A New Renormalization Method for the Asymptotic Solution of Weakly Nonlinear Vector Systems

Mudavanhu, Blessing; O’Malley, Robert E.

doi:10.1137/s0036139901394311

Cited by 19 publications

(16 citation statements)

References 28 publications

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“…Fundamental asymptotic analysis of the RG method has also been presented in [30][31][32] for systems subject to small-amplitude, time-periodic perturbations and for weakly nonlinear, autonomous perturbations of planar oscillators. In these works, a simplified version of the RG method is presented.…”

Section: Relation Of This Analysis To Other Results About the Rg Methmentioning

confidence: 99%

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

Deville¹,

Harkin²,

Holzer³

et al. 2008

Physica D: Nonlinear Phenomena

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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/ ).

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Section: Relation Of This Analysis To Other Results About the Rg Methmentioning

confidence: 99%

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

Deville¹,

Harkin²,

Holzer³

et al. 2008

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Chiba¹

2009

SIAM J. Appl. Dyn. Syst.

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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.

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“…A comparison of averaging and multiple timing by a number of important examples can be found in [10]. The relation between multiple timing and the renormalization method was discussed in [2], [3], and [14]. In [17], Perko established the equivalence of the averaging method and multiple timing for standard equations likė…”

Section: The Basic Idea Of Two (Or Multiple) Timescalesmentioning

confidence: 99%

“…In the same spirit, perturbation methods for ODEs can also be seen as normal form methods, for the idea is always to transform the ODEs to equations that are simpler and can be solved or at least partly solved. For ODEs, normalization has, for instance, been formulated in [2], [3], [14], and [20]. A number of examples will demonstrate this.…”

mentioning

confidence: 99%

Profits and Pitfalls of Timescales in Asymptotics

Verhulst¹

2015

SIAM Rev.

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Abstract. The method of multiple timescales is widely used in engineering and mathematical physics.In this article we draw attention to the literature on the comparison of various perturbation methods. We indicate where we can obtain an advantage from the concept of timescales, and we present examples where the anticipation of timescales makes sense and cases where, because of resonances or bifurcations, the analysis is less straightforward. In a number of problems second order approximations are essential to understand the phenomena. We will conclude from these examples that the anticipation of timescales as in the multiple timescales method may give misleading results; methods that do not anticipate timescales a priori such as averaging and renormalization should be used in research problems.

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A New Renormalization Method for the Asymptotic Solution of Weakly Nonlinear Vector Systems

Cited by 19 publications

References 28 publications

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

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

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

Profits and Pitfalls of Timescales in Asymptotics

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