Return Maps in Extended Phase Space for Non-autonomously Perturbed Equations

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Periodically forced second order equations have been studied extensively in history. When an autonomous system with a saddle point and a homoclinic solution is subjected to periodic perturbations, the stable and unstable manifolds of the perturbed saddle intersect each other transversally, creating a homoclinic tangle for the time-periodic map. Melnikov's method has been introduced for the purpose of detecting these homoclinic tangles for the time-periodic map in concrete systems of differential equations. This research project follows a new route. We construct a Poincare section that mixes the original phase dimensions with time in the extended phase space and computed explicitly the return maps induced by the differential equations. Consequently, we can apply various theories on non-uniformly hyperbolic maps developed in the last thirty years, such as the Newhouse theory on homoclinic tangency, the theory of SRB measures, the theory of Henon-like attractors and rank one maps, to the studies of periodically perturbed homoclinic solutions. We also extend our study to quasi-periodically forced equations.To mathematically study a problem of practical importance from a given science discipline, such as astronomy, physics, engineering and biology, we usually start with certain established natural laws and write them in mathematical terms. For instance, to study the motions of celestial bodies in the solar systems, we start with Newton's Law of Gravitations, and write a set of mathematical equations. Mathematician's task is then to solve these equations to predict the future positions of the celestial bodies, to discuss issues such as the long term stability of the solar system. We also write mathematical equations for electric circuits, for biological systems, and so on. One of the major discoveries in the studies of these mathematical equations is the common occurrence of a dynamics phenomenon that is called "Chaos", partly reflecting our inability in predicting futures of complicated systems. Many mathematical theories have been introduced to study chaos, and these studies have led to the discoveries of many complicated but understandable structures. This research introduces a new way of studying chaos and the theory developed can be used in analyzing many systems of classical and practical importance.