The Chaos of Dynamical Systems

Abstract

This chapter discusses the general concepts of chaotic and nonchaotic behavior, stability, and genericity for dynamical systems. To understand a dynamical system means to know how its states vary through time — at least to describe their variation, at best to predict it. Mathematical dynamical systems theory had its inception with Newton. Ever since G. D. Birkhoff, dynamical systems theory in the mathematical sense has meant the study of the long run behavior of solutions to differential equations in which one variable is thought of as time. There is an interesting class of dynamical systems which are guaranteed to be nonchaotic, and which can usually be easily recognized. A major challenge to mathematicians is to determine which dynamical systems are chaotic and which are not. Chaotic dynamical systems are sometimes said to have “strange attractors” or even “strange strange attractors”. Such systems abound in models of hydrodynamics, mechanics, and many biological systems.