New Development of Singular-Perturbation Method with Renormalization Group

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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.