Spatial and Temporal Scaling Properties of Strange Attractors and Their Representations by Unstable Periodic Orbits

Abstract

Scaling properties of chaos are studied from a dynamic viewpoint by taking invertible two-dimensional maps. A fundamental role is played by an evolution equation for the probability of a small box along a chaotic orbit. A new relation between the partial dimensions of strange attractors in the expanding and contracting direction is derived in terms of the local expansion rates of nearby orbits. When the Jacobians of the maps are constant, this relation can be written in terms of a potential Φ(q) for the fluctuations of the local expansion rate in the expanding direction. For hyperbolic attractors, this potential Φ(q) is related to the generalized entropies K(q) by Φ(q) = (q - 1) K(q), and the above relation reduces to a simple relation between the generalized dimensions and the generalized entropies. Furthermore, the generalized dimensions, the potential Φ(q) and the probability densities are expressed in terms of the local expansion rates of unstable periodic orbits within the attractors, leading to a new method of studying chaos.