Robust Chaos in Two-Dimensional Discontinuous Maps with One and Two Switching Manifolds

This dissertation concerns a new approach to determining the structure of robust chaos in two-dimensional (2D) piecewise smooth discontinuous maps and an extension to the case of two switching manifolds. The study of chaotic dynamics has found many applications in various disciplines in science and engineering. Extensive research has been carried out to identify and analyze the dynamical behaviour of chaos. This dissertation centres on those chaotic attractors which are robust to small parameter changes and do not contain periodic windows or coexisting attractors. This behaviour is termed robust chaos. In recent years, the problem of understanding and predicting robust chaos in the real world has attracted many researchers. In this dissertation, we focus on hyperbolic dissipative piecewise smooth discontinuous maps in the plane. First, we consider a 2D piecewise linear discontinuous normal form map with one switching manifold, restricting to real eigenvalues, and determine regions of parameter space where a bifurcation leading to robust chaos might occur. We study the intricate structure of the chaotic attractors and present detailed analytical heuristics for determining parameter values for boundary crises leading the onset or termination of robust chaos. We also show the existence of unusual basins of attraction for some chaotic attractors and provide a strategy to determine the basin boundaries.Next, we propose an extension to a new map with an added second switching manifold and three linear branches. This map has application to a simple financial market model. We discuss our normal form map, under the assumption that the outer branches are the same, and present a systematic analysis of bifurcations leading to robust chaos. We develop analytical procedures to identify the underlying generating mechanism of robust chaos in the vicinity of each of the switching manifolds and determine associated parameter space regimes. We also illustrate conditions under which robust chaos occurs near both the switching manifolds simultaneously.