRESONANCE CASE. In this caseL0= 21t and k differs from an integer. For sufficiently small ~ conditions (15.9) are therefore always satisfied with the inequality signs. Consequently, for stability it is necessary that only one condition of (13.18) be satisfied, and if it is satisfied wi th the sign of inequality the motion will actually be stable and moreover asymptotically stable. Taking (15.7) into account we arrive at the conclusion that in the nonresonance case it is sufficient , for the asymptotic stability of the periodic solution, that the inequality be satisfied ""/t

•

~ JF (I, Xo. Xo. 0)

)

I)

-~-=.~-..-:..

ar,

dt

< O.

(15.10)

We may remark that if the expression in condition (15.10) were positive the periodic solution for sufficiently small ~ would be unstable, while if the expression reduced to zero then it would be necessary, in judging the stability, to take into account higher order terms in the expansion of the magnitude B, and possibly also in the equation of the disturbed motion.

2. CASE OF SIMPLE RESONANCE. In this case W = 2n and k = n, an integer. Hence on the basis of (15.8)

Substituting these values in conditions (15.9) we see that for sufficiently small ~ the first of these conditions as before is satisfied with the inequality sign. As regards the second condition of (15.9), which corresponds to the second condition of (13.20), the term free from ~ contained in it red,uces to zero and therefore a more accurate computation of the coefficients A and B is necessary. We shall compute these coefficients with an accuracy up to second order magnitudes with respect to~. For the coefficient A we have, on the basis of (15.7) and (15.11):

For computing the coefficient B it is now more convenient to use not expression (15.7) but the second equation of (13.14). Substituting in the latter for Y1(£J), ;11(0) and Y2(T), Y2( CJ l their values obtained from (15.4) and taking into account (15.11) we obtain: B = 1 + [y~l) (w)+

yt (00)] p. + [y~!l (ill) + y~ln (ill) + + y;ll (w) !j~ll (m) _

y~l) (ill) y;ll (w)] (-12

+ ..... ,

and therefore, on the basis of (15.12), the second condition of (13.18) assumes the form

This condition will be satisfied for II sufficiently sme.11 with the inequality sign if y~l)

.

(lO) y~J) (ill)

.

y;ll «(I)) y~U (to)

> O.

(15.13)

We shall compute the magnitudes entering here. From the differential equations (15.5) and the initial conditions (15.6) we find: t .

1 III

Yl

= ~ (' [oF (-:, :to. :to. 0) n. ~

•

cos n't _ n of (-:. :t~, ro, 0) sin Il't] X

iJxo

o

0

Xo

X sin n (t - 't) d't,

[

of (.. , :to, ;0. 0)

sin n't + n i)F ("t,

oxo

x~. ;0, 0) COSlt't]

X

i}xo

X !3in It (t - 't) d't

and therefore

1 _ n of (-::, x~'

. Xo. 0) sin

n't ] sin n't d't,

oxo

y~l) (OJ) =

2'K

S [oF

('1:.

;;~ ~o,

o

0) cos n't-

.

n of ('t, 2'0. • Xo... . . . 0).] sm l1:t cos n't d 't,

-

~

oxo 2'K (1) (

y'!.

) It)

1 \

= - n2 J o

[aF ('1:, i)xoo• X

} (15.14)

• 2'0.

0) SI'n n't

+ n i)F (1:. x~'

" XO,

+

0) cosn't] sin n't d't,

oxo

"(}} (OJ) =~ Y2 11

2'K

•

\ [oF('I:,:t o, :to. 0) sinn't+ ~

i)xo " + n i)F ('t, :t~, :to. 0) cos n't ]

i)xo

cos n't d't.

J

The generating solution Xo entering these expressions, on the basis of the results of sec. 3, has the form XO

(I) = Mo cos nt+ Nosinnt

82

+,.

(t),

where ~ equation

is a particular solution of the generating

and Mo and No are determined by equations (3.11):

( 15.15)

whence we easily obtain (1) (

Yl

) l!)

11) (

Y'l.

) l!)

=

i iJP -n iJJ/0

= -

}

t

1 iJP

,,2

a-':;Vo

t

( 15.16)

and consequently condition (15.13) can be rewritten as a (P, Q) iJ(Mo. No)

< O.

( 15.17)

To this condition, as in the resonance case, it is necessary to add condition (15.10) corresponding to condition (13.19). Conditions (15.17) and (15.10) assure the asymptotic stability of the periodic solution in the resonance case. If at least in one of them the sign of inequality is changed into its oPPosite the condition of instability is obtained. The obtained conditions can be given another form. In fact, from (15.14) we readily obta.in:

and therefore on the basis of (15.16) condition (15.10) assumes the form

( 8

-

ap.

iJQ) < 0.

oJ10 iaN;,

(15.18)

This condition together with condition (15.17) shows that for the periodic solution to be asymptotically stable it is sufficient that the roots of the equation ap

---l.

a.M,

=0

{JP

(15.19)

- aN.

have negative real parts. We may note that the magnitudes P and Q differ only by a multiple of n from the coefficients of sin nt and cos nt in the expansion of the expression •

F(t, xo(t), xo(t), 0),

(15.20)

i.e. of the right hand side of equation (15.1) after substitution in it for the magnitude x the generating solution and for the magnitude ~ zero. Let us denote these coefficients respectively by p'and Q~ In place of equation (15.19) we may then consider the equation ap*

.

--" oJ/a ap* oNo

aQ* -

oAla

aQ* --J..

=0.

(15.21)

aNo

To avoid possible confusion in the practical set-up of this equation it is useful to note that the magnitudes Nand -M are the coefficients of cos nt and sin nt in tHe derivRtive dXoldt of the generating solution. Thus, the elements of the determinant (15.21) are the partial derivatives of the coefficients of the n-th harmonic of expression (15.20) with respect to the same harmonic in the expression dXoldt, the derivatives along the principal diagonal being with respect to similar coefficients (i.e. the coefficient of cos nt is differentiated with respect to the coefficient of cos nt and the coefficient of sin nt with respect to the coefficient of sin nt). k

=

3. CASE OF RESONANCE OF THE N-TH KIND. In this case lin and w = 2nn. Hence, on the basis of (15.8), as

84

in the preceding case, formulas (15.11) hold true. All the preceding calculations are therefore retained without any changes and we arrive at the conclusion that the investigated periodic solution will be asymptotically stable if both roots of the equation (15.19) have negative real parts and unstable if at least one root of this equation has a positive real part. The functions P and Q in equation (15.19) are now determined by the formulas (7.6):

16.

Stability of the Oscillations Considered in Sections 5 and 6

We shall apply the obtained results to investigate the stability of the oscillations of a regenerative receiver and of the oscillations in the problem of Duffing, that were considered in detail in sec. 5 and 6. Regenerative receiver. by the equation

The oscillations are described (16.1)

Let us con 2 ider first the nonresonance case, i.e. we assume that 4}2 is not near unity. In this case the generating solution is of the form A . xo= - ! - t smt (J)

(16.:!)

-

and for stability it is sufficient that there be satisfied the condition (15.10)

i.e. (16.3)

If this condition is not satisfied, as will be the case when the system is sufficiently far from resonance, a periodic solution will not be physically realized, although equation (16.1) will admit such a solution. As will be shown below in sec. 18 of chapter IV, if condition (16.3) is not satisfied equation (16.1) will admit a particular solution to which corresponds a form of oscillation more complicated than periodic, the so-called "combinational"or \'quasiperiodic/I oscillation in which, together with the frequency of the external excitation, there will figure also a frequency close to the self-oscillation frequency of the system, and this oscillation will be stable. Let us now consider the case of principal resonance, i.e. assume that the magnitude cu 2 - 1 is of the order of smallness of~. The oscillations will now be described by the equation (5.11): (16.4)

where w 2 - 1 = a~ is the mistuning. solution will be

The generating (16.5)

and the functions P(Mo,N o ) and Q(Mo,N o ) are determined by a(P. Q) equations (5.16). For the functional determinant a(Mo. No) we obtain formula (5.19) and therefore the stability condition (15.17) assumes the form ( 16.6)

As regards condition (15.18) it may, on the basis of (5.16), be written as follows: 2 - (M! + N!)

< O.

(16.7)

This condition does not evidently differ from (16.3) since the magnitude it/ (CO 2-1) is the amplitude of the solution (16.2) and corresponds to the magnitude ~ + N~ in the solution (16.5).

86

At

------------>-4

Fig.15 Let us consider the resonance curves giving the amplitude A2 = M~ + N~ as a function of the mistuning a and which are represented in fig. 2. The conditions (16.6) and (16.7) show that stable oscillations correspond to those parts of these curves which lie outside the ellipse (5.22) and on the straight line A2 = 2. In fig. 15 these parts of the resonance curves are represented by the continuous lines as distinguished from the dotted lines, to which correspond the unstable oscillations. From this figure it is seen that the stable periodic solutions are possible only for sufficiently small mistuning a. If the equation (16.4) admits three periodic solutions, which is possible only for i\ ~ < 32/27, one of them will always be unstable and at leas~ one stable. But in this case there are also possible two stable periodic solutions, as is seen from fig. 16, in which part of fig. 15 is shown to an enlarged scale. For i\.~ >·32/27, when equation (16.4) has only one periodic solution, the latter will always be stable if the mistuning a is sufficiently small. Thus, at SUFFICIENT NEARNESS TO RESONANCE EQUATION (16.4) WILL ALWAYS HAVE AT LEAST ONE STABLE PERIODIC SOLUTION.

87

----------~~----------A~l

",

'

.....

Fig. 16 The obtained results make it possible to compare the oscillations of the system (16.1) with the oscillations of the corresponding linear system. In the linear system, as a result of the unavoidable presence of resistances, the free oscillations rapidly die down and purely periodic oscillations with the frequency of the external excitations are established. The system (16.1) however is,as we have seen, a self-oscillatory one. In the absence of an external disturbance stable nondamping oscillations are established in it. In the presence of a disturbance these free oscillations will not, in general, dampen and oscillations with two frequencies will be established in the system. These oscillations however, as we shall see in sec. 18, chapter IV, will be stable only for a sufficiently far removal from resonance. Near resonance, as we have just shown, equation (16.1) admits a stable periodic solution. Consequently purely periodic oscillations with the frequency of the external excitation are established in the system. The self-oscillations will be suppressed by the external disturbance or, as we say, the frequency of the auto-oscillations will be ('trapped u by the frequency of the external disturbance. This essentially is the phenomenon of trapping of the frequency that is observed in the self-oscillating systems. Problem of Duffing. by equation (6.2)

The oscillations are described

~: + k 2x =

f'-'( x 3 +). sin t

88

('r

> 0).

(16.8)

In the nonresonance case the generating solution will be A

•

xo= k2 -1 sm

t

and for it the condition of stability (15.10) is satisfied with the equality sign. If we took into account resistance, i.e. considered the system

we should have for the corresponding generating solution in the nonresonance case tiF (t,

x~.

;0. 0) = _ 2n

axo

and therefore equation (15.10) would be satisfied with the inequality sign. As a result we must consider that the nonresonance solution is always stable. For resonance, when l-k 2 =1J.a, the equation of the oscillations will be of the form

~: + x = fl (~sin t + ax+ l'X3 )

(A = fl A').

(1.6.9)

The generating solution will be Xo = AI ocost

+ Nosin I.

for which P(Mo' N o)=)/ +aNo+

! l'N (AI:+ N!). o

Q(M o• No) = aM o+ ~"i T'Mo(M!+N:>,

as a result of which we have Mo = 0, No a root of the cubic equation (6.7):

= A,

S(A)=).'+aA+! j'A8=0.

where A is

(1.6.10)

Condition (15.10) as before is satisfied with the equality sign. As regards condition (15.17), it gives:

or, remembering that A is a root of equation (16.10), A' ciS (A)

0

Ad:J< .

(16.11)

The roots of equation (16.10) are graphically represented by the points P,Q,R in fig. 3 (p.31). Since 8(-00)"p' z~p) = CODSt,

e- >.." (tziP ) + z~» = const, .. . . .. .. . .. . .. . . .. .. ..

..... . . . e-'", I (,'1'1. . -1 + (np-2)1 ' + ••• + Znp (p») = p (np-i)J Z1 Z2 'I'Ip-2

(p)

(p=1,

(p)

"0,

k),

1 (3.9) ~

const

J

where (3.10)

(i=l, .•. ,np; p=l, ... ,k)

are linear forms of the variables ficients.

Xs with constant coef-

Relations (3.10) may be considered as the linear transformation of the variables Xs into the variables z(l~ The determinant of this transformation A(f)

it

A(I)

- J2

AU)

1 "1

1(1)

A(I)

A(1)

AU) ,.2

~

21

22

1(1) 2nl

.L

nl

...··· ... A(I) "nt . ··· .··· 4(11)

A(h) It

4fk) - 21

A(h)

AW ...

A(h) I "It

A(h) 2'lk

12

- nl

(h)

AII2

..· · Attl) ,In"

differs, as is easily seen, only by the multiplier

which never becomes zero, from the determinant consisting of the n 2 functions y~l). The latter determinant however,

112

in virtue of the independence of the solutions (3.8) is different from zero. From this it follows that the transformation C3.10) is not singular. Let us with the aid of it transform equation (1.1). For this purpose we form the derivatives with res~ect to t of the expressions C3.9). Then, taking into account the fact that these expressions are the first integrals of the system (1.1), we easily obtain (3.11)

(i=2,

"'J

IIp;

p=l, ...• k).

These will be the transformed equations. The form C3.11) of equationSC(Jol) is termed the CANONICA.L FORM, and the variables z CANONICAL VARIAB1ES.

1

The roots ~ may also be complex and the corresponding variables z( will then also be complex. It is sometimes desirable to give equations C3.11) a real form. This can easily be done in the following manner.

1)

Let i\. = '\.l + i~ p • We set zC~)= uCI:)+ ivCI:~ Then pp J J J separating in C3.11) the real and imaginary parts, we obtain in place of the np equations (3.11), the 2np real equations du(P) = dl

_1_

J

u ,,(m .p 1

dv(p) _.~l_

v

v(p) P 1 ,

dt

=

IL

v{p)

.p 1

+ vpll(P) I'

(p)

uU. ---= rpU'(p) de l It

(p)

'Ip V·l

-

( )

p u·1I,

The same 2np equations are obtained also for the root IIp - i"» p' conjugate to i\. p' so that the total number of equations in the real and complex form will be the same. REMARK. The system (3.11) has evidently n independent solutions, that break down into k sets such that the first solution in some p-th set has the form

113

(p) = Z'11

zW = 111' 1

(r

,

-leAp', ( _t)"p-i

z() 1>

Z(T) 11

e).

=

(llp-1)! Z(r) ~l

=

• • •

e)'pI

'

Z(r)

-

Tlp-i -

0

1, ... ,p-f,p+1, ... ,L:),

while the remaining solutions of this set are obtained from the first by successive differentiation of the coefficients of e ~pt. This p-th set consists of n solutions and has a standard form. From this it follows P that the roots ~ and - ~ of the fundamental equations of conjugate systems have not only the same multiplicity, as immediately follows from the very form of these equations, but in both systems there correspond to these roots the same number of sets of solutions with the same number of solutions in the corresponding sets. 4. Periodic Solutions of Nonhomogeneous Linear Systems with Constant Coefficients We proceed to the very important question, for what follows, in regard to the conditions of existence of periodiC solutions for a system of linear nonhomogeneous equations drs

dt = asl Xl ••. +a$"xn + l~ (t) {s = 1, ... , It},

(4.1)

where a sj are, as before, constants while fs(t) are continuous periodic functions of period ~ • Let us denote by xSj(t) the fundamental system of solutions of the homogeneous equations

(4.2) determined by the initial conditions (s,j=1 •..• ,n).

114

(4.3)

where 6 sj is the Kronecker symbol, i.e. 6 ss = 1 and (for s j j) 0sj = o. It is easy to show that the functions I

l::{t) =

n

S~ XS'l{t-t)la(t)dt

(4.4)

o a=1

are a particular solution of the system (4.1). In fact, we have •

n

"'" (dxs [ t = -'J a=1 t

=

xsa{O)/a ( t)+ n

+ ~ ~ [asIXja{l-t)+ '" +asnxna{t-t)]ja(t)dt= o a=1 la (t) +aslxj + . " +asnx!:.

Adding to this solution the general solution of the homogeneous system (4.2) we obtain the general solution of system (4.1) in the Cauchy form t

Xs

=C 1 XS1 (t)+ ...

+C,jX

s7I

{t) +

n

~ "!JIXsa{t-t)/a(t)dt,

(4.5)

where Cl, ••• ,Cn are arbitrary constants, which evidently are the initial values (for t=O) of the magnitudes ms. We shall now choose the constants Cs in such manner that the solution (4.5) is periodic of period oU. For this it is necessary and sufficient that the relations x s (4l)-x s (O)= = 0 be satisfied. This gives for the determination of 01 the following system of linear nonhomogeneous equations: (0

C1XS1 (ill)

+ ... + Cflx.m (w) -

Cs

n

+ ~ ~ xs

p.=o· iJ(Mlt···,M m) •

Equations (6.10) therefore assume the form 2ft n

Pi(M~, ... ,AI:)::::: ~ :ZFa {t,7,O)'f a d't)d't=O

(6.15)

o 4=1

(i=l, ... ,m), 1t'

since for !3n-m+l= ••• =(3n = 0 the magnitudes Ml become Ml and the functions x~(t,Ml' ••• '~) become the generating solution ~s(t). By the same considerations, taking into account the fact that the determinant iJ ({1,....m+h ••• , '11)

arM ••... ,M,...) ,

agreeing, according to (6.13) , with (6.4), is different from zero, we can satisfy ourselves that condition (6. ) is equivalent to the condition (6.16)

From all that was said above we obtain the following fundamental proposition: IN ORDER THAT THE SYSTEM (5.1) FOR SU:E'FICIENTLY SMALL VALUE OF ~ HAVE IN THE RESONANCE CASE A PERIODIC SOLUTION REDUCING TO THE GENERATING SOLUTION~(6.2) FOR u = 0, IT IS NECESSARY THAT THE PARAMETERS Ml IN THIS GENERATING SOLUTION SATISFY EQUATION (6.15). FOR EACH SIMPLE SOLUTION OF TRESE EQUATIONS,I.E. A SOLUTION FOR WHICH CONDITION (6.16) IS SATISFIED, THE INDICATED PERIODIC SOLUTION ACTUALLY EXISTS AND IS THE ONLY SOLUTION. For the above proposition to hold true it is sufficient that the functions (Ml, ••• ,Mm) admit derivatives of the first order with respect to the variables Ml, ••• ,Mm, for which it is sufficient that the functions Fs admit derivatives of the first order with respect to the variables xl, ••• ,xn'u. Thus, in the case under 133

consideration the number k, figuring in the conditions relative to the functions Fs and characterizing their degree of smoothness, can be taken equal to unity • .Jf

If for the solution Ml of equations (6.15) condition (6.16) is not satisfied the question as to the existence of a solution of equations (6.9) and therefore also of a periodic solution of the system (5.1), remains open. In practice the case may be of interest where equations (6.15) are satisfied identically. Assuming that the functions Fs have with respect to xl, ••• ,xn'~ derivatives at least of the first order, which condition the existence of derivatives of the first order of the function Q. with respect to ~, we can, for the condition Q.(~ 1 l' ••• ~ 0) ~ 0, write 1 n-m+ n Qi = ~Qi(~n-m+l, ••• ,Sn'~) and equations (6.9) assume the form Qi (~n-m .. l' • . . , ~71' p.) = O. In order that these equations admit solutions of the required form it is necessary that the parameters the generating solution satisfy the equations

Mr

Pi(M:, ... , Al~) = Qi(O, ... ,0) = O.

of (6.17)

If at the same time the condition is satisfied

a(Pi, ... , P;") a(Mr, ... , Mrn)

=1= 0 ,

(6.18)

such solutions will actually exist and will be unique. In this case there will as before exist a single solution of the system (5.1) reducing for ~ = to the generating solution. It is here necessary to assume that the functions F admit derivatives of the second order with respect to tRe variables xl' ••• '~' i.e. that k = 2, in order to assure the existence of the derivatives of the first order of the functions p! that figure in (6.18).

°

1

If equations we can divide them equations as with manner, we arrive

(6.17) are likewise satisfied identically by ~ and proceed with the obtained new the preceding ones. If, continuing in this at equations with respect to Mt not

134

identically satisfied and having a simple solution, system (5.1) will admit a unique periodic solution reducing for ~=O to the generating solut~on corresponding to the found values of the parameters

Mr.

We shall denote the case where equations (6.15) are satisfied identically as the SING~~R case.

7. Practical M.ethod of Computing the Periodic Solutions of Nonautonomous Systems at Resonance in the Case of Analytic Equations Let us consider the system (5.1) anx!

+ ... + asnxn + Is (l) +}l Fs (t, Xl' •••• XlI' p.)

(7.1)

(s=l, ... ,n)

for the assumptions of the preceding section but for the particular case where the functions Fs are analytic with respect to xl, ••• ,xn'~. We shall show how in this case to find practically for this system the periodic solution which for ~ = 0 reduces to the generating solution X~

x~ (I)

+M:1~1 (0+ ... + !r/! 1~m (t) = 1'$'

(7.2)

in which the parameters ~ satisfy equations (6.15). In this section we shall restrict ourselves to the consideration of the nonsingular case, i.e. we shall assume that equations (6.15) are not identically satisfied. We shall, moreover, assu.me that condition (6.16) is satisfied. Then, as was shown in the preceding section, system (7.1) admits for sufficiently small ~ a unique periodic solution reducing to (7.2) for ~ = O. It is easily seen that in the case under consideration of analytic equations this solution will be analytic with respect to~. In fact, tbe functions xs(t,Pl' ••• tPn'~) will be analytic with respect to Pl' ••• '~n'~. The magnitudes ~l, ••• ,Sn-m determined by formulas (6.8) will also be analytic with respect to Pn-m+l'··· ,t3n'~ and the functions ~n-m+l (~), ••• , Sn(V.), determined by equations (6.9), will be analytic with respect to~. But then, eVidently, the required periodic function

135

will also be analytic with respect to

~.

We shall therefore seek to obtain this solution in the form of the formal series (7.3)

with periodic coefficients. Such series necessarily exist and if they are unique they will represent the required periodic solution and consequently will converge for sufficiently small ~. Substituting the series (7.3) in (7.1) and equating coefficients of like powers of ~ we obtain(~te following equations for determining the functions xsJ): (1)

dxs

_

+ ... + a x n + Fs (t,
1, ')-, ... , m;

204

+ ' .. +. 'PIIJ :t,~ = const (PI.

I'-1'» - , ... , ... , np .

(5.4 )

Relations (5. 4 ) determine the linear substitution with periodic coefficients, the substitution not being singular for any values of t. In fact, the determinant made up of the n 2 functions (5.2) is different from zero since these functions constitute a fundamental system of solutions of linear equations. But this determincmt, as can easily be seen, differs by a multiplier which never reduces to zero, from the determinant of the substitution (5.4) and therefore the substitution (5.4) is not singular for a~y values of t. Having established this, we transform the system (2.1) wi th the aid of the substitution (5.4). We talce into account the fact that expressions (5.3) are first integrals of the system (2.1). Differentiating these integrals with respect to t and equating the derivatives to zero we readily obtain the following differential equations:

1

(p)

a,;''!. .

......... (p

. .

= 1, 2, ... ,m).

I

(5.5)

. . }

J

These will be the transformed equations possessing constant coefficients. Thus, with the aid of the nonsingula linear substitution with periodic coefficients (5.4) the system of equations (2.1) is transformed to the system of equations (5.5) with constant coefficients. The obtained system (5.5) will have complex coefficients, since the magnitudes a p will in general be complex. Hence, if we wish to deal only with real coefficients further transformations will be necessary. We shall show how this is done. The magnitude ~p will be complex either when the corresponding root of the characteristic equation is complex or when this root is real but negative. Let us consider first the former case. Assume that the root Pi is complex. Since the coefficients Psi are u

205

real, all complex roots of the characteristic equation and all complex roots of the system (5.1) break down into pairs of conjugates. Let Pk be the complex conjugate of Pi • The solution~ corresponding to the root Pk' i.e. the functions Y(-J~) will be the complex conjugates of the . S (i) solutlons y . ,and therefore n k = n. and the variables (k) . 11 bSJth . l y~ Wl e e comp1 ex conJugate of the variables y. (~~ ) J

u

Let

(Ii

.1I}i) =

and let us take

"i --/- V - 1 f1 i . u}i) + V· - 1 v~i),

=

(i) uj

"k - v-=t fl" '

y?) = u}1!l -

v=T VJh l

as new variables instead

and

of

y\i) and y\k). Then, separating out in the i-th and J J k-th sets of equations (5.5) the real and imaginary parts, we obtain, in place of the two indicated sets consisting of n.l = n,K e~uations each possessing complex coefficients, one set consisting of 2n equations with real coeffici ients, of the following form (i)

dllt - dt

(i)

(i)

= A.Ut 1

u·Vt , , I

(5.6)

(j = 2, ... , n i ).

J

Assume now that Pi is a negative real number. In this case, taking the arithmetical value of the logarithm, we can write ex. = ~ In ( _ 0.) + (2q + 1) ~ 1

O, o

211:

SF' (p) d't -

21t < 0,

o

2'IC

2'IC

(So F' (?)cos 'td" -'It) (S F' (,) sin! 'Cd't -1t)2

0

2",

- (S F' ('f) sin o

234

't

cos 't

d'i.)2 + 'lt b :> O. 2 2

If these conditions are satisfied with the inequality signs the motion considered, will for sufficiently small ~ , actually be stable, and furthermore asymptotically stable. We may note that the first of these conditions is always satisfied with the inequality sign since the function F( 0/) is from its physical meaning an always increasing function. 12.

Application to the Problem of the Stability of Oscillations of Quasilinear Systems

In the preceding section we have shown on a particular example the applic ion of t~e above described method of computing the characteristic exponents to the investigation of the stability of the periodic oscillations of quasilinear systems. We shall now consider this problem for an arbitrary quasilinear nonautonoHlOUS system. Let there be given the arbitrary nonautonomous quasilinear system

~ic'>::= aHYt+ ..•' +a Yll+!,.(I)+fl Fs(t, Y} • • " • YII' fl) fiTl

(12.1)

(s=1, , .. , n)

and let Ys(t,~)

be a periodic solution of this system

for which it is necessary to find the characteristic exponents. We here assume that the functions f s and F s are continuous and periodic with respect to t with period 2n and that F for sufficiently small ~ admit continuous partial sderivatives with respect to y. 1

and ~ in the neighborhood of the periodic solution under consideration. The equations in variations for the solution considered are of the form ( 12.2)

where

. =.:.'JFS(l.!!i (t.$J.) • ••• , Yfl{I,P.).f-L) f s, 0Yi 235

•

(12.3)

t us cons

the system (12.4)

and assume that we are dealing with the ncnresonance case. The fundamental equation of the system (12.4) will then have neither a zero root nor roots of the form + pi, where p is an integer. We shall assume that this fundamental equation does not have either multiple roots or roots differing from one another by a magnitude of the form + pi. Such restriction for the nonresonance case is no~, evidently, essential. Let A. be some root of the fundamental equation of 1 system (12.4). The corresponding characteristic exponent ( .2) will then be of the form (X.i of the (12.5)

We restrict ours s here to the computation of this characteristic exponent with an accuracy up to magnitudes of the t order with respect to ~, for which it is sufficient to the magnitude a.1 = a~1 (0). In the case considered the system

has one s e periodic solution corresponding to the zero root of its fundamental equation. Hence the number m in the case cons will be equal to 1. The indicated periodic solution, which in correspondence with the notations of the ceding section must be denoted by fsl' has the form •

a

Equation (13.3) actually has the degree m. fm In fact, the coefficient of a in this equation is equal to the determinantlArj' which, as was shown in the preceding section, for the conditions considered is different from zero. REW~K.

14.

Stability of the Self-Oscillations in Two Inductively Coupled Circuits

We shall apply the method of the preceding section to the investigation of the stability of self-oscillations in the two inductively coupled circuits that were 246

considered in sec. 13 of chapter

•

The equations of motion aftet their reduction to the new independent variables T, for which the period of the periodic solution under consideration does not depend on ~, are given by the formulas (13.9) of chapter II. But since according to formulas (13.13) of the same chapter the magnitude aItt is equal to zero we shall have (14.1)

where-the terms not written dovm are of an order higher than the first with respect to~. For definiteness we shall consider the self-oscillations corresponding to the frequency ~l of the linear system rPXo d .. 2 -

d~yo· #-t "1XO 2

Xl

rPYo • J2xo -:r-x .. CT' J7: - 7;-

0,

2

+ n·,yo -

0,

1

I

(14.2)

I

for which the equation of the frequencies has the form ( 14.3)

These self-oscillations are, in the zeroth approximation, described by the equations ( 14.4)

From (14.1) and (14.4) we obtain the following equations in variations: iZll cPv ~xI

+nt u= 1

I~.! • ') fllt l { ill. 1 WI sm ...{I)J'r' 'l(z

d 2u

+ n 2,.v =

u+ (t -

n;

n.

- l' ~

~ U

au

-J 't

247

au } + ... ,

11[*2 2 _) 1 cos WI' (it

II

+ ... ,

(14.6)

where the terms not written down are of an order higher than the first with respect to~. For ~ = 0 these equations reduce to (14.2). The fundamental equation of the system (14.2) has the roots :t iWl and :t iW 2 of which the first two are critical. We shall first find the characteristic exponents corresponding to the critical roots. For this purpose let us put in equations (14.6) (14.7)

where (14.8)

and ~ and 2n/ w 1. We obtain

~

are periodic functions of

t

of the period

~ - ' I ~--I- n:u~~ {M:""n,sin2"lt.~+ M!' cos! ool't) ~~ -2a* ~: +2x la* ~~} + ... ,

+n l (1

.tf2o/ d'tl -

di'f 'K. ll

1

d... i

I

t

I

+ nsv = u {_ ni ~

•

,I

nI

d.jl

d~

-2a* .

dojl d-r:

+2a*x

.l

dCP} d.

+ • ••

( 14.9)

j

We shall seek the periodic solutions of these equations in the form of the series

+fl?l + ... , } '1 = '~o + I''h + .. . 'P 0 must satisfy

II = fo

The functions ~o and equations (14.2). These equations, as follows from their general solution (13.5) of chapter II, have the periodic solution ~o = N 1 COS OI J "! + IV:! sin 00 1" '¥o = kl (N 1 COS oo l 't + N"J, sin OOl't~, containing the two arbitrary constants Nl and N2 • For

~l

and

~l

we have

248

(14.10)

or (14.11)

where the terms not written down do not contain cos WIT and sin ~IT while the coefficients P,Q,R,S have the values p=

{1I1Wl (

Q

{nlwl

R

= { -

S

= {

1-! Alf2') - 2a*w 1 (1- k 1x1) } N!.

(-!- Mf2 -

:: 6k t ll)1

-

1)

2a*w 1 (1- ktX t ) } Nit

2a*w 1 (k 1

~ll Ok t W + 2a*w (k Il 1

nl

x2)

}

iV2 ,

x2 )} N I'

In order that equations (14.11) admit periodic solutions it is necessary and sufficient, as was shown in sec.13 of chapter II, that the relations be satisfied (formulas(13.12)) (ni - wi)R - x2w~P= 0,

(ni-wi)S - x2w iQ=0.

These relations have the form (Aa*

+ B) M2 =0,

(Ca*

D) Nt = 0,

whence we find the two following values for B a *1 = -A'

a*:l

a:

D -C'

Computing the values of the coefficients A,B,C.D, after simple transformations we find: (14.12)

There were here used relations (14.5) and also relations

249

obtained from them

n' -+

1- k1x 1 =

(14.13)

w- , 1

Substituting (14.12) in (14.8) we obtain the first approximation of the characteristic exponents corresponding to the critical roots ~ i~l of fundamental equation (14.3). We shall restrict ourselves to this first approximation. One of the characteristic exponents was obtained equal to zero on account of the autonomous character of the initial system (14.1). We shall now find the characteristic exponent corresponding to the root i&J 2. For this purpose we transform, as before, equations (14.6) with the aid of the substitution (14.7) but where we now have a.=iUJ'2

+"a*--!-IL2a rr 2 + ••• I

(14.H)

where the terms not written down have an order higher than the first with respect to~. These equations we shall also try to satisfy by a periodic solution of the form (14.10). Now however there are obtained for ~o and ~ 0 the equations d210 _. -;;:;r"1. 1 .1

2

40

-:;-:;- a1:-

d~~o d'C~

I ')' d"f/o -;- ... llJ)2~-

2'.l'l.lUJ::.----cit d",o + (!! 1I

2 ,."I.., -c/d 'f>o" + 2'[Il)" d4'o d'fo (:. d - 2', l'l."IIl., - J + 1/ .. -

't-

-

1:

~)

1 -1IJ 2

-

-

't"

-

~." - 0 'fo --! -Xlll)~'fO•

2)f0 _L ,"I.'••.llI·,(OO ! 0 r - _. = •

W.. -

which admit a single periodic solution and in this solution the magni tudes ~ and q; 0 are constant. This solution will be 0 '-10 = - NXllU~.

~o = N (II~ - w~).

250

where N is an arbitrary constant. For the equations d~,?!.. _

d~'¥1

-d~:! •

-t- 2'It!)., -d'Pl ,- -

G-o/'-.

d 't~·,

xI d ., t~

a~'f'l

- x2 -d ~ T

-

+.). d'li, _IW., --i-- -

?

ri't

(i'fl

_lX.,W., - -

dt

-

(

::;;

d J'l

0)'

... lX1W., .:..I..!..-

at

-

!!

112 -

dr

-

!!) ! W~ ? 1

~iwt."l.2a*r.pO + ( -

i

~l

and Wl we have

+

1 (14.15)

+ + X2U)~'?1 ?

nl 0(1)2 -

2i~)2a*) ~o.

To set up the conditions of periodicity of the functions ~l and t/J 1 we may make use of the general formulas, which requires reduction of equations (14.15) to the normal form. It is possible however to proceed in a more simple fashion. The fundamental equation of the homogeneous part of the system (14.15) has the roots 0, -:2i0l 2• ± iOl}-iu);J' and therefore the resonating terms in these equations will be only the free terms to which correspond the periodic solution with constant values for - ~l and 'V 1" Hence, retaining in the right hand sides of equations (14.15) only the free terms we shall try to satisfy them by the solution l = A, \V 1 = D, where A and B are constants. For these constants we then obtain the equations

~

[( ni - (I);) A + xl(1);B]

_1 [_ 2 if l':Ztl)2 A

+

?

2

(ll'l.-- (')2)

BJ =

~

+ + 2iU)~xla* + 2ilOaXl (II: -

= - inlUl:x I ( 1 -

-

•

Z0)2

)).1: 2 )

w;) a* t

n:!Q (nll_w 2) j!

nIl

2_

The determinant of this system is equal to zero since w 2 is a root of equation (14.3). Hence, for the written down equations to be solvable for A and B it is necessary that the condition of the compatibility be satisfied, which is the required condition of periodicity. To set up this condition we multiply by n ~ and -lC. l 4>~ respectively and combine. In this

wf

251

way after simple transformations we obtain

a*=a*,3

nl (/I~-{O)D (2Mj2-Mr2)

( 11.16)

8 (wg -;~i) (f-1"IX~-'-

0.

( 15.U)

is always satisfied. But it is satisfied also for the second of these solutions provided .;

A2 • b2=~>? gJ, ..... ,'~t..

(15.10)

' '

i.e. if the frequency of the oscillations of the point of suspension is sufficiently large. We note that far the condition (15.10) there are obtained for ~o two further solutions determined by the formula .

2

cospo .. - 6' •

For the oscillations correspondjng to these two solutions condition (15.9) is not satisfied and these solutions will be unstable. If the condition (15.3) for equation (15.1) is not satisfied the characteristic exponents will be developable in powers of the magnitude V~. They can be computed by a procedure simiiar to the one just considered setting in (15.1) x = e a y, where (1.5.11 )

and seeking the periodic solution in the form of the series

o~

the oTItained eguation (15.12)

We shall not here dwell on the question of the convergence of the series (15.4) and (15.11). We may note, finally, that in determining t~e magnitudes Yl' Y2'.·· arbitrary additive constants enter both in series (15.5) and in series (15.12). ~This corresponds to the fact that the solutions (15.5) and (15.12) can be mul tiplied by an arbitrary constant., which, in particular, can be chosen in the form of a series in ~ or in Vll with constant coefficients.

'f"

.~y

s will have the same structure with respect to the variables t, Ml, ..• ,Mn , 11 as the functions Fs with respect to variables t, xl'·." x n ' 11 • Equations (7.6) satisfy all conditions of the preceding section. Hence on the basis of the results there obtained we can state that these equations have an almost periodic solution reducing for 11 = (; to certain constants M;. These constants must satisfy the equations corresponding to equations (6.7) of the preceding section. On the basis of (6.4) these equations have the form , Pj(Mi• ... , AI,,)

limi ~ ~dt, )JI

t-HO

0

I • ... ,

M". O)dt=O (7.7)

It is necessary here to assume that the fundamental equati?n of the system corresponding to (6.8) does not have root~ w1th zero real parts. This fundamental equation on the bas1s of (6.9) has the form aPt

iJMs -x .,.

=0.

apn

••• aMn -x

Substituting the values of M.1 in (7.4) we obtain an almost periodic solution of system (7.1). This solution for 11 = 0 reduces to the generating solution

M;

The parameters of the generating almost periodic solution, to which corresponds the almost periodic solution of the complete system (7.1), satisfies equations (7.7), the analogous corresponding equations in the problem of periodic solutions. It is however necessary to bear in mind that in contrast to the problem of periodic solutions conditions (7.7) are not necessary in order that to the generating solution there correspond an almost periodic solution of the complete system. From the results of the preceding section it follows further that the obtained almost periodic solution of

284

(7.8)

system (7.1) will be asymptotically stable if all the roots of equation (7.8) have negative real parts, and unstable if at least one of the indicated roots has a positive real part. 8.

Almost Periodic Oscillations of Nonautonomous Systems for the Case of the fresence of Critical Roots of the Fundamental Equation. General Case

Let us now consider the system (7.1) on the assumption that the characteristic equation of the system (7.2) has m < n critical roots. We shall here as before assume that the number of sets of solutions of the system (7.2) corresponding to each critical root is equal to its multiplicity. The system (7.2) will then have m almost periodic solutions which we shall denote by ~sl' ••• ' ~sm and the system conjugate to (7.2) will likewise have m almost periodic solutions which we denote by tlJ sl ' • •• 4J sm • The system (7.2) will not have solutions of the form t~si + fsi ,where fsi are almost periodic functions. As a consequence, exactly as in sec. 12 of Chapter III, it can be shown that the almost periodic solutions ~ si of the system conjugate to (7.2) can be chosen in such manner that the relations are satisfied n

~

1.1=1

!fai

'~aj = Sii t Su = 1, 0ilt = 0 (i 4: k) (i,

i=

(8. f)

1, ... , m).

We shall assume that the functions ~si are actually chosen according to these conditions. We shall assume further that the functions fsCt) satisfy the conditions (i = t, .. " m).

(8.2)

(s = t, ~ ..• n)

(8.3)

The generating system

285

will then admit the almost periodic solution

(8.4) depending on m arbitrary constants Ml, ••• ,M. Here ~ m x~ is some particular almost periodic solution of the system (8.3). For this, as for every other particular solution of the system (8.3), the relations hold n

n

~ ~ X:*~ai = ~ /fJl '~lJi' (1=

I

(8.5)

«=1

We shall now seek to obtain the almost periodic solution of the system (7.1) that reduces for ~ = 0 to the generating solution. For this it is necessary to transform equation (7.1) to certain new variables. We introduce first of all the m variables 'g l' • • ., i m with the aid of the substitution n

~i = ~ Xa~ai

(i

a=1

= t, ... , m).

(8.6)

The remaining k = n - m new variables we introduce in the following manner. The equation 4 11 -p

°Il

au

a 22 -p

aliI

ani

....

aIR a 2R

.... ... an,,-p

=0

(8.7)

has n - m roots with real parts different from zero. We note these roots by Pl, ••• ,P z respectively. Each root is here written out as many times as the number of its corresponding sets of solutions of equations (7.2). Thus among the numbers Pj there may be some that are equal but to each of them co.rresponds one set of solutions. The fundamental equation of the system fit + aUl x. + ...

dzs

286

+ ans xn =

0,

(8.8)

conjugate to (7.2) has the roots - Pl, ••• ,-Pz , to each of which likewise corresponds one set of solutions. We shall denote by np the number of solutions of system (8.8) corresponding to the root p p • Evidently we have

n1 +n2 + ... +n,=n-m=k. Then, as was shown in sec. 1 of chapter II, the system (8.8) will have k particular solutions of the form (p) X sl -

C(P) • -lp'

x~~) =

(IC!r) + C!~»

x

.....

,

t n p -1 (p) ( snp - (np-l)! C sl (p) _

+

t n P -z (p) (np-2)1 C.:!.

(p =

where

1

e-1,J,

) -J.' + ... + Csnp e p (p)

f (8.9)

t, ..• , I),

Cs~p) are constants.

With this establi~hed we now introduce into consideration k variables ~~p) with the aid of a linear substitution with constant coefficients 1jj(P) --

n ~ C(~) L.J a1 a=1

xa

(p =

t, ... , I; i = 1, ... , np)'

(8.10)

Together with (8.6) we obtain n new variables. We shall find the determinant of the complete substitution (8.6) and (8.10), which we shall denote by D(t). This determinant has the form ~11 i'2J ••• ~nt

D(l)=

tJllm

~2_

C(t) II

c1"21

e(l)

dl) Zftl

e(l)

c12.' )

Inl

tl

e(l) In,

i>l.1)

···.· ... IJInm · .. eel) nl ··· · · .. nnl • ···.· . nl · · ·pet)· I c(t)

"

e(l)

'-'Zn,' .. '-'nnt

On the other hand, the functions (8.9) together with the functions I{I sl ' ••• , '" sm .form a system of n independent particular solutions of equations (8.8). The Wronskian A(t) of this fundamental system, after

287

elementary transformations on it, can be represented in the form (8.11)

But by the formula of Liouville for the Wronskian we can write:

where iV l , iV2, ••• ,i~ m is the aggregate of all critical roots of equation (8.7) (among the magnitudes Vj there may be some equal to each other or equal to zero). Comparing with (8.11) we find:

From this it follows that the substitution (8.6) , (8.10) is not singular and that the coefficients of the inverse transformation are almost periodic functions of t , being finite trigonometric sums. We shall ~ow transform equations (7.2) with the aid of the substitution thus introduced. Taking into account that the right hand sides of (8.6) and also the expressions (1"11\") + .",':!:')e-l..J.

Tj~P) e- 1pf •

. ... . . . .. .. . ... . . ... .

t

n -I P

(p)

( (p-1)! Tjl

n -%

t + (np-2)t "1~ P

p) •

(p)

I'"

+ Tjnp

)

e

...

-l.pf

are the first integrals of equations (7.2) we easily find that these equations after transformation assume the form del_dt:! _ _d~m_o dt - dt - ••• - dt • d.,,\P)

dt

_"1

(p)

-"pTjl'

(8. f 2)

dTj(P)

-'-=). ..,(P)--n(P) dt p j "lj_1

(p = f •...• I;

i=

1, ... , n).

J

Among the variables n\p) there may be some that are J complex. The latter break down into pairs of complex conjugates. If we wish to retain the equations in the real form we take(i~ place of each pair of conjugate complex variables n Pj their real and imaginary parts as the

288

new variables. In this way we variables. With the object of shall renumber these variables them by nl, ••• ,nk. The last then be written in the form dlJr

Cit =

b rill. + ...

obtain k = n - m real simplifying the writing we in some order and denote k equations of (8.12) can

+ brk 11k

(r = 1, ... , k).

(8.13)

The roots of the fundamental equation bIll b2k

...

I !

b~'~A I

=

0

(8.14)

of the system (8.13) will evidently be the magnitudes P l , • • • , ~t and consequently this equation does not have roots with zero real parts. We shall express the initial variables Xs in terms of the new variables ' i ,ni. We can write: (8.15)

where F si ' fsr are certain almost periodic functions of t, developable in finite Fourier series. We shall show that Fsi ~ ~si. In fact, substituting (8.15) in (8.6) and equating the coefficients of ~j we obtain (i,

i= 1,

.... m),

(8.16)

where 0ij is the Kronecker symbol. From this, on the basis of the relations between the solutions of conjugate systems of linear differential equations, we conclude that the functions F . determine the particular solutions of the system (7.2) ~l But since these solutions are almost periodic we must necessarily have: (8.17)

where A .. are certain constants. Substituting these Jl relations in (8.16) and taking into account (8.1) we

289

obtain:

and therefore relations (8.17) give can write: X~ = 'fn ~1

Fcx.:i .

= A(XJ'. Thus we

+ ... +fsm ~m + Is]7)1 + ... + ISk "lk'

(8.18)

We now proceed to the investigation of equations (7.1). In place of the variables Xs we introduce the variables El , ••• , !m' ~l' ••• '~k. We then evidently obtain:

(8.19)

Here d t • ~t "" ~~, lj1. ... , lj%, O)dl=Pi,+u i , o

291

(8.25)

so that (8.26)

,

and P,(M1 ,

••• ,

Mm}=lim~ (cI>i(t, ~~, ••• , llt 1-+00

t ~

(8.27)

O)dt.

Taking (8.21) into account the functions P. can also be represented in the form 1 PdMl' .. ' f M llI)=lim+ 1-.00

1

n

b

«=t

~ ~

F«(t,

x~, ... , x~, O)ljIaidt,

(8.28)

since to the functions ; , l? and 11 r0 in the variables xl"",xn correspond the functions (8.4). The magnitudes u i will evidently be almost periodic functions of t, being finite trigonometric sums. Both the functions u i and the functions P. are polynomials with respect to M , ••• ,Mm' l

1

The transformation now under consideration is of the form

~, = ~y + p.ui = 1j, =

n

Mi +

~ X:*'fai + p.u" a=t

)

(8.29)

1j: + p.z"

the magnitudes M. and z being taken as the new variables. 1 r For the first group of equations (8.19) we find:

where the terms not written down are of an order higher than the first with respect to~. Whence, taking (8:25) into account , we obtain: (8.30)

For the second group of equations (8.19), taking

292

into account that the functions n~ are a particular solution of the system (8.22) ( these functions do not depend on M.), we find: ~

Solving (8.30) for dMi/dt ( which for sufficiently small ~ is evidently possible) and setting in (8.31)

we shall have dMi _ P (M M 2n dt -1-1 i 1"'" rn)+p.

ydt. M 1 " ' " Mm.z.,

~7 == br1z. + ... + brl,zh, + Z~ (I,

MI' ... , M m)

"'J

Ztt, (1),

1

+

+(1Zr(t, .ill.,

••• , AIm' ZI' (i = 1, ... , m; r = t, ... , k = n - m).

••• , Zit.

(1) (S.32)

Hence the functions Q.~ and Zr have the same structure as the functions Fs in equations (7.1), namely, they are in the form of series which converge for sufficiently small 1-L

(S.33)

in which Q~P)and z(q) are almost periodic functions of ~ r t (finite trigonometric sums) and polynomials with respect to the remaining arguments. The same is true for the functions z~. For equations (8.32) an almost periodic solution can be constructed in the following manner: Assume that the equations (i

admit a solution

1, •.. , m)

Mi = M~ for which equation 293

(S.34)

I iJP 1

iJPt

iJPt

iJM. iJP:

aM"

iiM;'

iJP~

l)/).

1-- --'It

-

7jlti-;

--It

ilM"

iJM,~

iJP rn

JP rn . iJMm

.....

iJP rn

a:v'J-

·iJM~-

=0

(8.3!)

-It

has no roots with zero real parts. Let us set (8.36)

Equations (8.32) then assume the form

d:'

= f1 (cuNi

+ ... + cirnNm )+ (8.37)

where

..

)II.

and the functions Qi and Zr as the functions Qi and Zr.

have the same structure

We shall now seek an almost periodic solution of the system (8.37) by the method of successive approximations. As the first ap{>roximation we shall take an almost periodic solution of the system d zr(t)

. (t )

(t)

(0)

•

tit = b,.,z. + ... +brhZh +Zr (I,ftl •• "" M !.).

dN U) ,

(I)

--;J;:- =f1(cu N •

(8.38)

+ ... +CimNm)+ (I)

+p.Q, (t. Mi •. ",

zlJ). 0)

and as the p-th approximation an almost periodic solution of the system

294

tll!~J)) iit -~ b,..z.(p) + ... + b,hZ"(pt + Zor (1/_ I. j J.

••• t

I

tI Z"tp-t» f1 + fl Z"r- (N{P-t) 1- CimN~ra» + IlQi (I. ,tit, ... zlP). 0) + j

~~ -it-

fl

(cifN~P) .,- ...

1

If·· ) +

Jr. m

•• • • •

t

t

t

I

(1'-1» I . . . . ,zlt tfl. J + 11.2Q-i (t • N{1'-t)

(8.39)

Since equation (8.14) and the equation I''t~ -

l!

•• ,

c~1Il

do not have roots with real parts equal to zero the system (8.38) admits one and only one almost periodic solution. In exactly the same way equations (8.39) also uniquely determine sequences of almost periodic functions Nip) , z~p) that determine a particular almost periodic solution of system (8.37~ Returning to the initial variables Xs we obtain the almost periodic solution of the system (7.1). As is seen from (8.36) and (8.29), this solution for ~ = 0 reduces to the generating solution (8.4) for which the parameters Mi have the values M~1 i.e. are roots of equations (8.34). This leads to the following theorem: 1 1 This theorem was established in the paper: Malkin I.G., 0 pochti-periodicheskikh kolebaniyakh nelineinykh neavtonomnykh sistem, ( On Almost Periodic Oscillations of Nonlinear Nonautonomous Systems), Prikl. matem. i mekh., vol. XVIII, no. 6, 1954. The problem considered in this section was investigated also from another point of view in the paper: Biryuk G.I., K voprosu 0 sushchestvovanii pochti-periodicheskikh reshenii nelineinykh sistem s malym parametrom v sluchae vyrozhdeniya, ( On the Question of the Existence of Almost Periodic Solutions of Nonlinear Systems with Small Parameter in the Case of Degeneration), DAN SSSR, vol. XCVII, no. 4, 1954.

295

THEOREM.

ASSUME THAT FOR THE EQUATIONS

t!ft = aux. + ... + aSllx + 13 (I) + fJ-Fs (it x fI

J'

••• t XfI'

p)

(7.1)

THE GENERATING SYSTEM

ADMITS AN ALMOST PERIODIC SOLUTION

x:

M1'PSl +

... + M m?sm + X:* (t)

AND THAT THE FUNDAMENTAL EQUATION OF THIS GENERATING SYSTEM HAS n - m ROOTS WITH REAL PARTS DIFFERENT FROM ZERO. THEN, IF THE PARAMETERS Mi SATISFY THE EQUATIONS

AN EQUATION (8.35) HAS NO ROOTS WITH ZERO REAL PARTS THE SYSTEM (7.1) FOR SUFFICIENTLY SMALL ~ ADMITS AN ALMOST PERIODIC SOLUTION REDUCING TO THE GENERATING SOLUTION FOR ~ = O. The theorem established is an immediate generalization of the corresponding theorem in the problem of periodic solutions. In particular, equations (8.40), which by assumption are satisfied by the magnitudes Mi' are an immediate generalization of the corresponding equations in the problem of periodic solutions. However we here have also some essential differences. First of all, in the theorem just proven an essential assumption is that the fundamental equation of the generating system has n - m roots with real parts different from zero, a condition not required in the problem of periodic solutions. Further, in the problem of periodic solutions it was sufficient to assume that the magnitudes Mi are a simple solution of equations (8.40) or what amounts to the same thing, that the equation (8.35) has no zero root. In the problem of almost periodic solutions however the stronger assumption is made that equation (8.35) has neither a zero root nor purely imaginary roots. Finally, in the problem of almost periodic solutions conditions (8.40) are not necessary. As a result of the special features noted above the fundamental results on periodic solutions that were 296

established in chapter II cannot be considered as a simple consequence of the results of the present section. The problem of periodic solutions for those general assumptions for which the problem was treated in chapter II, if it is considered as a special case of the problem of almost periodic solutions, belongpto those specific cases that require special investigation.

9.

Proof of the Convergence of the Successive Approximations

We now go on to the pro9f that the(s~quences of almost periodic solutions z~p) and NiP) considered in the preceding section converge and actually determine an almost periodic solution of the system (8.37). These sequences are determined, as we have seen, as the almost periodic solutions of the system of equations 4- UZ'·I (I N(P-O • r r tit' eN(P)

d:

•• ,

N(P- t)

m

t

..(p- f)

"I

t· • • t

= f.' (CilN~P) + .. , + CimN~» + p.Qi (t, :~P), it··· + ru!Q!" (t tN(P-t) I

aT (t) = Z~ (t,

M:

t

••• ,

t

N(P-I, ..(p-t) m t "'1 t'

..(p- t)

""It

••• ,

p. ) ,

%1P » +

",(P-t» •• , ...,\

t

1

t

(9. t)

P. ,

M:.).

Q, = Q, (t, Mr, ... , M~t %~P),

••• ,

ziP), 0).

)

where

(9.2)

Let fl(t), ••• ,fm(t) be arbitrary almost periodic functions and f the upper limit of their moduli for all real values of t. Since equation (8.35) has no roots with zero real parts, the system of equations

297

as shown in sec. 2, admits a single almost perlOQlC solution vi vi(L) and this solution satisfies t~e inequalities

where P is a certain constant not depending on the choice of the functions f i • Let us now consider the system (9.3)

Setting in it

L

= ~t

we obtain:

whence it follows immediately that the almost periodic solution viet) of system (9.3) satisfies the inequalities

/vd t ) I < ~p. PI·

(9.4)

Since equation (8.14) likewise has no roots with zero real parts we can say that, whatever the system of almost periodic functions Fl(t), ••• ,Fk(k) for which IFr(t)( < F, for the almost periodic solution of the system

(9.5) the estimates hold

I wr (t) I < QF,

(9.6)

where Q does not depend on the functions Let us now choose a positive constant the region determined by the inequalities

Iz, _z~u (I) I "SlDwlt} COS wIt.

(12.13)

with

and equations (12.6) determining the parameters Ml and M2 differ little from the corresponding equations (5.16) of chapter I in the problem of periodic solutions. These equations can be investigated in exactly the same way as in the problem of periodic solutions. We shall restrict ourselves however to the consideration of the case of exact resonance, i.e. we shall assume that a = 0. In this case the equations have the solution M2 = 0, Ml = M, where M is a root of the cubic equation

(12.14) The roots of equation (12.7) will now be the magn;tudes . dP (.'1) xl

=

dJI

1 BI ]If I 'A--' x2 = 2- - 4 (wi -wU· - 8 - 2wIM •

Hence, if equation (12.14) has one or three different real solutions equation (12.7) will have real roots different from zero. In this case equation (12.12) will have one or three almost periodic solutions, reducing for ~ = to the solutions

°

319

of the generating system. Since equation ( ) cannot have a trip root at least one almost periodic solution always exists. The conditions of stability of the obtained periodic solutions have the form ill

< 0,

dP(M) dM

al~ost

< 0.

Hence only those almost periodic solutions can be stable which correspond to negative roots of equation (12.14). But this equation, as is seen from the expression of its free term, always has at least one negative root. If there is only one such root the almost periodic solution corresponding to will actually be stable since p( -oq,.. 0 and therefore for this root p' (M) < O. If however equation (12.14) has three negative roots the almost periodic solutions corresponding to the least and greatest of them will be stable while the solution corresponding to the middle root will be unstable. 13.

Analysis of the Equations Determining the Parameters of the Generating System. Resonance and Nonresonance Frequencies

Let us consider again dz,

dt

= 4.1%1

th~

quasilinear system

+ ... + as"x" + Is (1)+ t'- Fa (t, Xl' . . . , X"' p.)

(13.1)

(s=t, ... ,n)

for the assumptions of sec. 8. The generating system will then have the almost periodic solution (13.2)

depending on the arbitrary sonstants Ml, .•. ,MM· The fundamental equation 4 1l -p

412

a lft

au

4 2t -p

42ft

aft.

4d

320

a",,-p

=0

(13.3)

of the generating system has m roots with real parts equal to zero. The remaining n-m roots of this equation have real parts different from zero. Let us assume that among the m roots of equation (13.3) with zero real parts there are 2q purely imaginary roots + Ali, ••• , + i\ i, q so that the magnitudes A. are the frequencies of the l. free oscillations of the generating system. Let VI"'" \) N be the aggregate of all frequencies figuring in the expansions in trigonometric sums with respect to t of the functions Fs(t,xl, ••• ,xn,O) and also of the functions f s (t) • We shall call i\.p a RESONANCE frequency if at least one of the magnitudes (13.4)

is of the order of smallnessof lJ.. Here ml , ••• , mil' np .•• , nN are positive or negative integers for which

I m.1 + ... +, mq I

+,

nll + ...

+ InN I 2:

,. ~ Pa {t, x~ .

.sl

... , x!. O)'~"i = ,~P (M ip cos vp + N il.sin vp) (13.9) (;=3, ...• m),

where Vp = m(p) (All

and

- ex)

+ (mg')A:! + ... + m~P)/Q+ n\")'#1 + ... + n~\'V)1

m(p) , m (p), ••• ,n (p) 2 N

are integers

Im(p)l+ Imlt)I+ ... +IIIW)I~r+2.

for

which

Since the frequency r...l is nonresonance the coefficient of t in v can become zero only for the condition that m(p) =PO. But then v will also not contain a •. From this it follows that tHe free tErms in (13.9), i.e. the left hand sides of the last m - 2 equations of (13.5) , will not depend on a and we can write ~hem in the following form:

323

Pi(M •• ... , M"J=Q.(Ms ' "" Mm. )1)=0 (i=:l, ..• , III).

(13.10)

Further we have : C. r.os A,' - D_ sin AI' = = [C .. ("·os (AI' -- 61 (OY) ( O~ ) 1";" + ••• + AO' p1>,p 1:; +

Xs -

+ A~"l?'.

2p+J

(l) + •••

+ A:'-p?sm (t) + x:* (I).

(15.22)

The system (13.1) thus admits in the first approximation a family of almost periodic solutions for which the 332

parameters Ag in the generating solution satisfy the equations (15.17). This family contains p arbitrary constants El, ••• ,E p ' i.e. as many constants as the number of nonresonance solutions possessed by the generating system. In the obtained almost periodic solution of system (13.1) there are present not only the resonance frequencies but also the frequencies (15.23)

(i=1 •... , p),

which differ by small magnitudes of the order of from the nonresonance frequencies. Let us now consider the question of of the family of almost periodic motions For this purpose we set up the equations motion for the magnitudes A and e. (15.16) g g Ag= A:+«g.

~

the stability thus obtained. of the disturbed Putting in

8/=0'+&1'

( 15.24)

where (15.25)

and Pg(al, ••• ,am_p ) are analytic functions of the variables a g (polynomials) the expansions of which in powers of these variables start with terms of not lower than the second order. We shall assume first of all that all roots of the equation (JR (JR (JR t

t

(JAt

oAt

---4,1)

t

• , • (JAm-p

(JR,

(JR:

(JR,

- - w ••• (JAm-p -dA t

OAt •

•

•

•

•

•

*

..

•

•

•

•

(JRm-p

•. '~A v m.... p

333

=0. •

•

-w"• 9 =Ao,

(15.26)

and also all roots of equation (15.20) have nega~lve real parts. Then from the first group of equations (15.24) it follows immediately that all the solutions a g (t) of these . equations exponentially approach zero as t increases without limit, provided the initial ~lues of the functions a g (t) are sufficiently small. But then from the second group of equations (15.24) it follows that lim &j (I) = 3;.

(15.27)

t-co

where

since the integrals figuring in OJ evidently converge. Turning now to the functions zr(t) of equations (15.16) we readily find that for arbitrary zr(O) the relations are satisfied lim zr(t) = %;.(t), l-+co

where zr(t) is an almost periodic solution of the equations

~; = br1Z1 + ... + brkzlt + Z~* (t, AO, flO + 3). From what has been said it follows that if all the roots of equation (15.26) and equation (15.20) have negative real parts each almost periodic solution of equations (15.16) belonging to the family (15.18) is stable. If the disturbances are sufficiently small any disturbed motion for t ~oo asymptotically approaches one of the almost periodic motions of the above mentioned family ( not coinciding in virtue of (15.27) with the undisturbed motion.) Let us now assume that either equation (15.26) or equation (15.20) has roots with positive real parts. Then either the trivial solution a l = ••• =am-p = 0 of equations (15.24) will be unstable or there exist solutions of the equations for zr with arbitrarily small values zr(O) for which l'1m (t + '" +ZIlt) = 00. 2:1 1..00

334

lience, in the case considered each of'the almost periodic solutions of the family (15.18) is unstable. From the results of sec. 13 it follows that, for the assumption made in the present section that the frequencies ~ , ••• ,~ are nonresonance frequencies, the first 2p functtons ofP (15.3) necessarily reduce to zero for Ml = ••• =M 2p =O. From this it follows that the first p equations of (15.17) are necessarily satisfied for Al =••• = Ap=O. It is easy however to satisfy oneself directly that for j = l, ••• ,p it is possible to write: Rj(Al' ... , Am_p)=AjRi(Al'

.'O.

Am-p)'l

(15.28)

Sj (A" ... , Am_p) = AJSf (AI' ...• Arn_p)

(j = I, ...• p).

where R I . , S I . are polynomials in Al , ••• ,A • In fact, J J ~p the functions P.(Ml, ••• ,Mm) are polynomials and consequently ]. rational ,which, on the basis of (15.8) and (15.12) is possible only when relations (15.28) are satisfied. Thus, the system of equations (15.17) can have a solution in which Al = ••• = Ap = O. This evidently is that solution to which correspond the almost periodic oscillations of the system (13.1) in which there are no nonresonance frequencies and which have been investigated in detail in the preceding sections. If we therefore restrict ourselves to the first approximation the. method of the present section makes it possible to determine not only the oscillations with nonresonance frequencies but also the oscillations investigated in the preceding sections. For investigating the stability of the above oscillaand also for clarifying the question of their actual existence it is necessary to know the signs of the real parts of the roots of equation (8.35). We shall show that this can be done with the aid of equation (15.26) (the degree of which is less by p units than the degree of equation (8.35)so that there is no need of setting up equation (8.35). tions~

In fact, on the basis of (15.12) and (15.28) we have:

335

OPi)=R:=(ORi) ( aMi . - oAj , ( app+i) = iJAlI

S~ 1

== ( iJAj' aS j )

I > I

(..!!:.L) = aMp+j

-SI"-_-(3..) _) aAj ,

(~Pp+j)

R'.= (fJRj )

iJM p+1

=

iJAi'

J-

(:~)=(~~/)=O (i4:i, i4:pti; i=I, ... , P; i-=I, ... , m),

(15.29)

J

where the parentheses denote that the derivatives have been computed for Ml = ••• = M2p = 0 and therefore for Al = ••• = Ap= O. From this it follows that equation (8.35) for Ml = •.. = M2p = 0 assumes the form

-( :!i)

p

D(x)=

IT

iJRi) -x X ( iJAj

i=t

ap'J.P+l"\ _ ( aAl P+l) 2

1l. • •

(ap!p+l) aMm

x ...... . aPm) - -x (aM".

and has therefore the

roots

and m - 2p roots determined by the equation

(ap!p+l) aM m . . . . . . . . . . .. . . . . . . =0. aPtP+1 ) _ ( aiLlsP+1 x ••. ".

•••

aPm) -x ( aMm

On the other hand, from (15.28) we have:

oR, ( aAIt

)=0

(i, k = I, ... , p;

336

i

=f:::

k),

( 15.30)

whence we easily find that equation (15.26) for Al= ••• =Ap=O has the p roots (j= It ... , p)

and m - 2p roots determined by the equation oRp+1 ) _ ( iJAp+l

(I)

•••

(oR p+l ) oAm-p

. . . . . . .. . ..... .. .. ( oR~) OAP.l

oRm-p) ••• ( oA _ m p

=0,

(J)

which agrees with equation (15.30) (since

Hence the real parts of the roots of equation (8.35) for Ml ••• = M2p = 0 agree with the real parts of the roots of equation (15.26). The problem of stability therefore for the oscillations corresponding to the solution of equations (15.17), for which Al =., •• = Ap = 0, is solved by the same equation (15.26) by which is solved the problem of stability for the oscillations corresponding to the other solutions of equations (15.17). We may note in conclusion that the results of the present section are equally applicable to nonautonomous as well as to autonomous systems. 16.

Oscillations with Nonresonance Frequencies. Properties of Exact Solutions

In the preceding section we considered in detail the oscillations with nonresonance frequencies determined by the equations of the first approximation. We showed that these oscillations are found to be almost periodic and can be described by equations (15.22). Let us now consider what character will be possessed by the oscillations described by the exact equations of the motion. We saw earlier that oscillations in which nonresonance frequencies are absent will be found almost periodic also in the case

337

where the exact equations of the motion are considered and we established a converging process of successive ap~roxi­ mations for determining these oscillations. The situation is otherwise in the case of oscillations with nonresonance frequencies. In this case the exact equations of the oscillations, in contrast to the equations of the first approximation, do not have a family of almost periodic solutions. Nevertheless, for sufficiently small ~, we can, with an accuracy sufficient for practical purposes, consider that the equations (15.22) describe actual oscillations of the system and therefore these oscillations can be considered almost periodic. Thus, letous assume that equations (15.17) have a solution Ag= Ag for which all roots of equation (15.26) have negative real parts. For the assumptions made, the system (13.1) admits in the first approximation a family of stable almost periodic solutions described by the equations (15.21), or by the more simple but for practical purposes sufficiently accurate equations (15.22). Let xs(t) be the exact solutions of equations (13.1). We shall show that the following theorem holds true: THERE EXISTS A NUMBER l-l~ AND A CERTAIN REGION G OF THE VARIABLES Xl' ••• ,xn SUCH THAT FOR t~ 0, lJ,$lJ,"" ,xs(O) .THE FUNCTIONS xs(t) CAN BE REPRESENTED IN THE FORM

)+ ... + Ap?sp (~: ) + + A ?. (I) + ... + Am-J,Tsm (I) + t4s (I), OJ (I) = OJ (0) + [A. Sj(A~, ... A?n-p) + f1~j (I) ] 1 XS

EG

(I) = Al~$l ( : : p +1

2p+l

j -

;,

(s= I, .,., n;

(16. 1)

t

i= I,

..

'I

p),

WHERE THE FUNCTIONS Ag(t), ~.(t), x (t) POSSESS THE FOLLOWING PROPERTIES: J s 1) HOWEVER SMALL THE POSITIVE NUMBERS El AND E2 TWO OTHER POSITIVE NUMBERS ~ AND l-l o CAN BE FOUND SUCH THAT THE FUNCTIONS A. g AND ~j WILL FOR ALL t> 0 SATISFY THE INEQUALITIES

I Clol = I All (I) -

A: I < ai'

I?, (I) I < a2 ,

338

( 16.2) ( 16.3)

PROVIDED THE INEQUALITIES ARE SATISFIED II.

< Po < 1'.'

(16.4)

Iag (0) I = I Ag (0) - A~ I < "I: 2)

FOR ALL t> 0

( 16.5)

THE INEQUALITIES ARE SATISFIED ( J6.6)

B IS DETERMINED BY THE EQUATIONS OF

WHERE THE CONSTANT MOTION.

PROOF. We shall start from the equations of motion transformed to the form (15.15). In these equations we put Ag = A~ + aj/'

On the basis of (15.24) we then obtain:

~~g

=

_dt, _ dt -

+ p.P (aI' ••• , Il + + p.2R; (I, 0, AO +«, Z, p.), brlz l + . • • + britz" + Zr. (i, A +a, '1) + p. (dg1a 1 +

... + dg , m_pa

lll _

p)

0

r

+I1Z~C:t dO; I' (Jt=Aj-y. J

0

[Sj(AI +a 1 •

_ ) tn p

y

••• ,

(16.7)

Ii.

6, AO + a,

p.),

(J 6.8)

+I1Sj(/, fJ, AO+a, z, p.»).

( 16.9)

I

Z,

0

Am_p+a p)+

We now form the two quadratic forms V(al, ••• ,am-p ) and W(zl, ••• ,zk) wi~h the aid of the equations m-p

~ :~

,,,_ p

(dOl

al + ... + dgt m_p(lm_p) =

g=f

-

~ a;, g=f

k

~ ~w uz,.

It

(b r1 z) +

... + brkz,,) = - .-'.J ~ z;" T=1

r.=f

Since by assumption the roots of equations (15.26) and (15.20) have negative real parts, both forms V and W will, on the basis of the theorem of Lyapunovmentioned in sec.ll, be positive definite. For the total derivatives of these

339

forms with respect to time, formed in virtue of equations (16.7) and (16.8), we shall have:

~= ~ (

m-p

m-p

-

~ ex; + L :~ P9 +!~ ~ :~ g=1

g=J

h

dW ~

= -

~ £.J

Z~

m-p

(16. to)

g=1

h

It

+ £oJ ""

R;) .

aJV chI"

Zt)* (t , AO + C{, r

~ mv Z·T' (I ('), II) O) + Il £oJ;r:.,.. r=1

r=1

Let there be assigned a positive number h and let us consider the region of variation of the variables ~l' ••• '~m-p bounded by the ellipsoid V(~l, ••. ,a ) = h. In this region o" 0 \ m-p the functions tZr (t,A + ~,e) have constant upper bounds. Taking this into account let us consider the expression dW/dt at the points (zl, ••• ,zk) lying on the ellipsoid W(zl, ••• ,zk) = C, where C is likewise a positive number. It can easily be shown that if ').1 0 or for ~ .. I+ ... O. We shall assume that this condition is satisfied. but in satisfying this condition it cannot yet be assured that in the above mentioned regions unstable oscillations

394

actually arise in the electrical circuit. The reason is that the variable x figuring in equation (1.3) is an auxiliary one. For the current q we have: -~,

q= e

2L

x.

Consequently, on the boundaries of the regions (4.11) and (4.12), where the real parts of the characteristic exponents of equation (4.10) are equal to zero, the oscillations of the current will be damped. If the magnitude R/2L is sufficiently small, namely, less than the greatest value of the real parts of the characteristic exponents of equation (4.10) in the regions (4.11) and (4.12), regions of instability will exist for the current q, situated within the regions (4.11) and (4.12). If this condition with regard to R/2L is not satisfied the current oscillations in regions (4.11) and (4.12) will be damped. We may remark that the width of the second region of instability (4.12) is of the order of smallness of ~2. As a result, unstable current oscillations can arise in it only for very small values of R/2L. The values of this magnitude must be still smaller for the possibility of unstable oscillations within the third region of instability since the width of the latter 0n the basis of (4.9) 3 Hence for practical is of the order of smallness of ~. purposes the first region of instability is of the most significance. Similar circumstances apply also in other cases of parametric resonance. Since in practice resistances are unavoidable, the regions of instability corresponding to n~l generally have no practical significance.

5.

Problem of Parametric Resonance for Canonical Systems with Many Degrees of Freedom

Let us now consider the problem of parametric resonance mechanical systems with many degrees of freedom. We shall assume that these systems are described by linear canonical equations of the form ~or

(k

t, ... ,m),

(5.1)

where H = H(~'Yl"",ym' zl, ... ,zm' ~) is a quadratic form of the variables Yl' .•. 'Ym' zl, ... ,zm whose coeffic~ents are continuous periodic functions of ~ of period

395

T = 2n/~ and analytic functions of the parameter sufficiently small values.

~

for its

Particular cases of the systems (5.1) are the systems

provided the conditions qki the system (5.1) for

= qik are satisfied.

In fact,

2lf has the form

and therefore coincides with the system (5.2). For m = 1 the system (5.2) reduces to one equation, of the second order the particular case of which was considered in detail in the preceding sections. Let us put in equations (5.1)

after which they will assume the form

ddYIl t

where H(t'Yl, ••• to t.

= J. °aBZit ,

,zm'~)

(k=l, ...• m).

(5.4)

will "have the period n with respect

The problem of parametric resonance for the system (5.4) consists in determining those values of the frequency GU or of the values of the parameter ~ for which the char-

396

acteristic equation of this system has roots with moduli greater than unity. ~his problem for certain rertrictions on the function H was considered by M. G. Krein , who established several important general theorems. In particular, Krein succeeded in extending to systems of the form (5.4) certain fundamental theorems of Lyapunov on a second order equation. We shall here consider the problem for more particular assumptions, which will permit us to obtain certain more concrete results and also procedures for practically determining the regions of stability and instability. We shall assume, as in the case of one equation of the second order, that the system of equations (5.4) little from a system of equations with constant coefficients. other words, we shall assume that H is of the form 11 = Ho+plll

+ . . 211 + ... .lo- Ho+ pH*. 2

(5.5)

where Ho' Hl, ••• are quadratic forms of the variables Yi' zi' the coe cients of the form Ho being constants, while the coefficients of the forms Hl , H2 , ••• are periodic functions of t of period n. In what follows a fundamental part will be played by the fact that for the system (5.4) the theorem of Lyapunov, proven in sec. 8 of chapter II, holds according to which for each root p of the characteristic equation of this system there is a root lip of the same multiplicity.

1

Krein M. G., Osnovnye polozheniya l-zon ustoichivosti kanonicheskoi sistemy lineinykh differentsial1nykh uravnenii s periodicheskimi koeffitsientami (Fundamental Aspects of the ~ Zones of Stability of a System of Linear Differential Equations with Periodic Coefficients). Collection of Articles, .. Pamayati Aleksandra Aleksandrovicl:a. Andronova Memory of Alexander Alexandrovich Andronov), Izd-vo AN SSSR, 1955. I/{

397

(

We can therefore assume that the characteristic exponents of systems of the form (5.4) are pairwise equal and opposite in sign. From this, in particular, it follows that all the roots of the fundamental equation of the system of linear equations with constant coefficients (5.6)

will be pairwise equal and opposite in sign. But then, evidently, the same will be true for any values of ~ for the fundamental equation of the system (5.1)

Having established this, let us assume that the fundamental equation of the system (5.6) has roots with real parts different from zero. Then this equation and together with it the fundamental equation of the system (5.7) will have roots with positive real parts. .PtS a result, for sufficiently small ~, the system (5.4) will for any values of A. have characteristic onents with positive real parts and for it instability will take place. This case is of no interest to us. We shall therefore assume that the fundamental equation of the system (5.6) has only purely imaginary roots. For this, as is known, it is sufficient l that the form Ho be of determinate sign. We shall assume that this condition is actually satisfied and for definiteness take Ho> O. For suffiCiently small ~ the form H will then likewise be positive definite, as we shall also assume. Let t LV 1 i, ••• '.± 4Jmi be the root s of the fundamental equation of the system (5.6), so that the magnitudes W. J are its proper frequencies. We shall assume that all the magnitudes ml WI + ••• + mmWm for any integral (positive and negative) ml, ••• ,mm are different from zero. From this it follows, in particular, that all the magnitudes W j are different.

1

This condition is not however necessary. 398

I

Fig. 29 Let us now pass on to the determination of those values of ~ near which can be found regions of instability. For this purpose let us assume that there exists a number ;t.- (lJ.) such that for A. = r-.'" all roots of the characteristic equation of th~ system (5.4) have moduli equal to unity while for~·-0(0, V-=t + pal + p2a 2 + ... =

= NY -1+pa1 + 1'2a2 + .. . To determine them, following the method of sec. 13 of chapter III, we put in the equations (6.10) (for A.:::: r..

0 + lJ.lC)

and determine the constants aI' a , •.• from the condition 2 of the existence of periodic solutions for the system thus obtained n

~-O+tn:) ~ (asll+I'/!!)+" ')Y(J-(r1al+" .)y,. (6.20) P=l

For our purpose it is sufficient to determine only the magnitude a l • If the periodic solution of system (6.20) is sought in the form of the series

410

where Po and Qo are arbitrary constants, we obtain for the equations n

y~l)

ft

d~U = i-o ~ a~~ y~l) + ~ P=I

+ >-OfW)(Po~~1 + Qo'i'P2)-

(M"

~=I

and the conditions of periodicity will give:

+ Bu - a1}Po+(xA:u + B21)Qo=Ot (xA 12 + B 12 J Po + (xA 22 + B22 -a 1) Qo =0. (xA ll

}

(6.21)

To simplify the computations we here assume that the functions'sl and I{J s2 are chosen such that the conditions are satisfied ft

~ ~d?ai = 1,

(6.22)

a=1

Equating to zero the determinant of system (6.21) we obtain a quadratic equation for ale One of the roots of this equation corresponds to the characteristic exponent reducing for 1.1. ::: 0 to +Ni and the other to the exponent reducing for 1.1. ::: 0 to -Ni. Since the system (6.10) is canonical the coefficient of the first degree of a l in the quadratic equation, determining this magnitude, should be equal to zero. Taking this circumstance into account we easily find from (6.21): (6.23)

where A, B, C have the same values as in (6.15). As was pointed out above, the regions of ~stability are situated on either side of the interval (~l' ;.: 2). As a result, if the magnitude ~ lies outside the interval (ail), ai 2 » both roots (6.23) must be purely imaginary and therefore the quadratic function kjc.2 + B)( + C is positive. But then for ai l )(W(ai 2 ) this function must be negative

411

since the magnitudes

a~l)

and

a~2)

it immediately follows that for

are its roots.

a~1~~j

"OWj

=

+ 2, + >-00>" + 2.

(8.5)

For the solution of the problem in the case (8.3) we shall apply the method of sec. 6 and in the cases (8.4) and (8.5) the method of the preceding section. In all cases we shall restrict ourselves to the first approximation. Let us first consider the case (8.3). Following the method of sec. 6 we seek the value ').. = 7L4', corresponding to the boundary of the region of instability, in the form of the series (8.6)

the coefficients of which we determine from the condition that for i- =;: the system (8.1) admits a periodic solution of period 2n, developable in the series

xs = x: + f1X~u + ...

417

For the functions

X

O

s

we have the equations (r::p J).

The periodic solution (of period 2n) of these equations has the form (r -::/= J),

where M and N are arbitrary constants. obtain ~or x~lf the equations tJl

From this we

z +. + .lot + (w'a +>"~/jj) (Mocost + Nosint) = 0, l.l)

1

d 2z ( l )

d,; + >"~w;x~n + 1..~/ii (Mocost + No sin t) = 0 (r* J).

The second group of these equations admits a periodic solution without any supplementary conditions. As to the magnitude x~l), in order that it come out periodic it is necessary that in the equation which determines it the coefficients of cos t and sin t reduce to zero. In this way, taking (8.2) into account, we obtain the following equations:

i >"!PW) Mo+ ! A!QWNo=O, ~ l.!Qi})Afo+ ( (1);«1 - ! A!Pj}') No 0,

(I)~1l1 +

=

whence we find:

Hence the required boundaries of the region of instability are determined in the first approximation by the following formulas:

418

(8.7) Thus in the first approximation the functions fs~ do not show either the mutual effect of the coordinates or the higher harmonics. Let us now consider the cases (S.4) and (S.5). Following the method of the preceding section we set in equations (S.l)

and seek the characteristic exponent of this system corresponding to the root ~ W.i. Since in the case cono J sidered the equations are analytic with respect to ~ this characteristic exponent may be sought by the method of sec. 13 of chapter III. For this purpose we make the substitution

and determine the coefficients a l , •.• from the condition that the obtained equations have a periodic solution of period n. These equations, if we discard in them all terms of order of smallness higher than the first, have the form . fJ:Jy~

d,e

+

'\2 "0

(t (I), -

~)

wI

Y,I +

21

' dy,.

"OwI'

dt + n

+ { (xw; + 2a.>-Owl i) Y,I +A! ~ /saY. + 2a ~8 1

}

11 = O.

111=1

We shall seek to obtain the periodic solution of these equations in the form of the series Y,I -- gil0 -t- P.YIIIt) + •••

For yOs we obtain the system (8.8)

419

which breaks down into n separate equations of the second order. The roots of the fundamental equation for the s-th equation will be the magnitudes + A (.c.) i-A. ",.i. -

s

0

J

0

Hence only the j-th and k-th equations have critical roots 0 and -2i. The periodic solution of period n of the system (8.8) will therefore have the form y~

0

(r

+ j,

r

'*' k),

where Mo and No are arbitrary constants. Having in this manner determined the functions y~ we obtain for y~l) the following equations: J!y'll

~ +).2 (Z -rrIl (tIS ~t

-

2)

I»j

dyH'

6 + YsII) + 2""000,'. - d I

All equations of this system, with the exception of the j-th and k-th, admit periodic solutions without any supplementary conditions on their right hand sides. In order that the j-th and k-th equations admit periodic solutions it is necessary and sufficient that there be no free term in the right hand side of the j-th equation and no term with e -2i t in the right hand side of the k-th equation. In this way, taking (8.2) into account, we obtain the following conditions of periodicity of the functions y~l): (Xoo} + 2a l Ao ooi) "/0 +

~ )~ (Pi:' + iQ}~)

11/0

! k! (Pi~) - iQiL') No = 0,

+(xoo: + 2a 1Aoooi -

4a l i) No = O.

whence we find: 16>-0001 (>'oOOj - 2) ai - Six/oOOj (001 + OOj (>'000 / - 2)] a l

+ A!

+

4x!OOjOO: (Pj~) 1 Qi~) I) = -

+

O.

(8.9)

Having established this, let us assume first that we are dealing with the case (8.4). In this case equation (8.9) assumes the form

420

One of its roots will have a positive real part when and only when the inequality is satisfied

From this we arrive at the following approximate values of 'A. :::; "/1... on the boundaries of the region of instabili ty

These formulas for j = k go over into (8.7). Let us now assume that condition (8.5) is satisfied. In this case equation (8.9) assumes the form

This equation has purely imaginary roots for any values of ~ • The obtained result is not accidental. It is in agreement with an important theorem established by M. G. Krein 1, which we here present without proof: IN ORDER THAT FOR ~HE SYSTEM (8.1) THERE EXIST A' REGION OF INSTABILITY CONTRACTING FOR ~ = 0 TO THE POINT "- = /I... 0 IT IS NECESSARY THAT THE MAGNITUDE ;t 0 SATISFY A RELATION OF THE FORM (i. k = 1• .. ... m; N = 1. 2, ... ).

1

See the work cited on p.397. This theorem was formulated by Krein in a more general form.

421

In other words, for systems of the form (S.l) the plus sign before the term ~o~k in relations (S.5) must be discarded. B.

};'OhCED OSCILLATIONS OF QUASIHARMONIC SYSTEMS 9. Conditions for the Existence of Almost Periodic Solutions of Systems of Linear Equations with Periodic Coefficients

We now go on to consider systems described by linear nonhomogeneous equations of the form (s=I, ... ,n),

(9.t)

where Fsj are certain continuous periodic functions of t of period W, and f s (t) are almost periodic functions of t. The general solution of system (9.1) is made up of the general solution of the homogeneous system dZ!I

dI .,.,. P"l X l + ...

+ Psnx,.,

(9.~

describing the free oscillations and investigated by us in detail, and some particular solution characterizing the forced oscillations. In this part we shall take up the computation of these forced oscillations, the explanation of their properties, the investigation of the possibilities of resonance, etc. In all these cases a question of fundamental importance is that of the existence for the system (9.1) of almost periodic solutions. To this question, we shall now turn our attention. In what follows we shall assume that the functions fs(t) in equations (9.1) are of the form N

fs (I) =

L eivj' fllJ (t),

(9

i=1

where '/ 1 ' • .• , V N are arbitrary real numbers and fsj continuous periodic functions of t of period", developable into absolutely and uniformly convergir.gFourier series Let us consider the characteristic equation of system (9.2)

422

We shall assume that this equation can have both simple and multiple roots, that t~ each multiple root can correspond one or several sets of solutions. We denote by ~l' ••• '~k all roots of the characteristic equation having a modulus equal to one and by ~l' ••• '~m all roots having a modulus different from one. We shall here write out each multiple root as many times as the sets of solutions that corrf'!spond to it. Therefore k + m,sn and among the magnitudes Ai and ~a some may be equal but to each of them corresponds only one set of solutions. Let, further,

;a..I =.! InAJ, (oJ

In..

=~ In fl. =~. + iT_

(j = 1, ... , k; a

(oJ

=

(9.4)

I, .. . , m)

be the characteristic exponents of the system (9.2). Here a j , ~a and Ya are real numbers, allP>a being different from zero. For the assumptions made, the system (9.2) has k and only k almost periodic solutions which we shall denote by Wsl' •• 'Wsk. Each of the functions Ws' is here equal to .J the product of a periodic function of period GU, developable into a uniformly and absolutely converging Fourier . 't series, and of a periodic function e 1aJ Together with the system (9.2) we consider the system dys

dt + PllS,Yl + ...

+ PlsYn = 0,

(9.5)

conjugate to it. According to the theorem of Lyapunov the characteristic equation of this system has the roots 1/ ~ l' ••• ,1/ ~ k' l/~l'··.' l/~m' the same number of sets of solutions corresponding to corresponding roots in the systems (9.2) and (9.5). System (9.5) will therefore have k and only k almost periodic solutions, which we shall denote by ~sl' ••• ' ~sk. The functions ~Sjhave the same structure as the functions ~sj. We shall now prove the following theorem: 1

-----r---See the work cited on p.295.

423

THAT SYSTEM (9.1) ADMIT ALMOST THEOREM. IN IS NECESSARY AND SUFFICIENT THAT PERIODIC SOLUTIONS THE FOLLOWING CONDITIONS BE SATISFIED:

(9.6,

(j = f. . . ., 1.-).

PROOF. Let us denote by Pj and qa the number of solutions corresponding in (9.2) and (9.5) to the roots A.j and lla so that Pl+··· + Pk + ql + ••• +qm ;:: n. The system (9.5) will then have n particular solutions of the form y~~l = ~'/' (j) = t-I, • + _,,(2) Y1i2 'rtll 'rsj • 111

•

«

•

•

•••

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

(9.7)

•

•

•

•

«

•

•

•

•

•

•

•

«

•

•

•

•

•

•

•

..

•

" (h+lI) _

Ysq'l

-e

-(»0.1

,q'l-I

(

(9.- 1)1

F

(ex=t, ... ,m;

where IV (~) SJ

,QIl-2

'''+(q«-2)1

F(2)

111

•

•

+

8'%

•

•

•

•

+ F(qIJ») •••

, ..

s=t, .... n).

are almost periodic functions having the same

structure as the functions Iu·, while F(P) are simple periodic TSJ sa functions of t of period w. Making use of these solutions, in the same way as in sec. 5 of chapter III, we transform the system (9.1) to equations with constant coefficients. For this purpose we introduce in the system (9.1) in place of the variables xl' ••• 'x the variables (j) , VI(a) , ••• ,v (a) Wl·th th e a ~dl o f th u l(j) , ••• , u pj e su b s t't 1 U t·10D qa n

n

u(j) - ~ t sj x ,,' I 8=1

n

n (II)

Vt

=

~

~=2,

2; F(l) sa xB

F611 x$ '

ri") l:=:

k; ... , Pj;

ex = t, ... , m; T = 2, ... , q•.).

8:=1

(j = t t

uV1 = _=1 ~ 4(~)X", 8/

••••

424

8=1

1 I

IJ

(9.8)

Then, taking into account that the expressions

and

are the first integrals of equations (9.2) we obtain: du(J)

_II dl

n

= _ lI(/) + L.J ~ IJ - I

'!.(~)I 'f' 81

•

• =1

(j = 1, ... , k;

(11

1, ... ,m;

P== 2,

... , Pi),'

(9J)

T=2, ... ,q.. ).

(9.10)

We shall denote by D(t) the determinant of the substitution (9.8) and by aCt) the Wronskian determinant of the fundamental system of solutions (9.7) of equations (9.5). By elementary transformations of the determinant A(t) we easily arrive at the relation

from which it follows that \D(t)l> 0, since the solutions (9.7) are independent and therefore the magnitude A(t) does not reduce to zero for any values of t. But, as follows directly from the form of the substitution (9.8), we can write

where W (t),

(t2.5)

are periodic with period

't' sJ

w.

We shall distinguish three cases: 1. We first consider the NONRESONANCE case. assume that the fundamental equation a u ->"

a l2

au

a22 ->..

. . . . . • .. a112

a,,1

We shall

a ln

...

a2n

...

• • aJlJl->"

.

(12.6)

=0

of the system (t2.7)

has no roots equal to iY j + 21tki/W (j = l, ••• ,N, k = 0,1, ••• ) In this case we can set p = and for the functions y~r),

°

in place of which we can now write x~r), we shall obtain the equations elx:

~

di = £.J all";c:,,, + /a .-0

,AI)

,

",=1 (r)

dxs "([t

n

n

""

= L.J 11=1

Q SIZ

(r) XII·

~

+ £.J

n

(I) qSII

(r-I) XII

+ ... + £J

11=1

(r = 1, 2, ... ).

435

~

Cl~1

(r) 0 q'lI X'"

+ ,-,Ar)

For the assumptions made on the roots of equation (12.6) both the eauations for' x sO and the equations for x(r) s (r = 1,2, ••. ) admit a unique almost periodic solution of the form (12.5). As a result, the series (12.4) will converge and actually represent an almost periodic solution of system (12.1). This solution will be unique if equation (12.6) has no purely imaginary or zero roots. 2. Let us assume now that some of the magnitudes i V . + 2TIki/W (j = 1, ••. , N, k = 0,1, ••• ) are roots of J . equation (12.6) so that the system (12.1) for ~ = 0 is resonance. We shall assume that the total number of sets of solutions of system (12.7) that correspond to these roots and by 0/ sl ,. • • , ~ sm the almost periodic solutions of the system conjugate to it. The system (12.7) may here have also other almost periodic solutions different form ~sl'.'.'~sm' since equation (12.6) may have, b8sides the above mentioned roots, also other purely imaginary roots. Futting (12.8)

where M.1 are arbitrary constants, we shall immediately have: n

(I)

dys

_

~

n

0)

'Tt - .4J aSIl Yu.

+

M O "" I

L.J

(I) qSIl

'fi«l

+ ... +

«=1

o

n ~

+M m .4J q""0) <pam +xis(0 ,

( 12.9)

t = 0 if p» 1. Which of there two cases holds we shall presently establish from a consideration of the obtained equations. The conditions of existence oflalmost periodic solutions of equations (12.9) are of the form 1

The remaining conditions (9.6), if the system conjugat to (12.7) has almost periodic solutions different from ~sl'.'.' ~sm' are satisfied in themselves.

436

(i = t, ., . Ill),

(12.10)

where t

t

1'1

'" J• T1 J\ ~ Aji = t~.~ (I

'I,

q~a

(I)

p='

faj

4~i dl.

I ,"1.i

=

].

Jill -

t-+c»

1 t

'ai

S£J '" /a

(I)

4)

(i,i

n

dl

a=j

1, .. qm).

Let us assume that the determinant \A .. \Of the system JJ.. (12.10) is different from zero. We put p = 1 and therefore X = 1. The system (12.10) then uniquely determines the constants M~. At the same time the system (12.9) will admit almost periodic solutions of the form (12.5) for which we can write: (I)

U,

u(l)

=JP'.Il

If",

+Y.(lh ... + MU) m 'Psm

t

•

where y~l) are almost periodic functions and Mil) are arbitrary constants. These constants, in exactly the same way as M~, are uniquely determined from the conditions of periodicIty of the functions y~2). Continuing in this manner we shall obtain an entirely definite system of series (12.4) with coefficients of the form (12.5). These series will converge and after dividing by ~ will give the required almost periodic solution of the system (12.1). This solution will thus start with the terms of the (-l)-th order with respect to ~.

3. Let us assume that for the same conditions of resonance as in the preceding case the determinant l of JJ.. system (12.10) is equal to zero. For definiteness we shall assume that tbe rank of this determinant is equal to k<m- 1 and that the system of equations (12.10) is incompatible. In this case it will be necessary for us to assume p~ 2 and therefore )C = O. For the determination of M~ we then obtain J.. a system of homogeneous equations from which it is possible to determine k constants M?J.. in .terms of the m - k remaining ones. For definiteness we shall assume that as the

[A ..

437

o

0

independent constants may be taken Mk+l, ••• ,Mm so that we can write:

where a qj are certain entirely definite constants. Having in this manner chosen the constant M~]. we obtain an almost periodic solution of equations (12.9) of the following form: y.(I) =

MU)

•

'¥$1+'" + 11/(1) m - 2. The conditions of periodicity of the function Yl have the f~rm

8.Mo+4No =x A,'4lt/o+2No = xB.

442

(13.5

It is here necessary to consider two different cases depending on whether the ma~nitude A - 2B is different from or equal to zero. Let us first assume that A - 2B p O. In this case equations (13.5) will be incompatible and it is necessary to put )( = 0, i. e. p) 2. On this assumptions conditions (13.5) will give only one equation (13.6)

'rhe second equation connecting the magnitudes M and N is o 0 obtained from the conditions of compatibility of the equations expressing the conditions of periodicity of the function Y2. Let us however continue our computations. we shall now have: Yl

=

From

(13.4)

M 1 COS t + NIsin t + cx I cos 3t + ~l sin 3/,

where (13.7)

and for Y2 the equation is obtained tf!y'l

dl'l

+ YI+

+(5 + 6 cos2t + 8sin 2t) (All cost + Nl sint +cx l cos 3t +- ~1 sin 31)= = 'It (A cost + Bsin t). (13.8) where k

=

1 if P

=

2 and

)( = 0 if p~ 3.

The conditions of periodicity of the function Y2 have the form

and from the conditions of compatibility of these equations we obtain: (t3. to) t la l - 2~1 = (A - 2B) 1t

443

or, taking (13.7) into account, 25Mo - 50No= 8(A- 2B)x.

This will be the second equation of lVl and N. It does o 0 not contradict condition (13.6) and we may therefore assume )c. = 1 and consequently p = 2. "Ie thus obtain: 8

11'0 = 125 (A - 2B),

(13.11)

and for Ml and Nl there remains only one of equations (13.9), which after substituting the values a l and ~l assumes the form 20M1 + 10N1 = 2A,+B.

(13'.12)

From (13.8) we now find: Y2 = M" cost+ N~ sin 1+ aa cos 31 + ~I sin 3t + 02 cos St + bll, sin St,

where 1

a2 =

~

(3Af 1 - 4NI

+ 5:2

J ),

P:l =8(41Jfl+3NI+5~1)'

J1

( 13.13)

and a 2 and b 2 are certain constants which we need not write out The equation for Y3 has the form

~~: +Ya+(5+6cos2/+8sin2t)y,=O, and the conditions of periodicity give:

whence we obtain

444

or, on the basis of (13.13) and(13.10), 5M l -10N 1 = -A+2B.

(13.14)

From (13.14) and (13.12) we find: t

Ml = 25 (A

t

+ 3B),

Nt = 50 (6A -7B).

Subsituting·the found values of Mo ' No' Ml , Nl , in Yo and Yl we obtain the following particular periodic solution of equation (13.3): X=p.-II

{1~5(A-2B)cost+ ti~(2B-A)Sint+ +p.

[~(A+3B)cost+ ~(6A-7B)Sint+

+ It~ (A -

2B) cos 3t

+ 1~5 (2B -

A) sin 3t ]

+ ... }.

(13.15)

Let us now assume that A - 2B = O. In this case equations (13.5) will be compatible and it is necessary to put X = P = 1. Instead of (13.6) we shall now have: (13.16)

The expression for Yl is found the same as before while for Y2 we obtain, instead of (13.8), the equation

1!s1 + 112 + (5 + 6 ('os 2t + 8 sin 2t) (All cost + Nl sint+ + CIt cos 3t + ~1 sin 3t) = O. As a result, in place of (13.10) we shall now have:

or, on the basis of (13.7), (13.17)

From (13.16) and (13.17) we find: i

Mo=SB,

44

1

NO=1O B .

The further approximations are computed in the same way as before. Thus, for the periodic solution of equation (13.3).for A - 2B 0 we have: x = p.-l

{! B cos I + 1~ B sin t +!.l ( ... ) + ... } .

It agrees with expression (13.15) if we put in the latter A

=

2B.

446

CHAPTEE VI APPROXIMATIONS TO ARBITlliiRY NONLINEAR SYSTEMS 1. Periodic Solutions of Nonautonomxous Systems in the

Case of an Isolated Generating Solution In the preceding chapters we considered systems approximating linear systems. Many general results there obtained may however be immediately extended also to systems of a more general form. Let us assume that the oscillations of a system are described by equations of the form drs _ ,r ( dt. - "\s I.

Xl' . . . ,

. ) +p. fsf, ( :r1 x"

, ••• ,

. po) x,.,.

(1.1)

(s= 1, ... , n),

where the right hand sides are defined for all real values of t, for values of ~ lying on a certain segment [0, ~J, and the values of xl,.,.,xn lying in a certain closed region G of the space of these variables. We shall assume that in this region of variation of the variables the right. hand sides of equations (1.1) are continuous and periodic with respect to t with period "', fs admit continuous partial derivatives of the first order with respect to the variables xl, ••• ,xn ' ~, while the functions Xs do not depend on ~ and admit continuous derivatives of the second order with respect to xl'·'. ,xn • Let us consider the generating system (1.2)

447

and assume that it admits a periodic solution x~ = C?s (l)

(1.3)

of period tV, lying in the region G. Taking it as the generating solution we shall seek the conditions for which there co.rresponds to it a periodic solution of the complete system (1.1). In other words, we shall seek the conditions for which the system (1.1) admits a periodic solution reducing for ~ = to (1.3).

°

For this purpose let us denote by

a solution of system (1.1) with the initial conditions

(0, 11> ...• Tn. !1) = 1s'

(1.4)

x" (t, TI (0) •...• 'PII (0). 0) - h (l).

( 1.5)

Xs

We here evidently have:

:For this solution to be periodic it is necessary and sufficient that the equations be satisfied

where the functions ~

= 0, Yl' ••• 'Yn '

(1.1)

~

s in the neighborhood of the point

~,

since the right hand sides of equations admit continuous derivatives with respect to

x l ,···,xn ,1J.· In virtue of (1.5) conditions (1.6) are satisfied for Y = ~s(o), since the generating solution is periodic. s Hence if also the condition is satisfied

~

= 0,

(1.7)

448

equations (1.6) will admit for sufficiently small ~L one and only one solution y s =Y s (u), for which y s (0) = ws (0). Substituting this solution in the functions xx(t'Yl' ••• 'Yn'U) we obtain the periodic solution of system (1.1) reducing for u = 0 to the generating solltion. We thus arrive at the following theorem, by Poincare: CONDITION (1.7) IS SATISFIED SYSTEM (1.1) FOR U SUFFICIENTLY SMALL ADMITS ONE AND ONLY ONE PERIODIC SOLUTION REDUCING TO THE GENERATING SOLUTION (1.3) FOR u = O. The theorem of Poincare can be formulated in still another way. For this purpose let us consider the system of linear equations with periodic coefficients (s=t, ... ,n),

(1.8)

which are the equations in variations of system (1.2) for the solution (1.3). Here the coefficients p . are continuous SJ periodic functions of t of period W, that are determined by the formulas (1.9)

where the parentheses denote that the derivatives have been computed for the generating solution (1.3). Since the functions xs(t, Yl' ••• 'Yn ' 0) are a solution of equations (1.2), depending on the n arbitrary constants Yl' ••• 'Yn and reducing to the solution (1.3) for Ys = ~s(O), then, as was shown in sec. 1 of chapter III, the derivatives of these functions with respect to the arbitrary constants, computed for Ys = ~ s (0), determine the uarticular solutions of the equations in variations (1.8). The system (1.8) thus has n particular solutions Ysl' ••• 'Ysn' determined by the formulas ,C

1

/

-

,

Poincare A., Les methodes nouvelles de la mechanique celeste, vol. 1, ch. III, Paris, 1892. Poincare assumed that the right hand sides of equations (1.1) are analytic with respect to xl, ••• ,xn ' U.

(s, i = 1, ... , n).

( 1.10)

Here, as usualfthe first index denotes the number of the function in the solution and the second index the number of the solution. From (1.10) and (1.4) we find: (1.11)

where DSj is the Kronecker symbol. the determinant :Yll(W)-P

From this it follows that

Y21 (m)

•..

Ylll

(m)

!

i

Y22 ( m) -? ... Yn2 (m)

D (p) = Yu (m)

(L 12)

I;h:{~)' .

is the characteristic determinant of the system (1.8). Further, from (1.6) and (1.10) we obtain: (1. 13)

Hence {

fJ('h •...•

a(Tt. •..•

'fn)} Tn)

Ti="i (0). 1'0=0 =

I Yu (m) -1

_ Yu (m)

Yn (m)

...

!lnl (m)

Y22 (m) -1 .•. Y,,2 (m) •

Y2n (m)

." * •

•

•

•

• •

=D(l)

(1.14)

••• Ynn (m)-1

and condition (1.7) is equivalent to the requirement that the characteristic equation of system (1.8) does not have a root equal to 1 or, what amounts to the same thing, that this system does not have periodic solutions. We shall say that the periodic solution (1.3) of system (1.2) is an ISOLATED solution if the equations in variations corresponding to it do not have periodic solutions. We can

450

· ' also in the following therefore express the theorem of POlncare form: FOR lillY ISOLATED GENERATING PERIODIC BOLUI'ION THE SYSTEM ONE AND ONLY ONE SOLUTION VJHICH REDUCES 'rO GENERA_TING SOLUTION FOR 11 = O. (1.1) FOR SUFFICIENTLY SIV'lALL

If equations (1.1) are analytic with respect to 11, Xl' ••• ' xn the above periodic solution, as is easily seen, will be analytic with respect to 11. 2. Periodic Solutions of Nonautonomous Systems in the Case of a Family of Generating Solutions From the theorem of Poincare established in the preceding section it follows that in the case of an isolated generating solution a complete correspondence exists between the complete system (1.1) and the simplified system (1.2). For the theory of nonlinear oscillations however precisely those cases are of greatest interest in which such correspondence between the complete and simplified systems does not exist, which is possible only when the generating solution is not an isolated one. The most important case of this kind, as we found in the theory of quasilinear systems, is that where the generating solution under consideration belongs to a family depending on a certain number of parameters. We shall now turn to consider this case. Let us assume that the generating system (1.2) admits the family of periodic solutions of period ~ (s=t, ... ,n),

(2.1 )

depending on k~ n arbitrary parameters hi and that the generating solution in question corresponds to the values hi = hi of these parameters. We shall here assume that the generating solution lies in the region G. correspondence with the assumed condition that the functions Xs in equations (1.2) admit continuous partial derivatives of the second order with respect to ~ ••• ,xn we shall assume that in the neighborhood of the generating solution the functions ws are twice differentiable with respect to the parameters h 1.• Moreover, we shall assume that the inde-yendence of these ~

451

parameters is assured by the circumstance that at least one of the determinants of the k-th order contained in the matrix

(2.2)

°

does not reduce to zero for h.1 = h., " t = and therefore 1 these parameters can be expressed in terms of k initial values of the magnitudes x~ We shall denote, as in the preceding section, by

the solution of equations (1.1) with the initi conditions (1.4). Then, as before, the conditions of periodicity of this solution wi have the form (1.6). These conditions will be isfied ffr the i~neratin? solution, i.e. for ~ = 0, Ys = ~s(O, hl, ••• ,hk ). But ln contrast to the case of the isolated generating solution conditions (1.6) will also be satisfied for ~ = 0, Ys = ~s(O, hl, ••• ,h ), since k all solutions (2.1) of the generating system are periodic. In other words, the equations (1.6) have for ~ = not one solution, as was the case before, but a family of solutions Ys = ws(O,hl, ••• ,hk ) depending on k arbitrary constants. As a result the functional determinant

°

necessarily reduces to zero. Moreover, all the minors of this determinant up to the (n - k + l)-th order inclusive must likewise necessarily reduce to zero. Of this we may convince ourselves also in the following manner. Since the system (1.2) admits the solution X O = ~ s s ( t, hI'·'.' h k ) depending on k arbitrary parameters, the functions

452

~ . (I)

= { ~s (t. hi •...• h/l) "h·,

• Sl

U

(S=t •...• n;

} h j-hf - 1

(2.4)

i=t ..... k)

determine k particular solutions of the equations in variations (1.8). But these functions are evidently periodic of period ~. Thus the system of equations in variations (1.8) has k particular periodic solutions. Consequently the characteristic equation of this system has a root equal to 1 which has a multiplicity not less than k and which reduces to zero all the minors of the characteristic determinant at least up to the (n - k + l)-th order. From this on the basis of (1.14) we convince ourselves of the correctness of our assertion in regard to the functional determinant

(2.3).

We shall assume that at least one of the minors of the (n-k)-th order of the determinant (2.3) is different from zero or, what amounts to the same thing, that the system in variations (1.8) has precisely k periodic solutions. Let us assume for definiteness that (2.5) For this condition the first n - k of equations (1.6) have a solution for Y1, ••• ,Y - k , in which these magnitudes n are functions of Yn-k+l' ••• 'Yn ' ~ reducing to

f.or

(j=n-k+1, ... ,n) and possessing in the neighborhood of this point continuous partial derivatives of the first order. Substituting these magnitudes in the last k of equations (1.6) we obtain for the determination of Yn- k + 1' ••• ' Yn a system of k equations. The left hand sides of the equations thus obtained will have in the neighborhood of the point ~ = 0, y. = ~.(O,h·l, ••• ,h~k) J J -

4

(j = n-k+l, ••• ,n) continuous partial derivatives of the first order. The above equations can therefore be represented in the form ~i = Fi (111-1I+l' ...• T1)

+P.lfi

(r1l-ll+1' ••• ,

Tn. p.) = 0

(2.6)

(i= 1, .•. , k), ~

t

~

where F. and . have near the values ~ = 0, y. = w.(O,h,~ •••• hk) l l . J J .J. ~ continuous partial derivatives of the first order with respect to the variables Yn-k+l"'.'Yn • Since the system (1.6) has for ~ (8 =

=

0 the solution

" ... , n).

depending on k arbitrary parameters the system (2.6) should also for the same condition have a solution for Y k l'.'.'Y n- + n depending on k arbitrary parameters, which is possible only in the case where the functions Fi reduce identically to zero. ThUS, equations (2.6) after dividing by ~ assume the form

For the existence of the required periodic solution it is necessary and sufficient that system (2.7) admit a solution with respect to yj(j = f-k+l, .•• ,n) in which these magnitudes reduce to ~j(O,hl, ••• ,hk) for ~ = O. And for this it is necessary first of all that the relations be satisfied Pj{II: ••..• it:) =

= Wi (,,.-/,

•••• Ph)

a(/;~ •.. .• hh) and therefore it is sufficient for us to show that the functional determinant

is different from zero. Of the correctness of the latter assertion we can convince ourselves in the following way. Let us consider the system of linear algebraic equations

a.;,s ) ( oh iJTI " J Al + ... + ( iJT"

AD = 0

(s=1 t

•••

,n). (2.ft)

where the parentheses denote that the derivatives are computed at the point 1J.. = 0, y s = q>s t 0 ,hi, • " ""h~) ( s = 1,.". ,n). Since the determinant together with all its minors up to the (n - k + l)-th order reduce to zero but at least one minor of the (n-k)-th order is different from zero, the system (2.11) admits k independent solutions Asl, •.. ,A sk " On the basis of (2.5) this system of equations can be chosen

455

in the following way. A

We set

t {0

-

lI-1f'+;,j -

f' r i _0 f"or'

= i,

t, 2, ... , If),

(i, j=

l-;-/.

~

(2.12)

and determine the magnitudes Al·, ••• ,A k . from the first J n- ,J n - k of equations (2.11). It is possible however to indicate another system of linearly independent solutions of equations

(2.11).

For this we recall that equations (1.6) are identically satisfied for ~ = 0, Ys = ~s(O,hl, ••• ,hk)' Differentiatin g + these identities with respect to h. and then putting h. = h. J 1 1 we shall have: i).rjI, (

= J. .

0

0'

i=

n;

I, ...• k).

But then we must have: II BSj

= ~ lZajA","

i=I. .... Ii).

(S=1. ... ,I1;

(2.13)

8:::1

where acx.j are certain constants. J:;'rom these relations it follows that all possible determinants of the k-th order of the matrix (2.2) are enual for t = 0, h.1 = h~1 to the ~ product of the determinant laajl by the corresponding determinant of the magnitudes A • But since by assumption at least one determinant of theS~-th order of the matrix (2.2) for t = 0, hi = h;is different from zero, the determinant [acx.jl is also not equal to zero. Putting noVi in (2.13) s = n - k + 1, ... , n and taking (2.12) into consideration we shall have (a.

456

i=

L

.0

••

k).

and therefore the determinant A is equal to (a~jl' and, as was stated,is different from zero. We shall now find the develoned form of equations (2.8). As was already remarked, the functions ~r admit in the neighborhood of the point ys = ~s(O,hi, ••• ,h~), ~ = 0 continuous partial derivatives of the first order. Hence, setting f's= Ys - ~S(O, hi, ••• ,h~), we can represent equations (1.6) in the form

where

(2.15) and Usi and Vs are continuous functions of ~l, ••• ,Sn' 11 reducing to zero for Sl = ••• = Sn = 11 = O. From (2.14) it follows immediately that equations (2.8) are obtained as a result of the elimination of the magnitudes Sl, ••• ,Sn from the linear equations

Let us consider these equations in more detail. the basis of (1.13) we can write:

usi =Y:i(W, O)-oSi'

On (2.17)

where y~(t,~) are a fundamental system of solutions of Sl equations (1.8) with the initial conditions lfor t = ~)

(2.18) As to the magnitudes vs' for them we find on the basis of (1.6)

(2.19)

457

But the functions xs(t, Yl, •••• Yn , ~) satisfy equations (1.1). Substituting them in these equations and differentiating the obtained identities with respect to ~ we shall have

...

~

Putting here ~ = 0, Ys = ~s(O,hl, ••• ,hk) and taking (1.9) into account we obtain: (2.20)

Moreover, form (1.4) we find:

or.. (0, rio .... rru 1') _ 0 01'

-

•

The functions ~xs(t'Yl' ••• 'Yn'~)/ a ~ therefore form a particular solution of equations (2.20) with zero initial values. But then, by simple verification, we easily convince ourselves that

( or$ (t, rho~.. ' Tn,

1'») = t

n

:::=) ~

[/.('t,

ifill' •. ,

lfIu' O))t='t.h,=htyt.(t,

't)d~.

o a=1

Having established this we can, on the basis of (2.17) and (2.19), write equations (2.16) in the following manner:

GI=' ft)

n

+1' ~ ~ o

[la(';, 1fI1"

··'~t•• O)]'='t.h,=htY;Il(m,

t:)d,;=O, (2.21)

11=1

and the problem reduces to the elimination of the magnitudes ~l' ••• '~n from these equations. For this purpose let us consider the system of equations (2.22)

458

conjugate to the system of variations (1.8). As we have seen, the system (1.8) admits the k periodic solutions (2.4). Hence the system (2.22) likewise admits k periodic solutions, which we shall denote by 'I'sl (t), • • • , 'P sk( t). From the relation between the solutions of conjugate systems we easily find on the basis of (2.18) the identities n

~ y:«(t, 't)1'llIi(t)=if'sd't),

(2.2:1)

ca=t

which are what we use for eliminating Pl, ••• ,P n from (2.21). Thus we obtain from (2.2]: n

n

U)

n

+1· ~ ~ [/.. (-: .. ··)]t=1:.h i =hTY:"(O), 't)i',dro)d't=O, o 8,4=1

and identities (2.23) give: U)

!l

n

~~

[/4 ('t •... )],=", hi=ht i'ai ('t) d-c = 0

(i = 1, ... , k).

o a=1

This will be the required result of the elimination of Sl, ••• ,Sn from equations (2.16). Equations (2.8) determining the parameters of the generating solution can thus be represented in the following form: Pi(h~, ... ,h:)= U)

n

=~~

o II=!

/4(/, (fh(t, h:, ... ,11:) ... ','finO, 11;, ... ,11:). 0)

~l2dt)dt=O (2.24)

(i= i, .. . ;k),

and we arrive finally at the following theorem: 1 1 Malkin I.G., K teorii periodicheskikh reshenii Puankare (On incare~s Theory of Periodic Solutions), Prikl. matem. i mekh., vol. XIII, no. 6, 1949.

459

THEOREJ.\il. IN ORDER THAT SYSTElVI (1.1) ADMIT A SOLUTION liEDUC FOR lJ. = 0 TO A GENERATING SOLUTION BELONGING '1'0 THE FAlv'.ILY (2.1) IT IS NECESSARY THAT THE rARANIETERS h.l OF GENERATING SOLUTION SATISFY THE SYSTEM OF EQUATIONS (2.24). TO EACH SIII&PLE SOLUTION OF T'HIS SYSTEk OF 'EQUATIONS, i.e. TO EACH SOLUTION FOR WHICH CONDITION (2.10) IS SATIS, THERE ACTUALLY CORRESl?ONDS ONE AND ONLY ONE PERIODIC SOLUTION OF SYSTEM (1.1). From the preceding analysis it folloVIS also that in the periodic solution thus obtained of system (1.1) the functions Xs will possess continuous derivatives of the first order with respect to '\1. If the right hand sides of eQ.uations (1.1) admit with respect to the variables x l , ••• ,xn ''\1 partial derivatives up to the order p the functions Xs in the periodic solution will admit derivatives 'with re t to lJ. also up to order p. If the right hand sides.of equations '(1.1) are analytic with respect to xl, ••• ,x ' lJ. the obtained n periodic solutions will also be analytic with respect to lJ.. In fact, in this case the functions x s (t'Yl' ••• 'Yn ,u) will be analytic with respect to Yl ••• 'Yn ' U and the solutions Ys = Ys(lJ.) of equations (2.7) and (1.6) will be analytic with respect to lJ.. In practice the case may also be of interest where equations (2.24) are satisfied identically. In this case the functions ~i equations (2.7) have the form

where p is a certain integer. It is here asslli~ed that the functions i have derivatives with respect to 11 of an order not less than p. For this it is sufficient that the right hand sides of equations (1.1) have partial derivatives with respect to lJ., , ••• ,xn up to order p + 1. Equations (2.8) are now replac by the equations

t

pr(h~, ... ,

II:) =

= Wr(Tn-h+I (0,

h:, .•. ,hk), ...• t'fn (0, h;, ... , It:), 0) = 0 (2.25) (i = 1, ... t k)

460

and for each solution of these equations for which

a (P: •...• Ph) a (h: •...

, h:)

0,

(2.26)

there will in fact exist a periodic solution of the system Cl.l), and which is moreover unique.

3. Case of Analytic Equations Let us assume that the right hand sides of equations (1.1) are ani'llytic with respect to xl' ••• ,xn ' 'jJ. so that these equations can be represented in the form (s= 1, ... ,n),

(3.1)

where the functions f(P) = f(P)(t,xl, ••• ,x ) are analytic s s n with respect to xl, ••• ,xn in the region G•. We shall assume the generating system (1.2) has a family of periodic solutions (2.1) depending on k arbitrary constants hl, ••• ,hk • Further, let h;1 be some solution of equations (2.24) for which condition (2.10) is satisfied. Then, as was sho~n in the preceding section, the system (3.1) admits only one periodic solution reducing for 'jJ. = 0 to the solution

of the family (2.1) and this periodic solution will be analytic with respect to 11. We can therefore write for this solution X~=CPs(l,

h:, ... , h:)+p.X!f)+p.2X!2)+

... ,

(3.2)

where x~p) are certain periodic functions of t of period w . We shall show that there exists only one system of series of the form (3.2) satisfying equations (3.1) and shall indicate the method of their comDutation. Substituting the series (3.2) in equations (3.1) and equating coefficients of like powers of 11 we obtain:

461

(3.3) where p

. are the coefficients of the equations in variations SJ determined by the formulas

(3.4)

and F(P) are polynomials with periodic coefficients of s x~l) , ••• , x~P-l) In particular,

F~t) =/~t)(t,!Pl(h~, ... , lin, ... , 'Pn(t,

h:, .... h:»).

(3.5)

In order that equations (3.3) admit a periodic solution for x(p) it is necessary and sufficient that the conditions s be satisfied IIJ) n

~ ~ F~P) ~lIi dl = 0 o

(i = 1• ... , k).

(3.6)

11=1

where ~sl' ••• ' fsk are periodic solutions of the system conjugate the system of equations in variations.

to

For p = 1 equations (3.6) agree with equations (2.24) determining the parameters of the generating solution. Since b) assumption the latter are satisfied, the equations for x~l admit a periodic solution. This solution is of the form X u) s --

where

M,u/D 1 T61

+

• • •

+ M(U k m + XU). TSIi.

.,

x(l)f is some particular solution of the equations s

for x~l), ~sl' ••• ' ~sk are the periodic solutions of the equations in variations, determined by equations (2.4), and Mil) are arbitrary constants.

These constants are determined

462

from the conditions of periodicity of the functions x~2) • We shall as.sume in general that all the functions x~l), ••• ,x~Z) have already been computed and turned out periodic.

The functions x~t) will here have the form

(I) _ M(l)

x, -

1 aM

Uln

('

~~i -

)

J

\)

(1)n axil>

=

I h a:! o 11= 1

=

n

I

{d'¥

axIl)

~ili- J ~ o:~ \)

Cii""

all"!

'=1

-.n \

J

(1)/

~

{aX~l> do/Pi

dt

p= I '

71

+ ~ P~«

a%~u oh"!

11=1 n

~Pi} dt =

I

+ ] PT~ hi} dt= T='

axil)

~ o,:tt 'i'~i'

o 11= I

I

We may remark that the functions

x~l)

are periodic only

for definite values of the parameters hj and therefore the functions

ax~ 1) /

0 h~

will not, in general, be periodic and

the expressions A .. will be different from zero. J1 can write: Ul

Aji =

But we

n

a!. I h xpl> 'i'llit J

O(i=1

will be periodic for any values of (l)"l aj~lli. a=1

The coefficients Aji agree accurately with the ssions (3.9) and therefore, as was shovm in sect 3, Tf.re can write: A .. = apt . ]1 ahj

ing the determinant of em (5.8) equal to zero we obtain the following equation: ap)

all; - a:I 21

ap,!

uh: - aa22

. ..

(5.9)

which determines the first terms of the expansion of k characteristic exponents of the system (5.3), provided 1 the roots of this equation are different, as we shall assume. Thus, for the stability of the neriodic solution (5.2) it is necessary that all roots of equation (5.9) have nonpos ive real pa~ts. If all these real p sure negative and if the re parts of the remaining n k characteristic exponents of tern (5.3) are also negat , then for sufficient solution (5.2) will actually be stable, and furtherasymptotically stable. The equation (5.9) exactly agrees 'with equation ( .8) of chapter III, tnat was established for quasilinear ems. If functions o/si are chosen such that the relations are isfied

n

n

~ 'fai1'ai = -1,

~

... =1

a=t

!fa icpaj =

0

(i

I') ,

eauation (5.9) can be replaced by the more s

ap,! .-Jhr

ap, alt* 11:

iJPt. _).. al,:

I .. I

..

..

(I, = am).

(5.10)

apR. , .• ah*--)..

iJPR. alt* 2

_ !P"iJh'{

=0

le equation

If.

remark in conclusion that for setting up equation ( .9) it is necessary merely to know the f~nctions Hence, as we saw in the prec ion, this eauation can be

ten also vvhen the general solution of equations

(5.5) is not kno'!;'!:. 6. Nonautonomous System with One

gree of Freedom

Approximating a Conservative System t us consider the special case where the system (3.1) has the form

where functions F(x) and fi(x, respect to their argunents x and F(x) not containing

x,. t) x

x.

are analytic with dx/dt, the function

generating system

(6.2) is conservative and may be written in the canonical form tlxo

.

all

(ft=-.-,

axo

where

474

tlxo tit

-

011 oXo '

The energy integral i • 2 :r~

+ V (.l'O) = E

(6.3)

determines the phase tranjectories of the system (6.2) and it possible to find its general solution with the aid of quadratures. Let us write this general solution of equation (6.2) in the form

ma~es

(6.1)

where the arbitrary constants the conditions are satisfied

hand

c

are so chosen that

•

Xo (Ot c) = O.

(6.5)

The function Xo is then determined by the equation

V2 (t + h) =

:co

\ dro _ Jc ± yv (c)- V (xo)

•

(6.6)

Let us investigatn this function more closely. We shall assume that the point Xs c is not a state of equilibrium of the system (6.2) so that F(c) J O. For definiteness we shall assume that F(c) '> 0 and consider the .point Xo moving along the x- axis according to the law (6.4). At the initial instant t + h = 0 the velocity of this point is equal to zero and its acceleration is negative. Hence the point x 0 for t +h> 0 moves at first to the left and in ". formula (6.6) it will be necessary to take the minus sign before the radical while the quantity under the radical sign will be positive. If at the same time the function

(6.7) to the left of the point Xo

=

c

in the region

G

does not

become zero the motion of the point Xo will continue to the left until it leaves the region G. Let us assume however that there is an interval of values of c for which the

475

function (6.7) has in the region G a root less than c and that the value of c under consideration belongs to this interval. Let c· c ~(c) be the root of the function (6.7) nearest to c and assume at that this root is simple. point x then attains the value c· after a finite interval of time o T/2, where c·

T

= T (c) = V2 \J I y V (c)dxo t• V (xo) c

(6.8)

In fact, the expression under the integral in (6.8) is continuous over the entire segment Cc~, cl with the exception of its ends where it becomes infinity of the order 1/2 (the~ root Xo = c of the function (6.7) is likewise simple since F(c) = V'(c) ~ 0). The integral (6.8) is therefore finite. After the -point x 0 atains the value c ~ , where nOVl evidently F(cjl; ) < 0, it turns to the right and will move thus unt it reaches the value c er an interval of time likewise equal to T/2. Fo~ this stage of the motion it will be necessary in (6.6) to take the minus s before the radical. After the point x re~ches the value c its motion o will repeat perioaically. Let us assume now that the root c~ of the function (6.7) will be mult leo Since we must then have F(c*) =IT' (c*) = 0, there will correspond to the value Xo = c'" the state of equilibrium of the system (6.2). V{r,l

~

.I I. I I II. I I. . I I . I I I.

z,

I

x,

Fig. 30 476

For this assumption the integral (6.8) will diverge and the point x will aSYffiptotically approach the state of equilib£'ium as t ..... co • Fig. 30 shows the phase trajectories of the system (6.2) for the case of the existence of values of c for which the function (6.7) has roots less than c. These phase trajectories are determined by equation (6.3) and will be symmetrical with re$pect to the x o - axis. On the sketch are indicated several trajectories corresponding to different values of E. A graph is also given of the function Vex ). o To the points where the function V(x ) has a maximum or o minimum correspond staces of equilibrium of the system, the state of unstable equilibrium corresponding to the maximum and stable equilibrium to the minimUm. To the stable states of equilibrium correspond the singular points of equation (6.2) of the type of a 'center' and to the unstable the singular points called ~saddle~ points. Thus, if there exists a region of values of c for which the function (6.7) has a root less than c (for F(c»O), we shall suppose that for these values of c the function (6.4) will be periodic. Hence, for the assumption made, equation (6.2) will admit a f~~ily of periodic solutions of equations (6.4) depending on two arbitrary constants. The period of these solutions, determined by formula (6.8), will depend on c. Let us denote by em the value of the equation

c

satisfying

(6.9) re m is an arbitrary integer, ~ a period of the right hand side of equation (6.1) (which is not necessarily the least period of this right hand side). For each value of c m we obtain a family of periodic solutions (6.10) of equation (6.2), having a period w depending on ONE arbitrary parameter h. These solutions can be taken as the generating solutions and for them can be found the periodic solutions of the complete equation (6.1). For this it is

477

necessary however to conS1Cler first in more detail the corresponding equation in variations. The equation in variations for any solution of the family (6.10) has the form (6.11)

This equation has the two particular solutions (6.12) (6.13)

since the function (6.4) is the generating solution of equation (6.2). The solution u is periodic of period W. to the solution it wi not be periodic due to the fact that the period of the function (6.4) depends on c. life shall explain in more detail the form of y

As

Y2'

i..

For this purpose let us replace in the function xo(t+h, c) the variable t by the variable 1: with the aid of the relation (6.14)

and denote the function thus obtained by x;(1:, c). This function will be periodic with resptct to 1: of period w . Differentiating this function with respect to c we obtain: ox: =.!..x ('t+h)T oc ac 0 ..,'

c)

oxo(t+h. e)+'t+hdT· (t+h )_ oe de Xo , C (II

= oXo (t

+ h, t') + ,+ h dT ;

oe

T

de

0

(t

_

+ h,

)

c,

where the left hand side is a ueriodic function of 1: of period 6J. rutting c = c m , we~find from (6.12),(6.13) and

(6.9):

y:= _

m ddT U)

c

{(t+h)rl(t+h)+v(t+h)}.

478

(6.15)

where v(t+h)=

_~{~~X;} m

dT i)c de c=:e

_~(_1_~ ('t+h)T m dT ae Xo

=

CAl

J

C

)}

de

m

e=cm

(6.16) is a ueriodic function of time of period~. From (6.15) and (~.12) we find that equation (~.ll) has also a cular solution

Y2

lu(t+h)+v(t+h).

(6.17)

which, together with (6. ), we shall take as the initi \Ve note down the following properi tes of the functiom u 1. The function u s isfies the equation in and the initial conditions (for t + h ~ 0) it

(0)

0,

ions

.

u (O) = - F (em)'

(6.18)

These conditions are directly obtained from (6.12), (6.5) and (6.2). 2.

The

fu~ction

v(t+h) satisfies the equation

. + h).

F' (xo) V = - 2u (t

(6.19)

This is directly obtained fromfue circumstnce that the function tu + v satisfies the equation in variations.

3. The equation holds v (O) =

-

---=-(d-==;:::--)m

de

fact, from (6.16) and (6.5) we V(

0) = _ ~ {_1_ QXo (0, C)} 111

dT de

QC

c=cm

c=c

. m

v.

(6.20)

4. The equation also holds

,,2 + ll~ -

~V = K = const.

(6.21)

In fact, the Ie hand side of (6.21) represents the Wronskian determinant

4 (t)

=

1:1 ~. j Yl

Y.

of the solutions Yl and Y2 of the equation in variations. Since the latter does not contain a term with first derivative, on the basis of the formula of Liouville, which for a linear equation of the second order has the form 1

-Sqdl'

A(t)=.1(to)e

1o

where q is the coefficient of the first derivative, we conclude that (6. ) holds true. From (6.18) and (6.20) we find directly that

(6.22)

Let us now consider the nonhomogeneous equation d2• dt;

+ F' (.To) Z = I (t).

(6.23)

As one can easily convince oneself by direct verificatiOL or by applying the method of variation of the arbitrary constants, the general solution of this' equation can be represented in the form

Kz

II

t

t

,

o

0

0

S(~llldl-Vf-~)dt+v(~ llldl-~ )+I1U,

480

(6.24)

where ~ and ex. are arbitrary constants. If the function f(t) is periodic of period~, then in order for equation (6.23) to admit a periodic solution it is necessary and sufficient that the condition be satisfied U)

~ u! dt = O.

(6.25)

o

In fact, when (6.25) is satisfied the coefficient of v in the expression (6.24) will be periodic. The expression under the outer integral in the first term will also be periodic. Hence if the constant 0 is so chosen that the mean value of this expression reduces to zero, i.e. if we assume that U)

~ = - ~ ~ vI dt + ~ ~ o

t

U)

dt

0

~ ul dt,

(6.26)

0

the solution (6.24) will be periodic. The constant ex. here remains arbitrary. Condition (6.25) may of course be obtained at once also from the general conditions of periodicity if account is taken of the fact that equation (6.11) is self-conjugate and therefore the function u is also a periodic solution of the equation conjugate to (6.11). With this established, we shall seek a periodic solution of equation (6.1) reducing for ~ = 0 to the generating solution (6.10). Following the method of sec. 3, we shall seek this solution in the form of the series

x = Xo (t + h, cm) + fiX! (I)

+ ...

(6.27)

For xl we have: (6.28)

and the condition of periodicity (6.25) on the basis of (6.12) gives: U)

P(II) =

~ Id .1'0 (I + h, crn)' ;~o(t + h, cm)' t];o (t + h, cm)dl = O. (6.29) o

This equation serves to determine the constant h. Let us assume that h has been chosen according to this

4.

equation, with P'(h) fi O. For xl' according to (6.24) and (6.26), we shall then have: x1 =

!

t'

[U

t

~

()

o

,0

'

UFldt-vFl-~I)dt+o( ~ llFtdt-~l)] + (llU, (6.30)

where w

~1 =

w,

~ ~ vFt dt + ~ ~ dt ~

uFt dt

000

and u l is an arbitrary constant. This constant is determined from the condition of periodicity of the function x 2 • The equation for u l ' as was shown in sec. 3, will be linear. The further approximations are computed in an analogous manner. We may remark that for the actual computation of the function xo(t + h, c) and its period T(c) there is no need of invariably using formulas (6.6) and (6.8). In practice it is usually more convenient to use approximating procedures of some kind that give the function Xo already developed in a Fourier series. One procedure of this kind for systems of special form will be discussed in the following chapter. A detailed investigation of the problem of periodic solutions of systems of the form (6.1) has been given by A. M. Kats 1, who considered in detail also the singular case when the equation (6.28) determining h is identically satisfied. Kats also gave stability criteria both for the nonsingular and singular cases. We shall now turn to a discussion of these criteria for the nonsingular case which we here consider.

1

Kats A.M., Vynuzhdennye kolebaniya nelineinykh sistem s odnoi stepen'yu svobody (Forced Oscillations of Nonlinear Systems with One Degree of Freedom), Prikl. matern. i mekh., vol. XIX, no. 1, 1955.

482

7. Stability Criteria of the Feriodic Solution Considered in the Preceding Section To solve the problem of "the stability of the perlodlc solution (6.27) we shall set up for it the corresponding equation,in variations. For this purpose, putting in equations (6.1)

and discarding magnitudes of gher than the first order with respect to y we obtain the following equation in variations

~~ +F'(Xo)Y+f1{[FW(XO)X1 (~;) Jy_(:~l)~~}+ ... =0, (7.1)

where the terms not written down are of an order higher than the first with respect to ~ and the derivatives of the function f 1 [ire computed for the generating solution x = xo(t + h, c m). Equation (7.1) for ~ = 0 reduces to equation (6.11) which, as we saw, has the two independent solutions (6.12) and (6.17). Hence the two characteristi~ exponents of equation (7.1) for 11 = 0 reduce to zeromd we must therefore compute them more accurately. We an not here make use of the results of sec. 5 since the Lorm of the solutions (6.12) and (6.17) shows that to a zero characteristic exponent of equation (6.11) corresponds, not two, but one set of solutions. As a result, the characteristic exponents of equation (7.1) will in general be developable in integral powers, not of ~,bu~of ~. (7.2)

be the expansion of one of the characteristic exponents of equation (7.1). It is easy to see that this expansion actually c:ontains fractional powers of "\.1 the expansion of the second characteristic exponent ~2 will have the form (7.3)

483

In fact, let

pt-2Ap+B=O be the characteristic equation for (7.1). Since for ~ = 0 this equation must have a double root equal to 1 the expansions must hold of the form

and therefore

where the terms not written dovm contain only intef1;ral powers of~. Hence, if these expansions contain fractional powers of ~, i.e. if the number k is odd, one expansion is obtained from the other by a simple substitution of -~ forin. But then the same will be true also forfue expansions (7.2) and (7.3) of the, characteristic exponents. From (7.2) and (7.3) it is immediately seen that for stability it is necessary that the real part of the coefficient a l reduce to zero. If this condition is satisfied the sign of the real parts of the magnitudes Al and fi2 will be determined by the coefficient a • From (7.4) it is seen 2 that the coefficient a will always be real while the coef2 either real or purely imaginary. ficient a will be l From this we arrive at the following necessary conditions of stability

(7.5) If these conditions are satisfied with the signs of inequality then for sufficiently small ~ stability will actually hold and will moreover be asymptotic. Let us now proceed to determine the coefficients a

484

l

and 0: 2 • For this purpose vve transform equation (7.1) with the aid of the substitution

and seek to obtain the magnitudes 0: 1 from the condition of existence of a periodic solution for the equation thus obtained

We shall seek this periodic solution in the form of the series

We shall then have:

~~ + F' (xo) Zo = 0, d%;;l

dt 2

+ F' ( Xo).

Zl

~; +F' (xo)Za -

2111

~l

? dZo = - "'«1 dt '

=, -[

(I) =

pw (xo) x 1 -

(~: )+cx~]

zo-

-[-( )+2«2] :~l

~7 .

For Zo and zl on the basis of (6.19) we find:

where Mo and Ml are arbitrary constants. The condition of periodicity of the function z2' which on the basis of (6.25) has the form Ul

Ul

~ ~V)udt== ~ ~(t)~odt=O, o

0

485

therefore gives w

a; ~

(il!

+ 2(1.11~) dt +

o

(7.6)

But on the basis of (6.21) II)

II)

~ (u2+2l~V)dt== ~ (lt2+11~ -;lV) dt=u>K.

(7.7)

Further, taking (6.28) into account, we can write: ill

II)

~ o

Ffl (xo) x.~; dt

=-

~

F' (xo)

(~1;0 +:1'1;0) dt =

0 It)

=- ~

[F' (:1'0) :1'1';;0 - F (xo) ;;1] dt =

o U)

-~;;o);l+F'(.rO).rl)dl= -~/l(XO' ~o,t);~dt.

(7.8)

Hence equation (7.6) assumes the form

whence, taking into consideration (6.29), we finally obtain 2

1

a.. = ."K

dP(h) dh

(7.9)

The coefficient u 2 is determined from the condition of periodicity of the fynction z·3. It is possible however to determine it more simply. In fact, the sum of the characteristic exponents of any system of linear equations with periodic coefficients

486

on the basis of formula (2.8) of chapter I by the equality U)

~ A« = S~ 11=1

is determined

p•• dt.

0 11=1

Applying this equality to equation (7.1) we obtain;

whence, comparing with (7.2) and (7.3), we find; (7.10)

Substituting (7.9) and (7.10) in (7.5) we finally obtain the following conditions of stability of solution (6.27) of equation (6.1): K

dP(h) dh

/0 ",,>,

(7.11)

(7.t2)

With this we finish our discussion of the theory of periodic solutions of systems of the form (6.1). A concrete example will be considered below in.sec. 7 of chapter VIII. 8. Fe

odic Solutions of Autonomous Systems

We shall now consider the problem of the periodic solutions of autonomous systems. Let there be given the system

487

J~3 =

X, (Xl'

••• , Xn)

+ p./$ (Xl'

••• , X n •

/-1)

(8.1)

(s=1, ... ,n),

where the functions Xs in a certain region G of the space of the variables xl,.,.,xn admit continuous partial derivatives of the second order with respect to xl, ••• ,xn and the functions fs in the same region G and for 0" 1-L~llo admit continuous derivatives of the first order with respect to xl' • • • ,xn ' '11. Let us assume that the generating system (8.2)

admits a periodic solution

X: =!PB (I)

(8.3)

of a certain period T. We shall seek the conditions under which the system (8.1) for '11 sufficiently small likewise admits a periodic solution reducing for '11 = 0 to the generating solution, and the period L of this solution. Since the system (8.1) is autonomous, the period L does not coincide in general with the period T of the generating solution and is one of the unknowns of the nroblem. We shall denote by Yl' ••• 'Yn the initial values of the required periodic solution. The magnitudes Ys are to be determined together with L. We may remark however that our problem contains a certain indefiniteness. In fact, if the system (S.l) admits some periodic solution then in virtue of its autonomous character it necessarily admits an entire family of such solutions which are obtained by replacing t by t+ h, where h is an arbitrary constant. Rvidently it is sufficient for us to find some one solution of this family. We can distinguish it 'bythe fact that we assign one of the magnitudes Ys • In fact, if this magnitude Ys is correctly given, namely if the corresponding coordinate X~ actually assumes in some solution of the required family iO

488

at some instant of time the value Y ' we can take this s instant of time as the initial one. Hence, if system (S.l) at all admits a ueriodic solution it also admits such solution for which one of the magnitudes x assumes for

s

t = 0 the initially given value and therefore the additional restriction made does not affect the generality of the problem. In practice difficulties never arise in the question of the correct choice of one of the magnitudes Ys since usually there are always knO\m in advance certain values which at least one of the magnitudes Xs must assume in the process of oscillation. We shall in what follows for definiteness assume that the magnitude Y is known. Moreover, we shall assume that n the magnitude ~n(O) is also equal to Y • This fixes also n the value of the constant h, which can be added to t also in the generating solution. The problem will thus contain n u~~owns T, Yl"'.'Yn-l" All these unknowns will play the same part in the further considerations and therefore we shall for convenience denote this set of unkovms by al, •.. ,a •

n

shall denote by x s (t, Yl""'Yn , ~) the solution of system (S.l) determined by the initial conditions

(8.4) The equations determining al, •.. ,a will then have the n following form: ~s (a. lt

••• ,

a. ,V

p.) =

Xs

(t,

Tl' , •• ,

1'11' }l) - T.$ = O.

(8.5)

o 0 Let al, ••. ,an be the values of al, ••• ,a corresponding n to the generating solution, i.e. the set of magnitudes T, wl(O)'"'"'~n_l(O). Equations (S.5) will then satisfy o for u = 0, as = as' since the generating solution is periodic of period T. Hence if the condition is satisfied (8.6)

489

equations (8.5) will admit for sufficiently small ~ one and 0

= a. s l~) for which a. s (0) = 0.s • Subs stituting this solution in xs(t, Yl' ••• 'Yn'~) we obtain the unique periodic solution of system (8.1) reducing for u = 0 to the generating solution, and its period 1:. only one solution a.

We shall say that the generating solution (8.3) is isolated if condition (8.6) is satisfied for ~t. We thus arrive at the following theorem of A. Poincare:

IF THE GENERATING SOLUTION IS ISOLATED THE SYSTEM (8.1) ADMITS FOR SUlfFICIENTLY SMALL 11 OlijE AND ONLY ONE PERIODIC SOLUTION REDUCING FOR ~ = 0 TO THIS GlENERATING SOLUTION. For the theory of nonlinear oscillations the greatest interest however is presented by the case where such unique correspondence between the complete and generating system does not exist. The most important case of this kind will be that for which the generating- solution belongs to a family depending on a certain number of parameters. This case we shall now consider. 1 Assume that the generating system admits a family of periodic solutions

(8.7) depending, besides on the constant h, which can be added to t, on k additional constants hl, ••• ,h • The period k T = T(h1, ••• ,h k ) will in the general case also depend on these constants. Let us assume that the generating solution under consideration corresponds to the values h.1 h~ of 1 the parameters. 0" " Denoting, as before, by o. o s = o.s(hl, ••• ,h k ) the values of a. in generating solution, i.e. the set of magnitudes .. s r 'II' *' .." ~ .. T = T (hl,.·.,h k ), ~l(O,hl,.·.,hk) '·.'~n_1(O, hl,···,h k ), 1

A very important special case of this problem was considered in the work: Zhevakin S.A., Ob otyskanii prede1(nykh tsik10v v sistemakh blizkikh k nekotorym nelineinym (On the Obtaining of Limit Cycles in Systems Approximating Certain Nonlinear Systems), Prikl. matem.i mekh, vol. XV, no.2, 1951.

490

we shall, as before, have the result that the system of equations (8.5) is satisfied for ~l = 0, ex.~o =ex.~. But in the ;::;, case considered the system (8.5) has for ~ = also the solution T = T.(h l ,··· ,hk ), Yl tpl (O,h l ,··· ,l>k)' Yn-l = wn_l(O,h1, ••• ,h k ), depending on k arbitrary constants. As a result, both the determinant (8.6) and all of it minors up to the order n - k + 1 inc ive reduce to zero and the que ion as to the existence of solutions of the system (8.5) requires further investigation

°

Let us assume that at least one of the minors of the (n - k)-th order of the determinant (8.6) is different from zero. Let this be the minor

(8.8) Then the first n - k of equations (8.5) can be solved for the magnitudes ex.l, ••• ,a _k so that we shall have ex.. n J 0 0 \ =ex..(ex. k l,.·.,an ,J.)(j=l, ••• ,n-k) with ex. j ( a nik+l, ••• ,an,Or J n- + = ex.~. Substituting these magnitudes in the last k of J equations (8.5) we obtain forme determination of ex.n_k+l, ••• ,an equations of the form (i= 1, ... , k),

where the functions F.1 reduce to zero for ~

(8.9)

= 0, ex.r = a ro •

As we already said, the system (8.5) admits for a solution depending on k arbitrary constants • Hence the system (8.9) must also admit for ~ = a solution containing k arbitrary constants, which evidently is possible only in that case in which for ~ = 0 they are identically satisfied. Hence we must have F i = ~ t i so that equations (8.9) assume the form ~

= 0

°

(i = 1, ..

't

k),

(8.9a)

where now the functions 1ri in general do not reduce to in zero for u, = 0 , an - k + l = a n0 - k +l ,···, a n = ex.n0 • lience H order that equations (8.9a) admit a solution a r = 0 = ex.r(~) (r = n-k+l, ••• ,n) for which conditions ex.r(O)=a r 491

are satis ed it necessary first of all that the parameters h. of the generating solution satisfy the equations 1

Pi (/ti, ... , hf) = =Wi«(1~-h+dh~, ... , hh), ... , (1~(h:, ... , Ilk). 0)=0

(8.10)

(i = 1, ... , k).

If, moreover, the condition is satisfied

(8.11)

then for sufficiently small ~ the system (8.9) will actually admit a solution for a of the required. form. Substituting r the found values of a l , ••• ,an the function Xs (t, il' ... , Tn' fl), we obtain a periodic solution of system (8.1) reducing for ~ = 0 to the generating solution ~s(t, h;, •.• ,h~), and s period 1:. From (8.10) we find:

Hence if the condition is satisfied

a(Ph'" , Pit) a(hr, ... , h:)

0,

(8.12)

condition (8.11) will so be satisfied. can be shown that, as in the case of nonautonomous systems, from condition (8.11) follows, conversely, condition (8.12), the proof of which however we shall not dwell on here. We arrive at the conclusion that to each generating solution for which the parameters hi are a simple solution of equations (8.10) there actually corresponds for small ~ a periodic solution of the complete system (8.1) .. the preceding considerations we have quently used the partial derivatives of the various functions with respect to the various arguments. It is easy to see that, as in the case of nonautonomous systems, for the assumptions

492

made with regard to equations (8.1) these derivatives actually exist.

9. Periodic Solutions of Autonomous Systems. Equations for the Parameters of the Generating Solutions Let us find the developed form of the equations (8.10). For this purpose we put

where Bn = 0 according to the condition on the choice of the magnitude y. Since the functions ~ s are continuously n differentiable and reduce to zero for 11 = a. = ~l =••• Pn _l=O, we can write equations (8.5) in the following form:

Here Usj ' Vs ' Ws are continuous functions of Sl, •.• ,Pn_l,a.,11, reducing to zero for Bl = ••• =B n _ a. 11 = 0, and the parentheses denote that the derivatives are computed for the generating solution, i.e. for 't T*, fl = 0, 1. -= ~s (0. hT •... , ht). From (9.1) it follows that equations (8.10) are obtained as a result of the elimination of the magnitudes a., Pl, ••• ,Pn-l from the linear equations

As a result of the fact that the functions ~s(~'Yl, ••• ,yn-l'l1) for l1=O,ys=~s(O,hl, ••• ,h~) reduce to ~s(~,h~, ••• ,h~)-Ys we have:

(a~s) =~s(T*. hT •... , h:).

(9.3)

Let us further consider the equations dy, dt = PslYl + ...

PsnYn

(9.4)

with periodic coefficients of period T

It

=

Ii

II>

T (hl, •.• ,h

"')

k

which are the equations in variations for the generating solution. We denote by Y;l(t, to)'."'Ys~(t, t o ) a fundamental system of equations determined by the initial conditions

(9.5) where ~sj is the Rronecker symbol. in sec. 2, we can write:

(:~;) =

Then, as was shown

Y:j (T*, 0) - SSj' T- n

(~~ ) = ~ ~ [f~ (111'

... , Tn' 0)]/=10. hi=h( Y:~ (T*, to)dto,

o 11=1

and therefore equations n-t

.l;

1=1

y:j(T*, O)~j T· n

+,.,. ~ ~

o 1'=1

(9.2) can be written in the form

~s+~s(T*, hT • ... , h:)tl+

[fp (~l' ...• 1'n> O)}I=lo.

hi=h(

y:~ (T*,

to) dto = 0, (9.6)

and the problem reduces to the elimination of the magnitudes ~l, ••• ,Pn~l,a from these equations. For the solution of this problem let consider more closely equation (9.4). These equations have k particular solutions (i=1, ... ,k).

Ysi

(9.7)

Moreover, since in the solution (8.7) an arbitrary constant h may be added to t, equations (9.4) have also the solution

Ys.

•

H1 = ~s (t,

hi, ... , h:).

(9.8)

This solution is periodic of period T~. As to the solutions (9.7), they will not in general be periodic. It here necessary to distinguish two cases: when the period T of the solutions (8.7) depends on the parameters h. and 1 when it does not depend on them. Let us assume first that hl, ... ,hk • All the solutions

T

does not depend on

(9.7) will then be periodic

of period '11+. Thus, in the case considered the system (9.4) will have k + 1 periodic solutions (9.7) and (9.8). We shall assume that the system (9.4) has no other periodic solutions. Let us assume now that the period T depends at least on one of the magnitudes hl, ... ,h . Let us consider the k functions

These functions will evidently be periodic with respect to t with a period T" not depending on hi" As a result the partial derivatives of these functions with respect to hi will also be periodic of period T~. Forming these

= h~

derivatives and then setting h j s) ( au ahi

• = (

a~~ (t. lIlt ... • ah,

hj=h

"I!.»)

we shall have:

•+

hi=hJ

+Ti. ( : { )hj-hj _ .t;,,; (t. "T ..... "n· I

Hence, the solutions

(9.7) have the form (9.9)

where

are periodic functions of t of period T". From this it follows that the equations (9.4) have, besides (9.8), additional k - 1 periodic solutions

os,. (

t) ( aT ) ah 1

0 hi=h; -

495

d

(t) (

aT ) ahr

• hj=hj

(r = 2 . ... . k) .•(9.10)

Thus, case where the od T denends on h.1 equations (9.4) have the k periodic solutions (9.8) and (9.10). We shall assume that system (9.4) has no other neriodic solutions. L

Vv'i th this established, let us proceed to the elimination of the magnitudes cx., ~l""'~n-l from equations (9.6). Vie shall assume first that the period T depends on the parameters hi' In this case the system conjugate to (9.4) will admit k periodic solutions, which we shall denote by 'i's l' ••• , \j1 sk. It is easy to see that n

~ fl'(T*. hl, ... , h:)hdT*)=O

(9.11 )

1=1

(i=t, ... , k).

In fact, since y

. and

sJ

~

. are solutions of

Sl

c on,iugate systems we must have: n

~ YTi (t) ~l'i (I) = const.

1'=1

Hence n

~ Yl'i (1'*)

n

hi (T*)

1=1

!!!II

~ Yl'i (0) ~,i (0), 1=1

whence onfue basis of (9.9) equations (9.11) follow, since the functions v sj are periodic. Let us now multiply equations

(9.6) by

..

tsi (T ) and sum with respect to the index s. Then, taking into account (9.11) and (2. ), we obtain, exactly as in sec. 2, after dividing by ~. the equations Pj(h!, ... ,ht)= T·

=~

n

~ fp[~l (t, hT, ... , h;:), ... , q:.n(t, hT, ... , Ii:), O];'(Ji(l)dl=O.

o (J=I

(9.12)

This will be the result of eliminating the magnitudes cx.'~l, ••• ,Sn_l from equations (9.6), i.e. the develop form of equations (8.10).

496

Let us now assume that the period T of the generating solution does not depend on the parameters h.. In this case 1 the system conjugate to(9.4) will have k + 1 periodic solutions, which we shall denote by 'fIs 1,···,·11 k l ' The T s, +_ identities (9.11) will not in general be satisfied and, proceeding in the same manner as in the previous case, we obtain the k + 1 equations n

•

Ql'(hi, ... , lit, a.)=a. ~~,(T*, ~=1

h!, ... , h:)~Pr(T*)+

T-

+ ~ !P[Tl(l, hl,

... , It;), ... , 1fn(/, ItT, ... , h:), 0] ~~r(l)dl=O

I)

(r =1, ... , k+ 1).

(9.13)

In order to obtain the functions Fi it remains only to eliminate the magnitude a from the obtained equations. There is however no need of this. In our case it is more expedient to consider directly equations (9.13) that permit determining both the magnitudes hi, ••• ,h~ and the magnitude a, the first ap~roximation of the correction of the period. Condition (8.12) is now replaced by the equivalent condition (9.14)

Equations (9.13) agree with those which we obtained for quasilinear systems for which the case under consideration precisely occurs where the period of the generating solutions does not depend on the parameters. We have thus obtained the equations determining the zeroth approximation of the required periodic solutions of the system (8.1). For computing the further approximations there may be used, as for quasilinear systems, either the method of successive approximations or the method of expansion into series in ~, if the equations under consideration are analytic. We shall not however here dwell on this question. 10. Almost Feriodic Solutions of Nonautonomous Systems in the Case of an Isolated Generating Solution The fundamentnl results of sec. 1 and 2 on the periodic

497

solutions of nonautonomous system can be generalized to 1

almost periodic solutions.

Let there be given the nonautonomous system d;s

Xs (i,

Xl' ••• ,

XII)

+ 'rls (I, Xl'

.•• , x n• 11)

(10.1)

(s=l, ...• n), where with regard tome functions X and f we make the s s following general assun:.tions. We shall assume that the variable t varies in the interval ( - m , + Q ) , the variables xl, ••• ,xn in a certain region G of the space of these variables and the param'eter 1J.. on the segment [0, 1J.. The functions X possess derivatives of the

1.

o

s

second order with respect to the variables Xl'.'.' ,and the functions fs derivatives of the first order with res~ect to the variables xl, ••. ,x and 1J.., and these derivatives n satisfy with respect to these variables the Cauchy-Lipschitz conditions with coefficients not depending on t. With resnect to t the functions Xs are continuous and periodic with given period W ,whi the functions fs are almost periodic and such that for any fixed u the functions fs(t, x l (t), ••• ,xn (t),1J..), where xs(t) are arbitrary almost periodic functions of t lying in the region G, will also be almost periodic. Let us consider the generating solution (10.2)

and assume that it admits in the region solution

x: = '$ (t).

G

the Deriodic (10.3)

Taking this solution as the generating solution we

1 Malkin I.G., 0 pochti-periodicheskikh kolebaniyakh neavtonomnykh sistem (On Almost Periodic Oscillations of Nonautonomous Systems), • matem. i mekh., vol. XVIII, no. 6, 1954.

498

shall seek the conditions for which the complete system (10.1) admits an almost periodic solution which reduces for ~ = 0 to this generat solution. Let

be the equation in variations for the generating solution. We shall say that the solution (10.3) is an isol ed one if the equations in variations (10.4) do not have almost periodic solutions, for which it is necessary that their characteristic equation have no roots equal to unity. rhis condition is more rigorous than in sec. 1, where we considered a solution as isolated if s equations in variations have no periodic solutions. This more rigorous requirement is connected with the fact that we are nov'! considering periodic solutions as a particular case of the wider class of almost periodic solutions. In the present section we consider the problem for the case of an isolated generating solution. The following theorem holds here: THE GENERATING SOLUTION ISOLATED THE SYSTEM (10.1) FOR SUFFIC SMALL ~ ADMITS ONE AND ONE ALMOST PERIODIC SOLUTION REDUCING TO THE GENEIi1\TING SOLUTION FOR ~ = O. J.J.J..;J'VJ.l...LJLU.

PROOF.

Let us put in equations (10.1) (10.5)

after which they will assume the form n

t/%t= ~ p$aYa+Y,,(L, Yl' ...• y,.)+f1/s(t, ~1' a=1

: •• t

+ f1F s (t, Yl'

...

where

Y, = X, (t, '1'. + y,. ... , '1'. + - Xs (1, F, =

0 two and only two solutions will be real, one of them being positive the other ne ive. Hence, for h2r> 0 we can give the number p any integral values less than -,.. while for h2r < 0 any integral values greater than "'}... Choosing the number p this manner we obtain for each p two generating solutions

{~(p) (I

Besides

+ h).

{~(p)(t

+

1jlp)

(t + It),

~~P) (I

+ h)}.

h), n(p)(t + h),

~~p)(t

+ h)}

a generating solution wi also be the trivial solution ~::= n ::= ~l ::= • • • ="5 ::= 0 of the system (1.4), which m can be considered as a periodic solution of period 2n. 2. The

riodic SOlut"ion

{x~ o)}

proceed to the determination of the periodic solutions of system (1.1). We shall first find a periodic solution reducing for ~ ::= 0 to the trivial solution x~ = ••• = x~ ::= 0 of the generating system. This solution we shall denote be the symbol {x~}. If we shall make use of the equations of oscillations in the form (1.7) we shall denote the same solution by {x(o), y(o), x~o)} The equations in variations of the generating system (1.2) for the trivi evidently the form

solution x~

= ••• = x~

::=

0 have

(2.1 ) Let us assume that the fundamental equation (1.3) of this system does not have roots of the form + pi, where p is an integer. In this case the system (2.1) will not have periodic solutions of period 2n and

549

therefore the generating solution considered will be an isolated one. On the basis of the general Poincare's theorem established in sec. 1 of chapter VI the system (1.1) will then have one and only one periodic solution reducing for ~ = 0 to the generat solution and this solution will be analytic with respect to u. For the practical computation of this solution we shall seek it in the form of the series

(2.2) with periodic coefficients, formally satisfying equations (1.1). For the coefficients of these series we obtain equations of the form dz;",

lit =

as1x 11l

+ ... + QsnX"t, + /$It ( f).

(2.3)

wher fSk are integral rational functions with periodic coefficients for all x . for which J'< k. These equations sJ

-

make it possible to determine successively the coefficients x sk in the form of periodic functions of time, entirely definite coefficients being obtained. If the fundamental equation (1.3) has purely imaginary roots of the form ± pi, where p is an integer, resonance will take place. We must also consider as resonance cases those for which the fundamental equation has roots differing from ~ pi by a magnitude of the order of smallness of~. The theory of resonance we shall take up in the following section.

3.

riodic Solution for Resonance

We proceed with the question of the existence of a periodic solution of system (1.1) that reduces for u = 0 to the" t al solution x~ ~ ••• = x~ = 0 of the ~enerating system in the case of resonance. Let us therefore assume that the fundamental equation (1.3) has a pair of roots differing from + pi, where p is an integer, by a magnitude of the order of smallness of~. We shall assume that equation (1.3) has no other roots of this type. In regard to the correction terms to the functions fs we can assume, as in the earlier investigated cases of

550

resonance, that the above mentioned critical roots are equal to ± pi. We shall assume that this pair of purely imaginary roots satisfies all conditions stated in sec. 1 of the preceding chapter so that these roots can be taken for the fundamental pair of purely imaginary roots which figures in the form (1.7) of the equations of the oscillatiions The equations of the oscillations can thus be represented in the form d:r:

di =

py+ X + p/ (I, x, y, xi' p),

-

dy

di

px+Y +pF(I, x. Y. xj • p),

\

I

(3.1)

j

(8=1 ..... m).

If we try to satisfy these equations by formal series of the form

x = p.x(1) (I) + l'-IX(I) (I) + ... , Y= flY(!> (I) + plyls) (I) + .. ., Xs = f1X~1) (I) + l'-!X~I) (I) + .. .

with periodic coefficients,we obtain for the coefficients of the first approximation the equations d:,;ll)

pym + /(t, 0, ... ,0),

(jI= -

dym _

~-px

dx'lI

d:

m

+ F( I" 0

.• "

0)

(3.2)

,

m

=~

bs'%x« +Fs (I. 0 .. ,., 0).

«=1

Since by assumption the fundamental equation (1.5) has no roots of the form! ki, where k is an integer, this system admits a periodic solution for

x~l).

As to

the magnitudes x(l) and y(l), in order that they be periodic it is necessary and sufficient that the two magnitudes

551

(3.3)

reduce to zero.

There has here been put

.!.

I (t. O•.••• 0) = A,. +

(A" cos itt + B, • Si.D nt), }

~

(3.4)

F(t, 0, ... ,0)=BI0+ ~ (A 2ncosnt+Bansmnt). n=1

We shall assume that at least one of the magnitudes In this case we shall say that PRINCIPAL RESONANCE takes place.

(3.3) is different from zero.

For principal resonance the following theorem is true: THEOREM. LET 2Z BE THE SJHALLEST DEGREE Oli' THE MAGNITUDE c IN THE EXPANSION OF THE PEHIOD

OF THE PERIODIC SOLUTION OF THE GENERATING SYSTEM

de

7i = -P1j+X(t

!ft

=

dE~ _ 7t -

p~ + y (~, lit

~ b

t

4.J SQ~II

Tj,

7],

(j)'

)

I

(j)'

(3.5)

+ ·los ,,.. (.~t 71, ~)

«=1

WITH THE INITIAL CONDITIONS qO) =

C,

lj (0)

=- o.

THEN, FOR PRINCIPAL RESONANCE, THERE EXISTS Olm AND ONLY ONE PERIODIC SOLUTION [x(res), y(res), x~res)-, OF EQUATIONS (3.1) FOR WHICH THE FUNCTIONS x(res) , y(res)~(res) s REDUCE TO ZERO FOR 11 = 0 AND THESE FUNCTIONS CAN BE EXPANDED IN SERIES OF POSITIVE INTEGRAL ?OWERS OB' 'THE M.AGNITUDE V = 11 ~ Z .;i , CONVERGING FOR SUFFICIENTLY SMALL VALUE OF 11.

552

PROOF. us consider the generating system (3.5). As was shown in sec. 4 of chapter VII, for this system there exists a periodic solution containing two arbitrary constants a and b, which are the initial values of the magnitudes ~ and n. The magnitudes ~s in this solution are functions of ws ( S, n) of ~ and Tl satisfying the partial differential equations

a;~s.

[- Pll + X (t 11, wj)1 + ~ [PlI + r (~, 11, w})] = ~

m

= ~

bsClWa

+ X,s (~, TI, w]) .

... =(

Let us in equations (3.1) replace the variables x s by the variables Us with the aid of the substitution

As was shown in sec. 4 of chapter VII, the expansions of the functions ws start wi~h terms of not lower than the second order and therefore the linear terms in equations (3.1) do not change. The functions X, Y, Xs ' f, F, Fs however are replaced by other functions of analogous form, which we shall denote respectively by x~ x;, f: F~ F;. Equations (3.1) thus go over into the equations

Y:

"(i(= - PY+ X* (x, h

y, llj)+ 11!* (I, X, y,

J,l).

)

F* (

y* ( .

(Iy -

lti'

; , -i:.x.~·":; ;~~: PUj);~:::l;:"~: ::'"" ,.).

(3.6)

Jt

'1:::(

and equations (3.5) assume the form

~;

= - I'll + X*

~/; = p~ + y* (e.

(~. "ti, ~i)' lj,

~),

(3.7)

til

~ = ~ b$a~ ... + x: (~. 11=1

11.

e

j ).

J

The transformation made is such that the problem of

seeking the periodic solution of system (3.1) is equivalent to the same problem for the system (3.6). We here evidently have:

/*(1,0,000,0)=/(/,0, 000,0), F*(I, 0, 000' O)=F(/, 0, 000,0),

so that the magnitudes (3.3) do not change and therefore principal resonance also takes place for the system (3.6). The periodic solution of the generating system (3.5) goes over into the periodic solution of the generating system (3.7). The latter solution possesses the property that for it the magni tudes ~ are identically equal to s zero. This follows immediately from the transformation formulas. Hence the solution of equations (3.7) with the initial conditions

where a and b are arbitrary constants, will be periodic. The period of the solution on the basis of formulas (4.13) and (4.15) of chapter VIr has the form

T = 2n: {I + H 2,( a 2+ b2 )' + .. o}, p

(3.9)

where the magnitude H2Z = h 2 Z by the condition of the theorem is different form zero. From the fact that equations (3.7) have a solution in which all ;s are equal to zero it follows also that (3. to)

Having established this, let us denote by x{t, a, b, ~i' (1),

y{t,· a, b, ~i' (1),

u~ (t, 4, b, ~i' (1)

(3.11)

the solution of equations (3.6) with the initial conditions X

(0, a, b, ~i' fl) = a,

y (0, a, b, ~i' fl) = b,

554

We have identically: {x (pT, a, b, ~i' O)hj=o

=

{y (pT, a, b, ~j' O)} Pj=o

= b.

a, )1

(3.12)

In fact, the solution under consideration for ~ = 0 reduces to the solution of system (3.7) which for the initial conditions (3.8) will be periodic with period T and therefore also with period pT. Further, on the basis of (3.10) (3.13) In order that the solution in question of the complete system be periodic of period ~ it is necessary and sufficient that the equations be satisfied [X]=X(2TC, a, b, ~j' p)-a=O,

I

[y]=y(2TC,a,b'~J,p)-b

[us] = Us (21C, a, b, ~J' t1) (s=1, ... ,m),

0,

~a = 0,

\

(3.14)

jl

where the notation was introduced

(fs (t)]

== f. (21C) -

fs (0).

Let us consider equations (3.14) in more detail. Writing out only the linear terms we have:

x (t, a, b, ~ j' ft) = Ala + BIb + C1ft

+ ... ,

y(t, a, b, ~j' ft) =Aza+Bab+Caft+···, lls

(I, a, b, ~j' ft) = AS1~1 +

... + Asm~m + Dsft + . ..

1 i (3.15) J

(the remalnlng linear terms, as can easily be seen from equations (3.6), identically reduce to zero). Hence the functional determinant of the last m equations of (3.14) with respect to the magni tudes ~ 1 ' • .• , ~ m for a = b = ~l :::; .•• :::; 6m :::; ~ = 0 has the value

All (211) - 1 A21 (211:) A= . .. . . .. Ami (211:)

An (211:) · .. AIm (2n) A Z3 (211:) - 1. .. Aam (211) . . . . .. . · . .. .. . . .. Am! (2n) · Amm (21t)-1

.

555

..

.

.

(3.16)

But the magnitudes

A

.

sJ

satisfy the equations

(3.17)

and the initial conditions

From this it follows that if the system (3.17) is considered as a special case of a system with periodic coefficients of period 21t and DC P) denotes its characteristic' determinant, we obtain from (3.16) ~= D(l).

But the characteristic exponents of the system (3.17), considered as a system with periodic coefficients, are the roots of its fundamental equation (1.5). Among the latter by assumption there are no magnitudes of the form ~ ki, where k is an integer, and therefore the characteristic equation DC p) has no roots equal to unity. Hence we have .~~ O. The last m equations of (3.14) are therefore solvable for the magnitudes ~s and give for them the analytic functions ~s =~: (a, b, "')

(s=l, ... ,m),

(3.18)

reducing to zero for a :: b = ~ = O. However the functions (3.18) will reduce to zero for ~ :: 0 and a and b different from zero. This immediately follows from the fact that the functions u (t, a, b, S .,~) on the basis of s .J (3.13) necessarily reduce to zero for 1 {!J m :::: ~ 0 whatever the values of t, a, b.

e

=... ::

Thus, ~1(atb,O)=O.

Let us now substitute the obtained magnitudes

556

(3.19)

Pj

in

the first two of equations

(3.14).

We shall have:

(2it, a, b, ~1 (a, b, p.), p.} - a = 0, y (2it, a, b. ~j(a, b, p.), 11) - b = 0.

X

Separating out the terms not depending on taking (3.15) into account we find:

~

and

(27t, a, b, ~10

•

(6.11 )

Substituting here for ~~ and 0/2 their expressions (6.7) and taking (3. ) into account we obtain:

+ 21)(a!+ b!)21 + ... = " + .... p2H;, (1 + 21)(a~O)2 + a.~O):t)!rp.t+2t

O(~l' 0/2) =i4'1t2p2H!, (1

o(a,

b)

=

4'1t2

570

where the terms not written down have a higher order of smallness. From this it follows that for sufficiently small ~ the inequality (6.11) is always satisfied. Hence there remains only the one condition of stability (6.10) and if it is satisfied with the inequality sign the solution {x(res), y(res)} will actually be stable and furthermore asymptotically so. .

7.

Application to Duffing's Problem

This section is devoted to a detailed investigation of the forced oscillations of a system described by the equation Tx3 = P. ( a l'Ol' lilt

(a

0, b >0, JJ

+ beos 111-2H ~:)

(7. t)

> 0, 7 > 0).

where m and n are integers. We are thus considering the problem of Duffing in the case where there is resist·ance and the disturbing force contains two harmonics. The case where there is no resistance and the disturbing force contains only one harmonic was considered by us in chapter I with the aid of the quasilinear theor~ We now consider the general case of equation (7.1) treating it as an approximation to a Lyapunov system. We shall investigate for equation (7.1) the solutions { x(m) and x(n)1 , the solution tx(o)j, the resonance solution, the stability of all these solutions and the character of the development of the oscillations. We shall not here consider the solutions {x(p)~ for p different from m and n, which evidently determine the subharmonic oscillations of various orders (integral and fractional). We shall consider subharmonic oscillations in the next section for another system.

J

1

We shall start with a consideration of the solution {x(m)). The solution {x(n)3 is obtained by a simple interchange of the numbers m and n. a) SOLUTION

{x(m)}

The generating equation d2! . k.... ~a -~ - 1" =

dt% T

571

°

(7.2)

was considered by us in the preceding chapter. The general solution of this equation is determined by the formulas (3.12) there obtained. Hence on the basis of (1.11) we have:

where

(7.4)

The period

T

is determined by the formula 2:t (1 T -- T

l. ) + l·:! ';!t + '-Ie +... ,

and therefore the equation (1.12) for c

m

assumes the form

(7.5) This equation has real solutions only for k> m. There will be two such solutions and they will be numerically equal to each other and of opposite sign. To find the solution {x(m)} we use the general formulas of sec. 6 of chapter VI. First, by the formulas (6.12) and (6.16) there established, we find the functions u and v, for which we obtain:

-Cmsrnm(t+Ii)+~:n~3[m(t+h)]+ 1 +C~i~5 (m (t + Ii)] + ..." v = A~m {cos III (I + h) + 3(';n~3 (Ill (I + It)J + + 5C:II~5 [111 (I + h»)} + ... ,

u(t+h)=

I

572

(7.6)

INhere (7.7)

and by formula (6.22) of chapter

we find the constant K,

(7.8) , setting

ermine the magnitude by the general formula (6.30) we of chapter VI. Carrying out computations with an accuracy up to magnitudes of the first order with re ct to c m we shall have: (7.10)

where *_

Il -

a cos mh (

2KA 2c;"

2) cosm(t+h)+

3 T2

-1 +32k2 em

+ mHcm :urx sm· m (t

h)

2

a bm + 4KA cos mt + (m!-n!) KA cos nt,

(7.ft)

and the constant a must be ermined from the condition l of pe odicity of the function x • 2 For the constant h we have on the basis of formula (6.29) of chapter VI the following equation: 21t

P (It)

~

[ a cos mt + b cos lit

28 d;(m) (t +h)] de

o b

d~(m) (t +h)

d,

Let us consider this equation more closely. s of (7.3) and (7.4) we can write: 00

~Oll) (t

Ii) = .~

J=O

A:?j+l

573

cos (2j + 1) In (t

+h),

dt- 0 -

.

On the

where

• We shall assume that the ratio n/m is not an odd integer. With this assumption the equation for h assumes the form co

P(It)

- attmA 1 sin mit - 2nHml ~ (2j j=O

+ 1)2A~J+l

O.

This equation determines two values for the angle TI. The magnitude P'(h) for these values of h is different from zero. If we now take into consideration that we have two different values for c of equal magnitude but opposite sign m we must conclude that tpere exist four periodic solutions { x(m)} • Actually however there are only two different solutions f x(m) In fact, since the function $ (m) (t+h) contains only odd powers of c m and only odd multiples of the angle met + h) the substitution of TI - mh for mh is equivalent, as can easily be noted from the form of P(h), to passing from the generating solution f"§. (m) (t + h) with the chosen value of c m to the generating solution

mh, the sum of these angles being equal to

1.

J

J

[ ~ (m) (t + h) with the value of c of opposite sign. m We can therefore assume that the angle mh is acute.

If only the first powers of c are retained in the m function P(h) we arrive at the following approximate formula: .

,

2mHcm . a

sm m,l =

(7.12)

This approximate formula will evidently hold also for the case that n/m is an odd integer-. a

l

•

We shall proceed to the computation of the magnitude For we obtain the equation d1x,-

dt Z

+ (k 2 _

3T~(m)2) x = 31~(m)x2 _ 2H 2

574

1

dXl

dt '

and the condition of periodicity of the function x

2

has

the form

Substituting for xl its value (7.10) and noting that the coefficient of a~, by what was earlier proven, neces reduces to zero and that

ly

we obtain:

If we now restrict ourselves in the computation to only the smallest term (with respect to c ) in the function m x~, i.e. the term -

acosmh ?K % cosm(t+h), _ ..12c'"

and the smallest terms in the expressions for we shall have: 4H a 1 =-3 2 1'C...

J(m) and

u

(7.13)

•

}!'ormulas (7.3) - (7.13) determine the solution lx(m),3 b)

SOLUTION [x(o)]. x(O)

For this solution setting = !lx1

+ !lIX:. + . . .

(7.14)

and proceeding as indicated in sec. 2 we find: XJ

a = k'l._m 2 cos

X2

= (k2_m2)2 sm mt

X3

2Hma.

mt

b kZ_n%


dz ) . (. 7 17) acosmt+ bcosnt-2Hr,+>-X

Following the method of sec. 4 we seek the solution of this equation in the form of the series 1

x~es)= Xd~3 + X2

2

113

+ Xal1 + ...

(7.18)

We have: d2z S ell2

+ m2Xa = jX: + a cos mJ + b cos nt, tPz. + dZ dt m x. = 3" .Xix, - 2H {it + "Xlt 2

t:JIz. dtZ

1

2

+ m2Xi = 3" 1Xixa+ 31X I X

:I 2-

\

2H Cit dZ 2

whence we find:

xl=Alcosmt+Blsinmt, } %2 = A2 cos mt + B2 sin mt,

576

(7.19)

where Al , A2 , Bl , B2 are constants. Equating the coefficients of cos mt and sin mt to zero in the equation for x~ we obtain: (7.20)

after which we shall have: a

b

xa=24m 2 COS 3mt+ m-n 2 2 cosnt

Ascosmt + Bssinmt. (7.21)

From the conditions of periodicity of the functions x 4 and x5 we find the following values of the coefficients , B2 , A3 and B3: B1_8Hml¥3y Y 4a' }

(7.22)

Ba == _ 16Hm). . 9ay

Formulas (7.18) - (7.22) determine the resonance solution. d) STABILITY OF THE SOLUTIONS. 'vVe shall investigate the obtained periodic solutions of equations (7.1) for stability. The equation in variations for the solution has the form

l x(o)}

where the terms not written down are of an order higher than the first with respect to~. Hence the characteristic exponents of the solution tx(O)} are determined by the formula

From thisrit ~ollows that for sufficiently small solution ~xlO/J is asymptotically stable.

~

the

For the solution{X(m)lthe conditions of stability,

577

as was shown in sec. 7 of chapter VI, are given by the inequalities KdP(h) ~O dh ~~ •

(7.23)

where f is the right hand side of equation (7.1). The first of these conditions is evidently satisfied with the inequality sign. Further, on the basis of (7.8) and the expression for P(h) we have: K dP (It) _ dh -

akrc cos mh

TIl

(2h z + 4h«c;" +

2

... ) (k -

2

TC m )

(

t Y

2

)

1 + 32 k2 Cm + . . . c,!t'

whence it follows that for sufficiently small c m the second of conditions (7.23) is satisfied with the inequality sign if c m< 0, and is not satisfied if c m > O. Consequently, of the two solutions x(m)) the solution corresponding to the negative value of c m is asymptotically stable while the solution corresponding to the positive c m is unstable •

1.

Let us consider, finally, the resonance solution. The .stability condition (6.10) established for it in the preceding section agrees in the case under consideration with the first of conditions (7.23) and therefore satisfied with the inequality sign. The resonance solution is thus asymptotically stable. e) CHARACTER OF THE DEVELOPMENT OF OSCILLATIONS. From all the foregoing it is now possible to draw definite conclusions in regard to the ciharacter of the development of the oscillations. Let n,.m and assume at first that k « m. The oscillations will then develop according to the stable solution fx(o)]. As k approaches m the series determining the solution [x(o)] begins to diverge. From this however it does not follow that the solution [x(o)] ceases to exist. It can be retained but has a different analytic expression. A more detailed analysis shows that as k approaches m the solution tx(o)J continuously goes over into the resonance solution (in the neighborhood of k = m) while

578

the latter, with further increase in k, continuously goes over into the solution {x(m)) , arising after the resonance, corresponding to the negative value of c m. For such value of k (or somewhat greater) there will also exist the solution corresponding to ±he positive value of c m' and again the solution! x( o)} • The former of these solutions, as has already been shown, in unstable. With further increase in k the magnitude c m' as follows from its defining equation, will increase numerically, i.e. the amplitude of the oscillations will increase, as a result of which these oscillations will at a certain instant become unstable. Let us assume that this occurs for k = k·. How will the oscillations develop for k >k·'G It is here necessary to consider two distinct cases de- • pending on whether the magnitude n - m is su~ficiently large or not. Let us assume first that the difference n - m is sufficiently large, namely we assume that n >k1l'.. In this case for k = k" the only stable solution is fx(o)J As a result the oscillations go over discontinuously into this solution. With further increase of k the solution {x(0)3 continuously gees over, through the resonance solution in the neighborhood of k = n, into the solution {x(m)] corresponding to the negative value of c , n and then, after loss of stability, again into the solution {x(o)J (discontinuously).

-,r/IIIJ

~----~m~------------~~~------------~-.

Fig. 31

579

Let us now assume that the difference n - m is small, namely we assume that n (k~. In this case the oscillations will develop according to the solution {x(m)j with negative c m for values of k not only

Irlmaz

_Zlflt/

L-------~m~--------------~n~------~H

Fig. 32 less than but even equal to and exceeding n. Hence, notwithstanding the fact that the second harmonic will be in resonance the first harmonic will prevail in the forced oscillations. For k >k~ the oscillations will break do~~ either' into the solution x( 0) or the solution 1x(n) ~ (with negative c n ), which can still be stab • All that was said above becomes very clear if we construct the resonance curves. Fig. 31 shows the relation existing between the different periodic solutions. The dotted curves correspond to the generating solutions I ~(m)(t + h)} and fs(n)(t + h)~ ( "the skeleton curves "). in fig. 32 the arrows indicate the

1 J

1.r11lJtt.:

~--~m~-------L----------=--~K

Fig. 33 580

development of the oscillations with gradual increase of k for-the case where the difference n - m is sufficiently large. In fig. 33 is shown one of the possible variants of the development of the oscillations for the case in which the magnitude n - m is small. 8. Example of Determination of Subharmonic Oscillations The problem which we have considered in the preceding section could have been solved also with the aid of the quasilinear theory. In determining however the oscillations near resonance it would have been necessary for us to solve a system of two cubic equations for determining the amplitudes of the generating solution and then to investigate the nine periodic solutions thus obtained. If the order of the characteristic of the nonlinearity in equation (7.1) were higher than three the degrees of the equations which determine the amplitudes in the generating solution, if this equation is regarded as quasilinear, would likewise have been higher than three. This would have resulted in the practical ineffectiveness of the method. On the other hand, an increase in the order of the characteristic of the nonlinearity in equation (7.1) in applying the method of the preceding section would in no way have complicated the computations. Examples may however be given for which the quasilinear treatment, in ~eneral. leads to erroneous results. Thus, in particular, we saw that if the equation (4.5) considered in sec. 4 is treated as quasilinear we arrive at the erroneous rewlt of the absence f"or this equation of a periodic solution, which however actually exists and can be computed with the aid of the procedure there indicated. We shall now, with the aid of the methods developed in this chapter, determine for a system of the type (4.5) the sub harmonic oscillations and shall then show that these oscillations cannot be determined with the aid of the quasilinear theory. We shall consider the equation d

2

x+ k x

dt 2

2

~X2 = flU cos

.

nt

(8.1)

and compute for it the solution x(l) ,which evidently determines the subharmonic oscillations of order lin. The general solution of the generating equation

581

~; -J- k2~ _~~2 =0 is given by the formulas (3.13) of the preceding chapter.

We therefore have:

00

~(t) (t

where

+ h) = m=O ~ Am cos m (I + 11),

.. _1~!

t~!3 k4 C1

+···, _..!..! ~ 29 fS! 3 k2 C + 144 C + . . .,

·>-0-2 kll

C1 -

A =

C

A..~ =

-..!.ti k~3 C~1 ..!. C -L 9 ~ k" 1

1

:I

1

1

kl

1

3

j

...,

1 SlI

A3 = 48 ok" c~ + ...

For the period T of the free oscillations we have:

and therefore the equation determining c l has the form 5 112

3

2

12 k" C1 -

fa5 13

11;6

3

C1

+ ...

r=

k - 1.

This equation determines two real values of c l if the condition is satisfied k-1

> 0,

(8.2)

these values being equal in magnitude and of opposite sign. We shall restrict ourselves to determining only the generating solution, for which it is necessary for us to compute only the value of the constant h. For this magnitude we have the equation 2111

P(h)=

-Q

Scosnt o

00

~

Ammsinm(t+h)dt

-anttAnsinnh=O,

111=0

from which we find for it the two values

582

To each of these values(~f) h there actually corresponds a periodic solution [x since

Since we have two different values for c l ' we thus obtain four different solutions (xCI)] , i.e. for equation (8.1), with condition (8.2) satisfied, there exist four forms of subharmonic oscillations of order lin for any n. For the magnit~de K, determined by formula (6. of chapter VI, we now have: K= _

)

k(kl_~l) 2h~+3h3Cl

+ ....

From this, taking (8.3) into account, we see that the second of the conditions of stability (7.23) is satisfied (with the inequality sign) only for two of the solutions (x(l)}. As to the first of conditions (7.23), it is always satisfied with the equality sign. Thus, of the four subharmonic oscillations found two are st,able and two unstable. Let us now try to solve the same problem with the aid of the quasilinear theory. For this it is necessary to put S = ~et and consider the equation (8.1) for the values of k near unity, i.e. it is necessary to put also k 2 = 1 - ~b. It is not necessary however to assume that the amplitude of the disturbing force has the order of smallness of~. Thus, with the quasilinear. treatment we must consider an equation of the form tPx dt Z

+x =

a cos nl

+ f1 (bx + W.:rl ).

Setting

we obtain for xl the following equation: (/2xJ J

dt2

T

_ Xl -

t

1=11

t'

(M2 + IV2 + (l-n!)Z a 2

0

. a' a + bNosmt+ l~n~

0

),

T

JlfoCOS (n -1) t

583

bM0 cos t + B'a. + l~n:l Nosm(n -1)t+ ... ,

where the terms not written down contain the cosines and sines of the arguments 2t, nt, en + l)t, 2nt. The equations determining Mo and No will therefore either have the form

if n

~

0, or the form

if n = 2. In either case these equations will have solutions different from the trivial solution Mo = No = only in case the condition ~s satisfied

°

~,Q) =0 iJ (Mo • .No) .

But with this condition satisfied the methods discussed in chapter I are inapplicable. Thus, at least with the of the general procedures of the quasilinear theory, we are not able to determine the subharmonic oscillations for the system considered. With the quasilinear treatment the problem requires special investigation since it belongs to the class of singular cases. 1

1

All the computations in this section are taken from the graduate thesis of the student V. M. Mendelson.

584

AUTHOR Andronov A.A. , (Andronow) Artemyev N.A. , Belyaev N.M. , Biryuk: G.I., Bogolyubov N.N., (Bogoliuboff) Bohr H., Bulgakov B. V• , Bulgakov N. G. , Butenin N. V., Chetaev N.G., Coddington E.A., Duffing G., Gorelik G.S., Goursat E., Kats A.M., Khaikin S.E. (Chaikin) Krein M.G., Krylov N.M. , (Kryloff) Levins on N., Lyapunov A.M., (Liapounoff) Loitsyansky L.G., Lurye A.I., Malkin I. G. , Mandelshtam L.I., Mendelson V.M., Merman G.A., Minorsky N.,

INDEX 1

3, 20, 42, 43, 62, 76, 165, 177, 219, 361, 519, 530, 540, 543 244 366 295 47, 273, 274, 275, 279, 330, 362, 517 263 143, 362 155, 165 143, 145, 350, 359 211 165 27 427 6, 242, 372 482 42, 43 397, 403, 421 47, 273, 274, 275, 362 165 68, 72, 73, 182, 209, 211, 216, 222, 311, 367 350 350 116, 203, 211, 220, 295, 459, 498 35, 362, 366 584 163 43

1 The parentheses indicate alternative transcriptions of the names of the more familiar authors that appear in the literature -Translator

585

Neugebauer 0., Papa1eksi N.D., Petrovsky I.G., Poincare H., Ryabov Y .A. , Shimanov S.N., Soko1ov V.M. , Stellmacher K., Stepanov V. V • , Stoker J.J., Van der Pol, Vitt A.A., (Witt) Zhevakin S.A.,

263 35, 362, 366 100, 449 62, 449, -490 524 155, 243, 244, 440 524 138 100, 431 41 177, 362 3, . 20, 76, 165, 177, 219 490

586

SUBJECT

INDEX

Almost periodic function, mean value of, Asymptotic stability, 183 Averaging principle, 359 Canonical equations, form of, 109 Canonical variables, 113 ~Center', 54, 477 Characteristic exponent, 191 Circuits, inductively coupled, 176, 246 Conjugate systems, 109 Critical case, 214 resonance, 9, 116 singular, 135 Critical roots, 273 Disturbance, 182 Duffing's problem, 32, 88, 571 Equation, characteristic, 71, 186 of distributed motion, 69 fundamental, 100 generating, 4 Mathieu, 388 in variations, 69 "Focus l1 , 60 Follower system, 143, 232 Fourier series, generalized, Gyroscopic car, 350 Interval, homogeneous, 374 nonhomogeneous, 374 Limit cycle, 60 stable, 61 unstable, 61 Longitudinal bending, 366 Method of Poincare~ 1 ff. Motion, disturbed, 182 undisturbed, 182 stable, 183 unstable, 183

587

261

262

Oscillations, forced, of quasiharmonic systems, 422, 427, 434 periodic, of autonomous systems approximating arbitrary nonlinear systems, 487, 493 quasiperiodic (combination), 85 of approximations to Lyapunov systems, 525, 530, 538 of nonautonomous approximations to arbitrary nonlinear systems, 447, 451 of quasilinear nonautonomous systems at resonance, 9,128 of quasilinear nonautonomous systems not at resonance, 3, 123 almost periodic, of quasilinear systems, 260 of nonautonomous approximations to arbitary linear systems, 497, 504 with nonresonance frequencies, 337 free, of quasiharmonic systems, 364 Parametric oscillations, excitation of, 364 in an electric circuit, 365 Pendulum with oscillating point of suspension, 364 Phase plane ( plane of state), 52 Phase trajectory, 52 Reduction of linear equations with constant coefficients to canonical form, 203 with periodic coefficients to equations with constant coefficients, 109 Regenerative receiver, 20, 37, 85, 314, 347 Region of instability, 367, 370, 376 of stability, 367, 370, 376 Resonance, 9 ff. combination, 401 curve, 25 parametric, 364 parametric, of canonical systems with many degrees of freedom, 395 of the n-th kind, 34, 37 principal, 85, 552 simple, 401 L'Saddle", 477 Self-oscillations, 60 in two coupled circuits, 176 of a vacuum tube generator, 63 588

Skeleton curve, 580 Small divisor, 272 Small parameter, 2 Solution, isolated almost periodic, 497 generating, 5 isolated periodic, 450 Solutions (see Oscillations ) almost periodic, linear nonhomogeneous systems with constant coefficients, 263 with periodic coeffiCients, 422 periodic, linear nonhomogeneous systems with constant coeffiCients, 114 homogeneous systems with constant coefficients, 108 Stabili ty, 181 asymptotic, 183 of oscillations, of periodic solutions, of almost periodic solutions, 74, 78, 85, 218, 235, 246, 282, 311, 470, 482, 505, 568 System ( see Oscillations ) autonomous, 1 approximating arbitrary nonlinear, 447 ff. approximating Lyapunov system, 545 generating, 2 Lyapunov, 519 with nonanalytic characteristic of nonlinearity, 91 with one degree of freedom approximating conservative, 474 nonautonomous, 1 standard, 275, 278 quasiharmonic, 364 quasilinear, 3, 235, 260 Trigonometric sum, 261 Vacuum tube generator, 63