Some tools for analyzing chaos

Although computer simulation and physical experiment are probably the most widely used techniques for understanding complex dynamical behavior in nonlinear systems, these are seldom adequate for full understanding unless they are used in conjunction with analytical techniques. In this paper we describe two such techniques, the Melnikov and Shiinikov bifurcation theorems, both of which deal with homoclinic orbits; that is, with trajectories which, in a suitable sense, contain both the unstable and stable dynamics of generalized saddle points. To understand why homoclinic orbits are important, we show how they are related to Smale's famous horseshoe map and thence to shifts operating on sequences of symbols. This makes it easy to understand why the Melnikov and Shilnikov results describe complex, unpredictable motion of the kind usually called chaotic. Our exposition of the theorems is aimed at someone who wants to understand why they are true, and to be able to use them, but is not concerned with the full technicalities.