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

周期强迫二阶方程在历史上得到了广泛的研究。当一个具有鞍点和同宿解的自治系统受到周期扰动时，扰动鞍点的稳定流形和不稳定流形彼此横向相交，从而为时间周期映射产生同宿缠结。Melnikov的方法已被引入检测这些同宿缠结的时间周期映射在具体的微分方程系统的目的。这个研究项目走了一条新的路线。 我们构造了一个庞加莱截面，在扩展的相空间中将原始相维与时间混合，并显式计算了微分方程引起的返回映射。 因此，我们可以将近三十年来发展起来的各种非一致双曲映射理论，如纽豪斯同宿相切理论、SRB测度理论、Henon吸引子理论和秩1映射理论，应用于周期扰动同宿解的研究。为了从数学上研究一个给定的科学学科（如天文学、物理学、工程学和生物学）中具有实际意义的问题，我们通常从某些已建立的自然定律开始，并用数学术语将它们写下来。例如，要研究太阳系中天体的运动，我们从牛顿的万有引力定律开始，写出一组数学方程。数学家的任务是解出这些方程，预测天体的未来位置，讨论太阳系的长期稳定性等问题。我们还为电路、生物系统等编写数学方程，在研究这些数学方程的过程中，我们发现了一种被称为“混沌”的动力学现象，这在一定程度上反映了我们无法预测复杂系统的未来。许多数学理论被引入研究混沌，这些研究导致了许多复杂但可以理解的结构的发现。本研究为混沌研究提供了一种新的方法，所发展的理论可用于分析许多具有经典意义和实际意义的系统。

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.