Bifurcations of invariant sets in a model of wild chaos

Abstract

In this thesis we study a two-dimensional noninvertible map that has been introduced by Bamon, Kiwi, and Rivera-Letelier [arXiv 0508045, 2006] to prove the existence of wild chaos in a five-dimensional Lorenz-like vector field. Wild chaos is a type of chaotic dynamics that can arise in a continuous-time dynamical system of dimension at least four. We are interested in the possible sequence of bifurcations that generate this type of chaos in dynamical systems. The map acts on the plane by opening up the critical point to a disk and wrapping the plane twice around it; points inside the disk have no preimage. The bounding critical circle and its images, together with the critical point and its preimages, form the so-called critical set. This set interacts with the stable and unstable sets of a saddle fixed point and other saddle invariant sets. Advanced numerical techniques enable us to study how these invariant sets change as the parameters are varied towards the wild chaotic regime. We find four types of bifurcations: the stable and unstable sets interact with each other in homoclinic tangencies (which also occur in invertible maps), and they interact with the critical set in three types of tangency bifurcations specific to this type of noninvertible map; all tangency bifurcations cause changes to the topology of these global invariant sets. Overall, a consistent sequence of all four bifurcations emerges, which we present as a first attempt towards explaining the geometric nature of wild chaos. Using two-parameter bifurcation diagrams, we show that essentially the same sequences of bifurcations occur along different paths towards the wild chaotic regime, and we use this information to obtain an indication of the size of the parameter region where wild chaos is conjectured to exists. In a different parameter regime, the map acts as a perturbation of the complex quadratic family and admits (a generalised notion of) the so-called Julia set as an additional invariant set. When parameters are varied, this set interacts with the other invariant sets, leading to the (dis)appearance of saddle points and chaotic attractors and to dramatic changes in the topology of the Julia set. We reveal a self-similar bifurcation structure near the period-doubling route to chaos in the complex quadratic family.