5 - Chaos in dissipative systems

Summary

We begin the detailed study of chaotic behaviour with dissipative systems. We consider permanently chaotic dynamics (cf. Section 1.2.1), and we start our investigations within the framework of a simple ‘model’ map, the baker map. The most important quantities characteristic of chaos will be introduced via this example. The simplicity of the map makes the exact treatment of numerous chaos properties possible, an exceptional feature in the world of chaotic processes. Next we turn to the investigation of a physical system, the kicked oscillator, with different kicking amplitudes. These functions will be chosen in such a way that, in the first case, the attractor is similar to that of the baker map. In the second, the attractor has a different structure and exhibits a general property of chaotic attractors: it appears to be a single continuous curve. The special form of the amplitude function continues to make its exact construction possible. This is no longer so, however, with the third choice, representing a typical chaotic system. The parameter dependence of chaotic systems will also be discussed within the class of kicked oscillators. Based on all these examples, we summarise the most important properties of chaos, first of all at the level of maps. As measures of irregularity, unpredictability and complex phase space structures, we introduce the concepts of topological entropy, Lyapunov exponents and the fractal dimension of chaotic attractors, respectively. Special emphasis will be given to the presentation and characterisation of the natural distribution of chaotic attractors.