Ralf Eichhorn
Simple Polynomial Chaotic Models
Supervisor: Privatdozent Dr. Stefan Linz [Theoretical physics I]
Date of oral examination: 07/14/2000
122 pages, english , Shaker Verlag, Aachen 2002, ISBN 3-8322-0317-6
The necessary requirements for the occurrence of chaotic behavior in autonomous, time-continuous dynamical systems are well-known: The phase space dimension has to be at least three and the vector field must be nonlinear. However, the question which functional structure of a dynamical system is sufficient to generate chaotic behavior still determines a largely unexplored problem. This thesis is devoted to the study of simple chaotic models that have polynomial nonlinearities only. Here, the term simplicity refers to the functional form of the model equations. We show that explicit third-order differential equations (so-called jerky dynamics) constitute a particularly simple subclass of three-dimensional dynamical systems. Developing a systematic transformation method, we present several classes of three-dimensional dynamical systems that can be transformed to equivalent jerky dynamics. Moreover, exploiting the special algebraic form of jerky dynamics, a simple criterion is derived to exclude chaotic behavior for certain parameter regions or even entire classes of jerky dynamics. These results are applied to the chaotic models of Sprott and the toroidal R\"ossler model. It turns out that these models can be classified according to the functional structure of their jerky dynamics. One obtains seven basic classes of chaotic jerky dynamics with increasing functional complexity. The dynamical properties of the two simplest jerk classes are studied in detail. In particular, we investigate the dependence of the long-time dynamical behavior on system parameters. To this end, Lyapunov exponents are calculated numerically. As a result, we obtain ``charts'' of the parameter spaces which display regions of qualitatively different dynamics. The concept of Lyapunov exponents is also used to study the transformation properties of chaos. For bounded long-time dynamics, the occurrence of a positive Lyapunov exponent is commonly regarded as a signature of chaos. We demonstrate that the signs of the Lyapunov exponents are invariant under a wide class of transformations.