弧形结构振动系统吸引域多重突变机理研究

The number of basins of attraction is usually not more than three when basin metamorphoses have been studied in the past three decades. The multitudinous basins with multiple Wada property or hybrid boundaries have been found in arch vibratory systems. We discovered double metamorphoses of basin boundaries and this phenomenon can not be understood by the saddle-node bifurcations and homoclinic tangency which are crucial methods to analyze the single metamorphose of basins. It indicates that the new mechanism for multiple metamorphoses of basins may exist. We intend to develop generalized basin cell theories and provide some criterions for the metamorphoses of multiple Wada basins. By comparing the manifolds on the basin boundaries before and after metamorphose of multitudinous basins, the basins of multiple metamorphose can be found and the different topological basin boundaries are verified. The mechanisms for different topological basin boundaries will be explored by expanding the heteroclinic internal and external tangent theorems combined with critical curve technique in irreversible systems, which will lay the theoretical foundation for the analysis of complex basin boundaries and catastrophe phenomena of massive basins in dynamical systems.

动力系统吸引域单重突变问题所研究的吸引域个数通常不超过三个，弧形结构振动系统存在众多吸引域且具有众多Wada域边界结构或混杂拓扑边界结构，申请者已发现该类型域边界具有不同寻常的二重突变新现象，而吸引域边界上的鞍结分岔和同宿正切理论等引起的单重突变已不能解释这种现象，预示着可能存在着决定吸引域边界多重突变的新机理。本研究拟发展广义域胞理论并给出判定系统多Wada吸引域结构发生突变的主要定理或判定准则，通过探寻系统众多吸引域突变前后边界流形之间的内在联系，筛选发生多重突变的吸引域并证实突变前后域的拓扑结构类型。通过拓展异宿内部正切和异宿外部正切定理，联合非可逆系统的关键曲线技术，阐明不同拓扑结构类吸引域的多重突变机理，为分析动力系统众多吸引域复杂边界结构及复杂域边界上的动力学突变现象奠定理论基础。

鉴于目前国际上对动力系统吸引域结构的多重突变问题的机理尚不清楚，本项目开展了关于弧形结构振动系统吸引域多重突变的机理研究，发现了吸引域边界集的多重突变现象, 揭示了非光滑异宿正切导致的吸引域多重突变机理；发现了一种新的类型的吸引域多重突变现象-吸引域反转，通过鞍点流形分析与不稳定极限环的追踪，揭示了吸引域反转的机理；通过分析吸引域突变前后的流形结构，发现了新颖的吸引域边界多层嵌套结构；发展了广义域胞定理并给出了系统Wada域存在的判定准则，证明了弧形结构振动系统广义域胞的存在性；给出了具有切换非光滑系统分形吸引域的拓扑学描述方法，给出了吸引域突变的Prime end方法；提出了一种联合庞加莱映射与无约束优化技术检测系统不变集中的不稳定周期轨道的方法，提出了通过暂态序列提取不稳定概周期轨道的方法。在完成计划书中预期理论成果的基础上，扩展了关联奇异非混沌吸引子(SNAs)吸引域的全局动力学问题，发现了三种类型的SNAs诞生新机理。研究成果丰富了以吸引域为代表的非线性全局动力学内容，为分析动力系统众多吸引域复杂边界结构及复杂域边界上的动力学突变现象奠定了理论基础。