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An analysis of the renormalization group method for asymptotic expansions with logarithmic switchback terms. (English) Zbl 1292.34051

Authors’ abstract: The renormalization group (RG) method of L.-Y. Chen, N. Goldenfeld and Y. Oono [“Renormalization group theory for global asymptotic analysis”, Phys. Rev. Lett., 73, No. 10, 1311–1315 (1994; doi:10.1103/PhysRevLett.73.1311); erratum ibid. 74, No. 10, 1889 (1995; doi:10.1103/PhysRevLett.74.1889.2); “Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory”, Phys. Rev. E, 54, No. 1, 376–394 (1996; doi:10.1103/PhysRevE.54.376)] offers a comprehensive approach to formally computing asymptotic expansions of the solutions to singular perturbation problems and multi-scale problems. For problems in which the expansions are in powers of the small parameter, it has been shown that the RG method leads to uniformly valid asymptotic expansions of the solutions. However, there has not yet been an analysis of the validity of the RG method, i.e., a demonstration that it leads to uniformly valid asymptotic expansions, for problems in which the expansions also involve logarithms of the small parameter as gauge functions. The aim of this article is to provide this justification of the RG method. In particular, we extend the approach developed in [R. E. L. DeVille et al., Physica D 237, No. 8, 1029–1052 (2008; Zbl 1145.34331)] from the classes of autonomous and non-autonomous perturbations considered there to include the non-autonomous systems subject to singular perturbations for which the solutions involve logarithmic gauge functions. This framework is built upon the relationship between the RG method and normal form theory. We apply the RG method to three successively-more complex examples and also elucidate the common general features.

MSC:

34E05 Asymptotic expansions of solutions to ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34E15 Singular perturbations for ordinary differential equations
70K45 Normal forms for nonlinear problems in mechanics
37G05 Normal forms for dynamical systems
81T17 Renormalization group methods applied to problems in quantum field theory

Citations:

Zbl 1145.34331
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Full Text: Euclid