Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Abstract

For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502-4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré-Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré-Birkhoff normal forms for these systems up to and including terms of O( 2 ), where is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O(1/ ).

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

“…If the assumption (A) is violated, the error estimate (2.27) for approximate solutions may get worse, such as ||x(t) − α (m) t (y(t))|| < C √ ε, even if the KBM condition is satisfied and thus the RG equation is defined. See Sanders et al [47] and DeVille et al [32] for such examples.…”

“…After their works, many studies of the RG method have been done [10][11][12][13]16,21,22,[26][27][28][29][30][31][32]35,36,42,45,46,48,49,53,[55][56][57][58][59]. Kunihiro [28,29] interpreted an approximate solution obtained by the RG method as the envelope of a family of regular perturbation solutions.…”

“…Nozaki et al [45,46] proposed the proto-RG method to derive the RG equation effectively. Ziane [53] , DeVille et al [32] and Chiba [10] gave error estimates for approximate solutions. Chiba [10] defined the higher order RG equation and the RG transformation to improve error estimates.…”

“…Ei, Fujii and Kunihiro [16] applied the RG method to obtain approximate center manifolds and their method was rigorously formulated by Chiba [12]. DeVille et al [32] showed that lower order RG equations are equivalent to normal forms of vector fields, and this fact was extended to higher order RG equations by Chiba [11]. Applications to partial differential equations are appeared in [16,31,42,45,46,[55][56][57][58][59].…”

On the Renormalization Group Techniques for the Cubic-Quintic Duffing Equation

“…In recent times the method of renormalization group (RG) has been employed [1]- [4] through the introduction of a set of modified variables to arrive at the elimination of secular terms. The theory of RG has rich connections with quantum field theory and is considered to be a very powerful tool to handle the so-called 'divergences' of quantum electrodynamics [5].…”