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The renormalization group (RG) method is one of the singular perturbationmethods which is used in search for asymptotic behavior of solutions ofdifferential equations. In this article, time-independent vector fields andtime (almost) periodic vector fields are considered. Theorems on errorestimates for approximate solutions, existence of approximate invariantmanifolds and their stability, inheritance of symmetries from those for theoriginal equation to those for the RG equation, are proved. Further it isproved 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 singularperturbation method and the phase reduction. A necessary and sufficientcondition for the convergence of the infinite order RG equation is alsoinvestigated.

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