New dynamical insights on the global behavior of chaotic attractors
Authors:   , Jones Timothy Douglas , Gilmore Robert 
Published: July 2021

A paraphrase of Tolstoy that has become popular in the field of nonlinear dynamics is that while all linear systems are linear in the same way, all nonlinear systems are nonlinear in their own ways. Despite this being quite true, there can be found a number of universal features in nonlinear systems which unify them in ways that enhance our understanding of their behavior. That nature is replete with nonlinear systems has proven to be a great challenge to our scientific understanding of the world. And while mathematics has proven to be apt at describing a multitude of physical phenomenon in the form of deterministic equations which describe future behavior based on a system's current state, it in and of itself held a rather shocking surprise which is now called Chaos. In Chaos we find deterministic systems which, due to our lack of omniscience, and the physical impossibility of building computers with infinite precision, become wildly unpredictable as they evolve in time. A number of new tools were developed to understand these systems, including a powerful program of topological analysis which has been completed for three dimensions. Yet, there still remains a number of unanswered dynamical questions about chaotic systems. Two such questions are the primary focus of this thesis. The first question we will address is regarding the general shape of the strange attractor. Specifically, what can we learn about the shape of strange attractor from the dynamical equations without numerically integrating them? For example, the Rössler and Lorenz attractors have remarkably similar dynamical equations, and yet are topologically very distinct. There is no self-evident relation between the dynamical equations that describe a strange attractor and its shape in phase space. Previously, we only had the fixed points to act as general guides as to the shape of the attractor, but these point sets are not exceedingly descriptive. We will outline work done to find more interesting sets of points from the dynamical equations themselves, sets of points which provide a sort of skeletal-structure for the strange attractors. We have examined these structures for a large number of strange attractors of varying topological nature. The second question is one which has been treated by only a small number of researchers, and not as descriptively as we do here. When viewed in co-dimension-2 space (that is, two parameters of the attractor varied, all others kept constant), one finds some remarkable regular patterns in the mapping of the intensity of Lyapunov exponents. While progress has been made in addressing the origin of these shapes, there has not yet been a full explanation to the simple question: where do these patterns come from, and what do they tell us about the dynamical system? We will examine these patterns in detail and provide a broad explanatory mechanism for them, with a particular focus on the Rössler attractor. We will also show how these findings can be used to predict the occurrence of super-stable periodic orbits as the parameters of the attractor are varied, predictions that were previously unobtainable except by purely numerical means.