动力系统的对称、可积与动力学

动力系统的对称存在于力学等很多实际问题之中，它的研究既是经典的又是困难的。对称的存在与所述系统的守恒量与动力学密切相关，在微分系统动力学的研究中起了重要的作用。该方向已有很多重要的研究成果，但仍遗留下一些长期没有解决的公开问题。本项目拟在该课题上展开攻关，解决其中的部分问题。已有的结果主要针对具体的系统寻找对称的存在，本项目拟对微分系统在奇点邻域给出局部对称的存在性及其形成的线性空间的结构、维数和表示，并利用其研究微分系统的可积性及其局部动力学，如Jacobi乘子的存在性、极限环分支和重数等。在一般理论的基础上研究具体微分系统对称的存在性及其动力学，具体为复系数四元数微分方程对称的存在性和约化问题，并运用其研究所述微分方程的局部和整体动力学。解析微分系统的具有一个零特征值其余特征值非共振的奇点邻域对称的存在性及其守恒量的存在性和正则性问题长期没有得到解决，本项目希望在该方向取得突破。

Symmetries of dynamical systems exist in many realistic disciplines, such as mechanics, researches on them are both classical and difficulty. The existence of symmetry is strongly related to the existence of invariants and their dynamics, and it plays key roles in the study of dynamics of differential systems. As we know, there are lots of important results in this direction, but there still remains some long standing open problems to be solved. This proposal will focus on parts of these open problems, and hope to get answers to them. Most of the known results put attentions on searching symmetries of concrete systems, here we present a complete understanding on the existence of symmetries and the linear space formed by them (structure, dimension and its expression) of differential systems near a singularity, and apply these results to detect existence of invariants and the local dynamics of the systems, for instance the existence of Jacobi multipliers and its associated limit cycle bifurcation and multiplicity. Under the guide of general theory on symmetry, we turn to study existence of symmetry of differential systems and their dynamics, especially the existence of symmetry of quaternionic differential equations with complex coefficients and the reduction of the equation, then apply these results to investigate local and global dynamics of the mentioned equations. Finally, we tackle the problem on the existence of symmetries, of first integrals and their regularity of differential systems near a singularity with one zero eigenvalues and others nonresonant, which is a long standing open problem.