翻转对称性是自然科学的一种基本对称性质，起源于物理学中的动力学模型，特别是经典力学和量子力学的模型. 该项目拟研究翻转映射的动力学行为，包括翻转映射的不动点、Aubry-Mather集、不变环面的存在性，以及应用到翻转微分方程得到其周期解、拟周期解、广义拟周期解的存在性，最后研究系数关于时间周期或拟周期的翻转微分方程的平凡解的Liapunov稳定性问题. 这些问题的解决，不仅可以丰富翻转系统的理论成果，而且也必将推动我们在应用领域新的发现.

Reversal symmetry is one of the fundamental symmetries discussed in natural science. Consequently, it arises in many physically motivated dynamical systems, in particular in classical and quantum mechanics.The aim of the project is to study dynamical behaviors of reversible mapppings, including the existence of fixed points, Aubry-Mather sets and invariant tori. Applying these results to reversible differential equations, periodic solutions, quasi-periodic solutions and generalized quasi-periodic solutions are also obtained. Finally we will investigate the Liapunov stability of the trivial solution for reversible differential equations with periodic or quasi-periodic coefficients. The study can not only enrich its own theories, but also be applied to some new reversible differential equations.