Oscillations in Nonlinear Systems

Jack K. Hale

• Verlag: Dover Publications
• Jahr: 2015
• ISBN:
  9780486803265
• Thema: Mathematics
• Thema: Differential Equations

By focusing on ordinary differential equations that contain a small parameter, this concise graduate-level introduction to the theory of nonlinear oscillations provides a unified approach to obtaining periodic solutions to nonautonomous and autonomous differential equations. It also indicates key relationships with other related procedures and probes the consequences of the methods of averaging and integral manifolds.
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.

Part I

Introduction and Background Material

1: Introduction

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), . . . , x_n(t)) of the physical system at time t is a point along the solution of the differential system

which passes through the point

, at time t = t₀.

In general, the functions f_i are nonlinear functions of the state variables x₁, x₂, . . . , x_n. For the sake of simplicity in analyzing (1-1), the functions f_i 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 = · · · = x_n = 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

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

whose trajectories in the

plane are shown in Fig. 1-2. The closed curve C has the property that all other trajectories approach it as t → ∞ except, of course, the trajectory which passes through the equilibrium point

. 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.

Fig. 1-1

Fig. 1-2

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 i...