Toward Analytical Chaos in Nonlinear Systems

Chapter 1Introduction

In this chapter, a brief literature survey of analytical solutions of periodic motions in nonlinear dynamical systems will be presented. The perturbation analysis has played an important role in such an approximate analysis of periodic motions in nonlinear systems. The perturbation method, method of averaging, harmonic balance, and generalized harmonic balance will be reviewed. The application of perturbation method in time-delayed systems will be discussed briefly.

1.1 Brief History

Since the seventeenth century, there has been interest in periodic motions in dynamical systems. The Fourier series theory shows that any periodic function can be expressed by a Fourier series expansion with different harmonics. In addition to simple oscillations, there has been interest in the motions of moon, earth, and sun in the three-body problem. The earliest approximation method is the method of averaging, and the idea of averaging originates from Lagrange (1788). At the end of the nineteenth century, Poincare (1890) provided the qualitative analysis of dynamical systems to determine periodic solutions and stability, and developed the perturbation theory for periodic solutions. In addition, Poincare (1899) discovered that the motion of a nonlinear coupled oscillator is sensitive to the initial condition, and qualitatively stated that the motion in the vicinity of unstable fixed points of nonlinear oscillation systems may be stochastic under regular applied forces. In ...