Chaos, strange attractors and bifurcations in dissipative dynamical systems
Degree GrantorUniversity of Canterbury
Degree NameMaster of Science
In this dissertation a study is made of chaotic behaviour, the bifurcation sequences leading to chaos and the manifestation of chaos in the form of a strange attractor in dissipative dynamical systems. In chapter 1 we provide an overview of the material covered in this review and introduce several concepts from the basic theory of dynamical systems, such as Poincaré return maps and simple bifurcations. After introducing the concept of chaos and strange attractors in dissipative dynamical systems, we divide higher dimensional systems into three categories in chapter 2. Each is illustrated with examples. Central to the discussion is the well studied Lorenz system. Other important mathematical models are looked at, in particular the Rössler model and the two-dimensional Hénon map. The various measures of dimension, in the fractal context, and the numerical methods currently in use for determining these quantities are presented in chapter 3. In view of their relative computational simplicity and direct relevance to chaos, one-dimensional mappings are looked at in chapter 4. In chapter 5, the idea of the transition to turbulence being a chaotic regime is introduced and the various routes to turbulence are examined in turn. In chapter 6, we present a Fourier series method for approximating the phase-space trajectories of a dynamical system. We illustrate the technique by carrying out the calculations required on the equations describing the evolution of the spherical pendulum model of Miles (l984b). No attempt is made to cover the whole field of research chaos. The use of symbolic dynamics is avoided wherever possible for simplicity and brevity in this review.