Dynamical properties of chaotic systems

Abstract

In Chapter 3 we have concentrated on geometric aspects of chaos. In particular, we have discussed the fractal dimension characterization of strange attractors and their natural invariant measures, as well as issues concerning phase space dimensionality and embedding. In this chapter we concentrate on the time evolution dynamics of chaotic orbits. We begin with a discussion of the horseshoe map and symbolic dynamics. The horseshoe map and symbolic dynamics The horseshoe map was introduced by Smale (1967) as a motivating example in his development of symbolic dynamics as a basis for understanding a large class of dynamical systems. The horseshoe map M h is specified geometrically in Figure 4.1. The map takes the square S (Figure 4.1( a )), uniformly stretches it vertically by a factor greater than 2 and uniformly compresses it horizontally by a factor less then ½ (Figure 4.1( b )). Then the long thin strip is bent into a horseshoe shape with all the bending deformations taking place in the cross-hatched regions of Figures 4.1( b ) and ( c ). Then the horseshoe is placed on top of the original square, as shown in Figure 4.1( d ). Note that a certain fraction, which we denote 1 − f , of the original area of the square S is mapped to the region outside the square. If initial conditions are spread over the square with a distribution which is uniform in the vertical direction, then the fraction of initial conditions that generate orbits that do not leave S during n applications of the map is just f n .