The characterisation of chaos in low dimensional spaces

• Main Author: McCreadie, Geoffrey Alexander
• Published: University of Warwick 1983
• Subjects: 530.1; QC Physics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.349677

This work attempts to characterise some of the complicated behaviour that is observed in many non-linear systems. For example, the frontispiece was generated by iterations of a two dimensional area-preserving mapping (the Chirikov map) that is typical of the systems studied herein. It will be shown that area-preserving deterministic mappings can be accurately characterised by a diffusion constant i.e. a quantity associated with random systems. In addition it shows how perturbation theory has a greater range of validtty than might be expected. The first three chapters introduce a number of physical systems that exhibit this chaotic behaviour and describe useful analytical techniques. Chapter 3 derives a general expression for a diffusion constant for 2D maps of the torus and shows the very good agreement between theory and numerical simulation for two example maps. Chapter 4 shows analytically how this type of deterministic system can be equivalent to a random system without the addition of external noise. Chapters 5 and 6 extend the theory to parameter values where chaos and order coexist and where the dynamics modulate an imposed noise. Chapter 7 calculates the Lyapunov exponent for the one dimensional logistic map. Chapter 8 examines the accuracy of computer models and perturbation schemes via the shadowing property of hyperbolic systems.