7 - NONLINEAR DYNAMICS

Summary

CHAPTER OVERVIEW

This chapter is devoted almost entirely to nonlinear dynamical systems, whose equations of motion involve nonlinear functions of positions and velocities. We concentrate on some general topics, like stability of solutions, behavior near fixed points, and the extent to which perturbative methods converge. Because nonlinear systems involve complicated calculations, the chapter stresses mathematical detail. It also presents results of some numerical calculations. The importance of numerical methods can not be overemphasized: because nonlinear systems are inherently more complicated than linear ones, it is often impossible to handle them by purely analytic methods. Some results depend also on topics from number theory, and these are discussed in an appendix at the end of the chapter.

We have already dealt with nonlinear systems perturbatively (e.g., the quartic oscillator of Section 6.3.1), but now we will go into more detail. It will be shown, among other things, that in the nonperturbative regime nonlinear systems often exhibit the kind of complicated behavior that is called chaos.

The first four sections of the chapter do not use the Lagrangian or Hamiltonian description to deal with dynamical systems. In them we discuss various kinds of systems, largely oscillators, which are often dissipative and/or driven. We go into detail concerning some matters that have been touched on earlier (e.g., stability, the Poincaré map), introducing ideas and terminology that are important in understanding the behavior of nonlinear systems of all kinds. In particular, considerable space is devoted to discrete maps.