中心型系统及退化系统的正规形及分岔

Along with the deep investigation on dynamical systems, more attentions are paid to complicated topological structure of orbits near nonhyperbolic equilibria, including center-type equilibria and degenerate equilibria. Concerning distribution of periods near a center-type equilibrium and distribution of orbits near a degenerate equilibrium, there is still a lot of unsolved problems. In this project we investigate normal forms and bifurcations of systems near the two classes of equilibria. We intend to study isochronicity and bifurcation of critical periods in nonhomogeneous polynomial systems having a center by decomposing algebraic varieties and employing corresponding transversal commuting systems and make advances to Jarque-Villadelprat’s open problem. We intend to weaken the condition of independence of period quantities to enlarge the lower estimation of the number of critical periods. We intend to give criteria for center, multi-center and bifurcation of critical periods for higher dimensional systems and nonsmooth systems. We intend to develop a new method to determine the qualitative properties of higher order degenerate equilibria which the Generalized Normal Sector method is not applicable. We intend to give normal forms for Lienard system and some special classes of systems and employ them to give restrictive versal unfolding for higher order degenerate equilibria in those classes. We intend to use the Generalized Normal Sector method to give monodromic conditions for degenerate polynomial differential systems, give normal forms for those systems and identify between center and focus. We intend to investigate invariant manifolds and invariant foliations near degenerate equilibria and use them to simplify systems.

随着动力系统研究的深入，人们愈加关注非双曲奇点附近轨道的复杂拓扑结构以及分岔。非双曲奇点包括中心型奇点和退化奇点两大类。中心型奇点附近的周期分布，退化奇点附近的轨道分布，仍有大量问题尚待解决。本项目将研究系统在这两类奇点的正规形及分岔。拟利用代数簇分解和横截交换系统来研究非齐次系统的中心等时性和临界周期分岔，推进Jarque-Villadelprat公开问题的解决，并设法降低周期量独立性条件，以获得更大的临界周期的个数。拟对高维系统、非光滑系统给出中心判据、多中心条件并研究临界周期分岔。拟对广义正常区域仍不能判定的高阶退化奇点发展新的奇点定性判定方法。拟在Lienard等限定系统类中研究正规形，并用以对高阶退化系统研究限定类中的普适开折。 拟利用广义正常区域方法给出退化多项式微分系统的单值性条件，进而研究其正规形并研究其中心-焦点判定。拟研究退化奇点的不变流形及不变叶层，进而用于系统简化。

本项目按原计划，研究了系统在非双曲奇点（包括中心型奇点和退化奇点两大类）的正规形及分岔。.关于中心型奇点, 在没有周期量独立性的情况下给出了更弱的“限制独立性”判据，从而给出了临界周期极大个数的可达性；对(1,3)-切换的FF型非光滑系统利用平均法给出了不同环数的参数条件,证明该平衡点在3次齐次扰动下的环性数至少是3。关于退化奇点，给出了奇Lienard 向量场族的正规形，并用于对具有幂零退化性的Lienard 奇点在奇Lienard 向量场族中给出普适开折。给出了强单值系统的正规形，并用以对具有高阶退化性的三次齐次等时中心给出在强单值向量场族中的普适开折，进而给出其中心-焦点判定。.进而，研究高维微分系统退化同宿轨分岔，给出由单一参数变化产生的多重混沌的条件；对非自治系统在线性扰动渐近于0的条件下证明非一致指数二分粗糙性的二择一结果；基于二分指数同号、异号的情况，证明了无穷维随机动力系统光滑不变流形和不变叶层的存在性、可测性以及不变叶层对基点的光滑性；在非一致法向双曲情况下证明无穷维系统不变流形的保持性；将C^1线性化对非自治系统实现，在强指数二分条件下给出了保证C^1线性化的谱间隙条件；解决了30年来未解决的可微与Holder线性化同时实现的问题并把结果推广到非自治情形；将著名的Hartman的C^1线性化定理推进到随机情形；研究了迭代算子本身的动力学性质，证明其所有轨道有界、大多数不动点不稳定但不产生混沌。.发表论文23篇，包括在Adv. Math.、Proc. London Math. Soc.、Math. Z.、J. Differential Eq.等国际著名杂志。获教育部自然科学一等奖。.