An Introduction to Chaos: Four Examples

Abstract

In this chapter we introduce four nonlinear systems which possess fascinating properties and which are still improperly understood. We have chosen two periodically forced single degree of freedom oscillators, a three-dimensional autonomous differential equation, and a two-dimensional map. The oscillators of van der Pol [1927] and Duffing [1918] originally arose as models in electric circuit theory and solid mechanics, respectively, while the Lorenz equations (Lorenz [1963]) represent a truncation of the nonlinear partial differential equations governing convection in fluids. Finally, our map models a simple repeated impact problem (Holmes [1982]) and, in a slightly different form, resonance problems in atomic physics (Chirikov [1979], Greene [1980]). In fact the conservative, area preserving version of this map has been studied intensively as a canonical example of the transition to stochasticity and chaos in Hamiltonian systems. The range of applications for the models outlined here should suggest the pervasive importance of nonlinear systems.