Exponentially Decaying Discrete Dynamical Systems

Yogesh      Joshi; Denis      Blackmore

Abstract

There are numerous phenomena in the space, biological and ecological sciences whose discrete evolution can be effectively modeled by exponentially decaying discrete dynamical systems, which can be represented as follows: Let F : Rn x P → Rn be a smooth map, where Rn is Euclidean n-space representing the state variables x and P is a subset of Rm representing the parameter space comprised of m-vectors denoted by µ . The discrete dynamics of the system is given by the iterates of the map fµ : Rn → Rn , where fµ (x) := F(x,µ), with the exponentially decaying behavior embodied in the property that there exists a positive constant M such that |f µ(x)| ≤ Me -|x| for all (x,µ ) ∈ Rn x P. It is not difficult to show that these exponentially decaying dynamical systems have attracting sets that can have remarkable properties. For example, the attractor may be a strange attractor – a set with non-integer fractal dimension on which the dynamics is chaotic - and variations in the parameter can cause bifurcations that change the fundamental nature of the chaotic regimes. Known results on these strange attractors will be discussed, several new developments will be presented and some challenging open problems will be outlined. AMS Subject Classification: 37C05; 37G10; 37N25; 92D25.

Keywords: Bifurcation, chaotic dynamics, discrete dynamical system simulation, fractal dimension, Lyapunov exponents, strange attractors, orbital trajectories, Dynamical Systems, exponentially decaying