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相互横向相交，从而产生时间周期映射的同宿纠缠。为了检测具体微分方程组中时间周期映射的这些同宿纠缠，引入了Melnikov方法。这项研究项目遵循了一条新的路线。我们构造了一个庞加莱截面，它在扩展的相空间中混合了原始的相维度和时间，并显式地计算了由微分方程组诱导的返回映射。因此，我们可以将近三十年来发展起来的关于非一致双曲映射的各种理论，如纽豪斯同宿切理论、SRB测度理论、Henon-like吸引子理论和秩一映射理论，应用于周期扰动同宿解的研究。我们还将我们的研究扩展到准周期强迫方程。为了从特定的科学学科，如天文学、物理学、工程学和生物学来研究具有实际意义的问题，我们通常从某些既定的自然法则开始，并将它们写成数学术语。例如，要研究天体在太阳系中的运动，我们从牛顿引力定律入手，写出一组数学方程。然后，数学家的任务就是解这些方程，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、生物系统等写出数学方程式。在研究这些数学方程方面的一个重大发现是，一种被称为“混沌”的动力学现象普遍存在，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来研究混沌，这些研究导致了许多复杂但可以理解的结构的发现。这项研究介绍了一种研究混沌的新方法，所发展的理论可用于分析许多具有经典和实际意义的系统。

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.