New Development of Singular-Perturbation Method with Renormalization Group

利用多种奇异摄动方法，如多重时间法、规范型理论和约化摄动法，研究了一类非线性动力系统的长时间渐近行为。微扰重整化群方法(RG方法)的最新发展统一了这些不同的微扰方法，其中系统的长时间渐近行为由重整化群方程来描述。1.建立了保辛RG方法，并将其应用于二维辛映射的Poicare-Birkhoff分支。提出了一种RG方法的原型RG方法，避免了RG方法中求显式长期解的步骤。将具有Lie对称性的RG方法推广到具有Lie对称性的有趣物理系统，如重力作用下的气球和绝热理想气体，这些系统只具有平凡的Lie对称性。

The long-time asymptotic behavior of a nonlinear dynamical system has been studied by various singular perturbation methods, such as the multi-time method, the normal form theory and the reductive perturbation method. Those various perturbation methods have been unified by the recent development of the perturbative renormalization group method (the RG method), where the long-time asymptotic behavior of a system is described by the renormalization group equation. In this work, the RG method is developed as follows.1.A symplecticity-preserving RG method is formulated and is applied to the Poicare-Birkhoff bifurcation of a two-dimensional symplectic map. We obtain analytical expressions to the resonant island structure.2.A proto-RG approach of the RG method is proposed to avoid the step of obtaining explicit secular solutions in the RG method. Various phase equations are systematically derived as RG equations from a general reaction-diffusion equation.3.The RG method with the Lie symmetry is extended to apply to interesting physical systems such as a gas sphere under gravity and adiavatic perfect gas, which have only trivial Lie symmetries.