The structure and evolution of chaos in dynamical systems

An in depth study of temporal chaotic systems, both\ud discrete and continuous, is presented. The techniques for\ud the characterization of chaotic attractors are: Lyapunov\ud stability, dimension spectra and unstable periodic orbits.\ud Comparison of numerical and analytical methods clarify some\ud of the limitations of these techniques. A two dimensional\ud hyperbolic baker map with a complete set of unstable orbits\ud is examined. The evolution of structure and changes in the\ud f(a) spectrum for this map are related to changes in an\ud underlining Cantor set. Numerical calculation of unstable\ud periodic orbits for a related baker map with an incomplete\ud set of unstable orbits allow the investigation of the\ud properties of a pruned Cantor set. The effects of the\ud pruning on the associated f(a) spectrum are investigated.\ud It is also shown that the unstable manifold of a hyperbolic\ud toral map does not wind densely around the torus, but\ud consists of an infinite number of line segments. This\ud facilitates the efficient computation of the dimension\ud spectrum through a rotation of this manifold. A new\ud structure not previously observed m discrete systems is\ud characterized. Intermittency theory previously applied to\ud dissipative systems is applied to a variety of two\ud dimensional non-dissipative systems. A new type of\ud intermittency is found from a detailed comparison between\ud existing theory and numerical experiments. The important\ud and unresolved problem of the correspondence between\ud continuous and discrete systems is investigated using\ud analytical and numerical techniques. Properties of the\ud chaotic attractors of infinite dimensional delayed\ud differential equations are examined as a function of the\ud time delay and nonlinearity parameters