基于移动坐标架的最小切环分支问题

项目名称： 基于移动坐标架的最小切环分支问题

项目编号： No.11201360

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 刘丹

作者单位： 西安电子科技大学

项目金额： 20万元

中文摘要： 异宿环及其分支问题在各种复杂的物理、生物等学科及相关研究领域都占有重要地位。异维环作为更一般的异宿环，其分支现象充分揭示了具有异维环的系统的结构不稳定性和其轨道拓扑类型的多样性，以及不同类型的轨道在演化过程中相互融合、转化、分叉的复杂性等等。目前异维环分支问题的研究是异宿环分支的一个热门问题。最小切环在异维环中具有一般性。然而由于最小切环本身的余维数分布的不均匀性给分支的讨论带来了很大的困难，故已有的研究结果不多，尤其对于那些退化程度较高的情况还没有得到系统的分析。本项目拟通过在最小切环的局部管状邻域内构造移动坐标架对高维微分系统中几类退化的三鞍点最小切环分支问题给出系统的、独创性的研究。这样的移动坐标架本身既继承了相应不变流形的几何不变性，又精确地反映了系统内在的线性压缩性和扩张性的动力学性态，由此得到的后继函数和分支方程的形式会变得相当简单而精确。

中文关键词： 分支；最小切环；移动坐标架；庞加莱映射；异宿环

英文摘要： Heteroclinic cycle and its bifurcations play an important role in many complex research fields such as physics, biology. Heterodimensional cycle behaves as a general type of heteroclinic cycles, which bifurcations reveal unstability of structure, diversity of topological types of orbits, and complexity of corresponding integration, translation and bifurcation in the evolution process of different orbits in higher dimensional systems with heterodimensional cycle. In present, bifurcations of heterodimensional cycles are attaching a lot of attention in many applications. Minimum tangency cycles exhibit generality in heterodimensional cycles. However, the asymmetrical distribution of the codimensions brings great difficulty in analysis on bifurcations of minimum tangency cycles, so there are not discussions enough in the existing results, especially in the higher degenerate cases. We plan to study some kinds of degenerate minimum tangency cycles with three saddles detailed and creatively by setting up a moving coordinate frame in a tublar neighborhood of the minimum tangency cycle in this research. Since such a coordinate frame itself not only inherits the geometric invariance of the corresponding manifolds, but also reflects accurately the dynamical properties including the intrinsic linear contraction and expansi

英文关键词： Bifurcation；Minimum tangency cycle；Moving coordinate frame；Poincare mapping；heteroclinic cycle