Spatial Extent of an Attractor

Abstract

Lyapunov exponents characterize the rate of approach or recession of nearby trajectories in a dynamical system defined by differential equations or maps. They are usually taken as indicators of chaotic behavior. The density of orbits in the state space or equivalently, the Poincare map is usually taken as another such indicator. Although these indicators usually give correct results, there are instances in which they can lead to confusing or misleading information. For instance, a system of three linear differential equations can have three positive eigenvalues $\lambda _i$ leading to a solution $\exp {\lambda _i t}$. The Wolf-Benettin algorithm [4] would report three positive Lyapunov exponents, in spite of the fact that the system is not chaotic. Another example is the Khomeriki model [1] or even the usual Bloch equations that would report a spectrum of all negative Lyapunov exponents but produce completely full state space plots, if the AC field is sufficiently strong. We will consider the class of systems proposed by Sprott [3] consisting of three-dimensional ODE’s with at most two quadratic nonlinearities as examples. Many of them obey two scenarios one of which is Lorenz model like behavior where an unstable linearized fixed point is surrounded by two stable fixed points so that the unstable fixed-point leads to a throw and catch behavior. The other is Rössler-like behavior whereas the system moves away from a weakly unstable linearized fixed point, nonlinear terms return it to equilibrium with a spiral out catch in mechanism. Since the presence of an attractor may involve structural stability, these two mechanisms are expected to produce different spatial extents for the attractor. Although Lyapunov exponents indicate time dependent behavior, spatial extent would complement this as a spatial measure of localization, thus complementing the Lyapunov exponents that characterize horizon of predictability. Direct numerical simulation and where feasible, the normal form approach will be used to investigate selected examples of the three degree of freedom systems.