Bouncing ball dynamics

Abstract: One important aspect of dynamical systems is the study of the long-term behavior of a set of ordinary differential equations (ODEs) In recent years many systems that are simple to write down have been discovered whose solutions are chaotic They oscillate irregularly, never settling down to a regular pattern Two trajectories which start close together will separate quickly Systems whose time evolution is governed by a parameter p can undergo intriguing variations in the behavior of trajectories In many cases, there are values p* such that the long-term behavior of typical trajectories of p p* For example, the system may go from stable periodic behavior for p p* Such sudden, discontinuous changes or "bifurcations" are quite common Research in chaos and bifurcations in dynamical processes has advanced at a rapid pace during the past decade, acquiring an extraordinary breadth of applications in fields as diverse as fluid mechanics, electrical engineering and neurophysiology The new results interest a wide spectrum of the scientific community, many of whose members, however, lack the mathematical background necessary to decipher the literature Accordingly, Guckenheimer and Holmes have written their book as a "user's guide" to an extensive and rapidly growing field The book surveys the theory and techniques needed to understand chaotic behavior of ODEs The first chapter contains a brief introduction of the theory of ODEs; it is a review of topics usually found in a standard text like Hirsch and Smale (1) The second chapter considers four examples of chaotic systems: the forced van der Pol oscillator, Duffing's equation, the celebrated Lorenz equations, and Holmes' "bouncing ball map" (perhaps more familiar as the map which describes the motion of a periodically forced, damped planar pendulum in the absence of gravity) These examples