A Renormalization Group Method for Nonlinear Oscillators

Mudavanhu, Blessing; O’Malley, Robert E.

doi:10.1111/1467-9590.1071178

Cited by 18 publications

(8 citation statements)

References 22 publications

(28 reference statements)

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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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“…determined by the nonsecular partỹ(t, A, , ) of the naive expansion y (t, A , ψ ), where the constants A and ψ are replaced by slowly varying functions A and , [49].…”

Section: Renormalizationmentioning

confidence: 99%

“…Such an ansatz is also central to averaging. Moreover, it allows one to renormalize, bypassing both the identification of the secular terms of the naive expansion and their subsequent removal [49]. Such an ansatz is also made in Lighthill's method [50].…”

Section: Amplitude Equationsmentioning

confidence: 99%

Working with multiscale asymptotics

2005

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This paper surveys, compares and updates techniques to obtain the asymptotic solution of the weakly nonlinear oscillator equationÿ + y + f (y,ẏ) = 0 as → 0 and for corresponding first-order vector systems. The solutions found by the regular perturbation method generally feature resonance and so break down as t → ∞. The classical methods of averaging and multiple scales eliminate such secular behavior and provide asymptotic solutions valid for time intervals of length t = O( −1 ). The renormalization group method proposed by Chen et al. [Phys. Rev. E 54 (1996) 376-394] gives equivalent results. Several well-known examples are solved with these methods to demonstrate the respective techniques and the equivalency of the approximations produced. Finally, an amplitude-equation method is derived that incorporates the best features of all these techniques. This method is both straightforward to automate with a computer-algebra system and flexible enough to allow the forcing f to depend on the small parameter.This paper is dedicated to Jerry Kevorkian with sincere appreciation for his long career of dedicated teaching at the University of Washington and for his substantial contributions to multiscale asymptotics.

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A Renormalization Group Method for Nonlinear Oscillators

Cited by 18 publications

References 22 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

Working with multiscale asymptotics

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