Computational Tools for Dynamical Systems: An Overview

this paper, dynamical systems are mappings or systems of ordinary differential equations defined on a finite dimensional Euclidean space. For example, a mapping F (h; p) of the plane to itself can serve as a model of an interacting host and predator with discrete generations. The map F "updates" current host and predator populations from generation n to generation n+1. As has been appreciated for a long time [2], the sequence of predicted populations from such a model (the model trajectories) can be very complicated, sometimes displaying what is now called chaos. Dynamical systems theory has emphasized the study of qualitative properties of trajectories and how the underlying phase space of the system is filled by trajectories. Using the mathematics of transversality (a part of differential topology), the theory seeks to classify generic properties that are found in "typical" systems. Putting rigorous mathematical foundations under these fleeting images is a substantial undertaking [7, 14]. The reader is warned that this mathematical theory requires care in its use and that there are many pitfalls that easily lead to its misapplication. We shall only discuss deterministic systems, though we shall touch upon the issue of "noise" when we discuss data. To simplify the discussion, we shall discuss henceforth continuous time systems rather than to make statements that apply uniformly to both discrete time and continuous time systems. There is analogous theory (usually a bit simpler) for discrete time systems, but the terminology gets confused a bit when discussing both simultaneously. Phase Portraits The fundamental problem about a dynamical model is the determination of trajectories from their initial conditions. Generally, the only way to compute this information is by..