Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

Abstract

The renormalization group (RG) method is one of the singular perturbation methods which is used in search for asymptotic behavior of solutions of differential equations. In this article, time-independent vector fields and time (almost) periodic vector fields are considered. Theorems on error estimates for approximate solutions, existence of approximate invariant manifolds and their stability, inheritance of symmetries from those for the original equation to those for the RG equation, are proved. Further it is proved that the RG method unifies traditional singular perturbation methods, such as the averaging method, the multiple time scale method, the (hyper-) normal forms theory, the center manifold reduction, the geometric singular perturbation method and the phase reduction. A necessary and sufficient condition for the convergence of the infinite order RG equation is also investigated.

publised as SIAM j. on Appl. Dyn.Syst., Vol.8, 1066-1115 (2009)

References in corpus (2)

• Stability of an [N/2]-dimensional invariant torus in the Kuramoto model at small coupling
• Renormalization Reductions for Systems with Delay

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• The renormalization method from continuous to discrete dynamical systems: asymptotic solutions, reductions and invariant manifolds
• A remark on renormalization group theoretical perturbation in a class of ordinary differential equations