Part I of the text features introductory material, including discussions of matrices, linear systems of differential equations, and stability of solutions of nonlinear systems. Part II offers extensive treatment of periodic solutions, including the general theory for periodic solutions based on the work of Cesari-Halel-Gambill, with specific examples and applications of the theory. Part III covers various aspects of almost periodic solutions, including methods of averaging and the existence of integral manifolds. An indispensable resource for engineers and mathematicians with knowledge of elementary differential equations and matrices, this text is illuminated by numerous clear examples.
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About the Author
A longtime professor at Brown University, Jack K. Hale is Regents Professor Emeritus at the Georgia Institute of Technology and the author of Dover's Ordinary Differential Equations.
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Oscillations in Nonlinear Systems
By Jack K. Hale
Dover Publications, Inc.Copyright © 1991 Jack K. Hale
All rights reserved.
Most physical systems are nonlinear. We shall assume the evolution of the physical system is governed by a real ordinary differential equation; that is, the state x(t) = (x1(t), x2(t), ..., xn(t)) of the physical system at time t is a point along the solution of the differential system
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-1)
which passes through the point x0i; i = 1, 2, ..., n, at time t = t0.
In general, the functions fi are nonlinear functions of the state variables x1, x2, ..., xn. For the sake of simplicity in analyzing (1-1), the functions fi are frequently replaced by linear functions. In many cases this is sufficient, but there are phenomena which cannot be explained by analysis of the linear approximation.
The purpose of the present book is to concentrate on some aspects of differential equations which depend very strongly upon the fact that (1-1) is nonlinear.
The basic quality of a linear system (1-1) is (1) the sum of any two solutions of (1-1) is also a solution (the principle of superposition) and (2) any constant multiple of a solution of (1-1) is also a solution. Consequently, knowing the behavior of the solutions of (1-1) in a small neighborhood of the origin, x1 = x2 = ... = xn = 0, implies one knows the behavior of the solutions everywhere in the state space; that is, globally. Furthermore, if one has a periodic solution of a linear system (1-1), then it cannot be isolated since any constant multiple of a solution is also a solution.
In nonlinear systems none of the above properties need be true. In fact, there is no principle of superposition, the behavior of solutions is generally only a local property, and there may be isolated periodic solutions (except for a phase shift). A simple example illustrating the local property of the behavior of solutions is
[??] = -x(1 - x)
whose solutions are shown in Fig. 1-1.
The most classical example of a system which has an isolated periodic solution (except for a shift in phase) is the van der Pol equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-2)
whose trajectories in the (x, [??]) plane are shown in Fig. 1-2. The closed curve C has the property that all other trajectories approach it as t [right arrow] ∞ except, of course, the trajectory which passes through the equilibrium point x = [??] = 0. This is a phenomenon which is due to the nonlinear structure of the system and could never be explained by a linear analysis. Such an oscillation is called self-excited.
Another interesting phenomenon that may occur in nonlinear systems is the following: Suppose system (1-1) is linear and apply a periodic forcing function of period T to (1-1). If the unforced system has no periodic solution, then there can never be an isolated periodic solution of any period except T. In nonlinear systems, this is not the case and isolated periodic solutions of period mT, where m is an integer greater than 1, may even occur. This phenomenon is known as subharmonic resonance.
Consider the forced van der Pol equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-3)
Is it possible to obtain solutions of this equation which oscillate with two basic frequencies, one due to the forcing function cos ωt and one due to the basic frequency of the self-excited oscillation discussed before? If such a phenomenon occurs, we say the solution has combination tones. As we shall see, such solutions may exist.
Our knowledge of nonlinear systems is still far from being complete. For the case where the system of differential equations has order 2 (that is, one degree of freedom), much more is known than for higher-order systems. The reason for this is that analytical-topological methods may be applied very nicely for systems of order 2, whereas for higher dimensions, the techniques of topology are not sufficiently developed. For systems of order greater than 2, the differential equations are usually assumed to contain a given parameter, and some type of perturbation technique is used to discuss the behavior of solutions. By using these perturbation techniques, one can build up a catalogue of phenomena which may occur in higher-order systems in the hope that one can use this experience as a guide to the eventual development of topological techniques which are applicable for higher-order systems. If we let the given parameter in our differential system be &8364;, then our knowledge of oscillatory phenomena for systems of differential equations lies almost entirely in the shaded regions of Fig. 1-3. As we shall see below, some techniques are available for discussing periodic solutions in the unshaded region of Fig. 1-3.
The techniques explained in this book will be methods of successive approximations and most of the emphasis will be on equations which contain a small parameter. The main reason for doing this is that the analytical-topological techniques would require more space than a book of this size, and also these techniques are covered very well in other places (see, for example, Cesari  and Lefschetz ).
Chapters 2 to 4 are summaries of required material on matrices, linear systems of differential equations, and the basic stability theorems of Liapunov.
Part II is concerned mainly with periodic solutions of differential equations, although Chap. 11 does contain some material on generalized characteristic exponents. We attempt to give a unified presentation of the problems of finding periodic solutions of nonautonomous differential systems, autonomous differential systems, and characteristic exponents of linear periodic differential systems.
Many methods have been devised for solving these problems, and it is perhaps worthwhile to mention the general ideas involved. The reader who is not already familiar with at least one of these methods should most probably read Chap. 6 before attempting to read the discussion in the next few paragraphs.
Let us consider a nonautonomous differential system
[??] = Ax + [member of]f(t,x) (1-4)
where x is an n vector, A is a constant matrix, f is periodic in t of period T, and [member of] is a small parameter. We wish to obtain a periodic solution of (1-4) of period T which, for [member of] = 0, is a solution of the linear system
[??] = Ax (1-5)
Let φj, j = 1, 2, ..., k, be a maximal linearly independent set of periodic solutions of (1-5) of period T and for any constant vector a = (a1, ..., ak), let
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-6)
The basic problem is to determine some vector a = (a1 ..., ak) so that there is a corresponding periodic solution x(t,a,[member of]) of (1-4) of period T with x(t,a,0) = x0(t,a). This is accomplished by determining functions x(t,a,[member of]), Pj(a,[member of]), j = 1, 2, ..., k, which are defined for all t, - ∞ < t < ∞, a in some set U, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which have the following properties: for any a in U, x(t,a,0) = x0(t,a), x(t,a,[member of]) is periodic in t of period T and if there exists an a = a([member of]) so that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-7)
then the periodic function x(t,a([member of]),[member of]) becomes a solution of (1-4). Thus the problem of existence of a periodic solution of (1-4) is reduced to the problem of finding a solution of a system of transcendental equations. Equations (1-7) are usually referred to as the determining equations or bifurcation equations. Most of the methods that have been devised for obtaining periodic solutions of (1-4) differ only in the manner in which these equations are derived. A few of these ideas are discussed below.
If one wishes to solve (1-4) by successive approximations, taking x(0) = x0 (t,a), and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-8)
with each iterate x(r)(t) periodic in t of period T, then each step of the approximation consists in finding a periodic solution of the nonhomogeneous linear system
[??] = Ax + g(t) (1-9)
where g(t) is periodic of period T.
Unfortunately, system (1-9) has a periodic solution of period T if and only if
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-10)
for every periodic function of period T which is a solution of the system
ψ = -ψA (the adjoint equation) (1-11)
Consequently, the method of successive approximations (1-8) will not give the desired result. However, one might suspect that it is possible to modify the algorithm (1-8) by subtracting from f(t,x(r)(t)) some factor which results in equation (1-10) being satisfied. The successive approximations then take the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-12)
where Mr(t,a,x(r),[member of]) is to be determined so that (1-10) is satisfied with g replaced by f(t,x(r)) - Mr. If the process (1-12) converges for [member of] small to a function x(t,a,[member of]), then x(t,a,[member of]) satisfies
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-13)
If, in addition, there exists a constant a such that the determining equations M(t,a,x(t,a,[member of]),[member of]) = 0, then x(t,a,[member of]) is a solution of (1-4).
One method for choosing the Mr in (1-12) began with the paper of Cesari  concerning linear systems with periodic coefficients. This method was subsequently developed by Cesari, Hale, Gambill, Fuller, and Thompson and involves carrying out the above process with
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-14)
where the Dr(a,[member of]) do not depend on t. By proving that x(r), Dr(a,[member of]) converge to x(t,a,[member of]), D(a,[member of]), respectively, for e small, the determining equations become
D(a,[member of]) = 0
References to the papers of the above authors dealing with this process may be found in the book of Cesari . Even though many results were obtained using this method, it has certain theoretical shortcomings and cannot be generalized directly to arbitrary nonlinear systems. A more elegant method has been devised by Cesari  which does generalize to arbitrary nonlinear systems. We shall return to this below.
If (φ1 ..., φk is a basis of periodic solutions of period T of (1-5), then one can define a method of approximations (1-12) with
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-15)
The Crj(a,[member of]) depend only on a, [member of] and can be chosen in such a way that (1-10) is satisfied with g replaced by f(t,x(r)) - Mr. The convergence of the Crj(a,[member of]) to Cy(a,[member of]) for [member of] small implies that the determining equations are
Cy(a,[member of]) = 0 j = 1, 2, ..., k (1-16)
This method has been discovered independently by Friedrichs , Lewis [1,2], Malkin , and Sibuya .
The procedure just outlined can be interpreted in another way. For simplicity only, assume that all solutions of (1-5) are periodic; that is, there are n linearly independent periodic solutions φ1, ..., φn of (1-5) of period T. Then eAt is periodic of period T and the transformation
x = eAty
applied to (1-4) yields
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-17)
For [member of] = 0, (1-17) is self-adjoint. If we wish to solve this by successive approximations taking y(0) = a, a given constant n vector, and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-18)
with each iterate y(r) periodic in t of period T, then we must have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-19)
for each r. For the particular case under consideration, it is easy to see that the vector Cr = (Cr1, ..., Crn) defined in the previous paragraph is equal to the right-hand side of (1-19) except for a multiplicative constant. Since (1-19) will not necessarily be satisfied for every r, the algorithm (1-18) is modified to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-20)
The fact that y(r) converges to y(t,a,[member of]) for e small implies that the determining equations are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-21)
This method is a very special case of a general procedure developed by Cesari  and is the one that will be employed in this monograph. As mentioned earlier, Cesari's method also may be applied to arbitrary nonlinear systems, and a discussion of this point is contained in Chap. 11. In Chap. 11, we also discuss generalizations of these basic ideas to a perturbation problem considered by Bogoliubov and Sadovnikov , and the generalized characteristic exponents considered by Golomb .
If y(t) is a periodic solution of (1-17) then y(t) has a Fourier series
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
One could therefore determine a periodic solution of (1-17) by requiring that the coefficients ak be of such a nature that all the coefficients of the periodic function
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-22)
be equal to zero. This yields an infinite set of equations for the coefficients ak. To obtain the determining equations, one shows that, for any given a0, one can determine the ak as functions of a0, [member of] for [member of] small in such a way that all the Fourier coefficients of h(t) in (1-22) vanish except the one corresponding to the mean value of h(t), which is given by b0. The vector b0 is then a function of only a0, [member of] and the determining equations are b0 = 0. This procedure has been applied by Bass [1,2], Golomb [1,2] and Wasow . Bass [1,2] has also considered a generalization of this procedure which is applicable in certain cases to equations not containing a small parameter.
Part III deals with the role of integral manifolds in nonlinear oscillations. In particular, in this part, the combination tones mentioned above are treated and the effect of high-frequency forcing functions on oscillatory behavior is discussed. In systems of autonomous differential equations of order greater than 2, examples are given to show that there may be interesting oscillatory behavior which is much more complicated than periodic phenomena. The treatment in Part III is based very strongly upon the work of Krylov, Bogoliubov, and Mitropolski (see the Bibliography for references). The method of averaging is explained and examples are given as illustrations of the method. After reading Part III, it will be apparent that the theory of integral manifolds and its application is not nearly completed and there remain many interesting unsolved problems.
In Part III, we do not discuss the important research on integral manifolds which has been done by Diliberto  and his collaborators mainly because it would require a complete redevelopment of the theory. There is some overlap between the results, and the interested reader is referred to the original papers, most of which are contained in volumes II and IV of the "Contributions to the Theory of Nonlinear Oscillations," Princeton University Press, Annals of Mathematics Studies, and the papers of Diliberto mentioned above.
Throughout this monograph, many examples are given but the physical motivation is not included. The examples are given only to illustrate some of the different types of phenomena that can occur in nonlinear differential equations. If one understands well the methods that are being employed, the author feels that many other types of phenomena can be discussed by using the same procedures.
This book does not claim to cover all the aspects of nonlinear oscillations. In fact, because of lack of space, we have omitted almost all reference to the important subject of relaxation oscillations (singular perturbations). The subject of asymptotic expansions and differential-difference equations is not even mentioned, nor are those problems in oscillations which deal with whether or not a solution has zeros.
Excerpted from Oscillations in Nonlinear Systems by Jack K. Hale. Copyright © 1991 Jack K. Hale. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Part I. INTRODUCTION AND BACKGROUND MATERIAL,
1. Introduction, 3,
2. Matrices, 11,
3. Linear Systems of Differential Equations, 16,
4. Stability of Solutions of Nonlinear Systems, 22,
Part II. PERIODIC SOLUTIONS,
5. Noncritical Cases, 27,
6. Periodic Solutions of Equations in Standard Form—Critical Cases, 34,
7. Practical Methods of Computing a Periodic Solution and Examples, 46,
8. Characteristic Exponents of Linear Periodic Systems, 68,
9. Periodic Solutions of Nonautonomous Systems, 82,
10. Periodic Solutions of Autonomous Systems, 89,
11. Generalizations, 95,
Part III. ALMOST PERIODIC SOLUTIONS AND INTEGRAL MANIFOLDS,
12. Almost Periodic Functions and Multiply Periodic Functions, 113,
13. Almost Periodic Solutions—Noncritical Case, 121,
14. Periodic Solutions Revisited, 130,
15. Integral Manifolds—Averaging, 134,
16. Integral Manifolds—Noncritical Case, 141,
17. Almost Periodic Solutions—Critical Case, 154,
18. Integral Manifolds—Critical Case, 160,
Appendix: Principle of Contraction Mappings, 171,