Explosion Points in Skew Maps

EXPLOSION POINTS IN SKEW MAPS Anne Stefanie Costolanski, MS George Mason University, 2007 Thesis Director: Dr. Evelyn Sander This paper describes various chaotic behavior present in skew maps. Some aspects common in the study of dynamical systems are explored: the existence of homoclinic orbits, the behavior of manifolds, chaotic attractors, and period doubling cascades; and for each, the similarities and differences between the behavior for one-dimensional maps, two-dimensional diffeomorphisms, and skew maps are given. While it is known that chaotic attractors and period doubling cascades are present in one-dimensional maps and two-dimensional diffeomorphisms as well as in skew maps, the existence of bifurcations that result in drastic changes in a chaotic attractor, namely crises and explosions, has not yet been shown for skew maps. Numerical examples are provided to demonstrate their existence, with comparisons to two-dimensional diffeomorphisms that exhibit similar behavior. Chapter 1: Introduction 1.1 Dynamical Systems At its most general, the definition of a dynamical system is a set of possible states together with a rule that determines the present state in terms of past states. Dynamical systems model a variety of behavior, from simple biological systems to more complex meteorological phenomenon, and as applications have spread into other scientific disciplines, the importance of studying and quantifying the behavior of dynamical systems has increased. A mathematical definition of a discrete dynamical system is the process X 7→ F (X) where F : R → R acts on a variable X ∈ S ⊂ R. X represents the present state of the system, and F represents the rule that determines future values. Although dynamical systems can exist in any n-dimensional space, a great deal of information has been discovered on the behavior of one-dimensional systems, as well as two-dimensional systems for which F is a diffeomorphism. Thus, we will limit our analysis to these dimensions, and will detail the similarities and differences in the behavior of maps in these dimensions. To determine how a particular state of a dynamical system evolves over time, it is necessary to know the asymptotic behavior of the system as a whole. It is also useful to know how changes in the system, or changes in the initial state, will affect future states. Our analysis of future states of a system begins by finding those whose 1