Averaging Methods in Nonlinear Dynamical Systems

Sanders, Jan A.; Verhulst, Ferdinand; Murdock, James

doi:10.1007/978-1-4757-4575-7

Cited by 1,237 publications

(1,432 citation statements)

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“…It offers a versatile alternative to classical perturbation methods, such as the Poincaré-Lindstedt method [26,34], the method of matched asymptotic expansions [15,18,25,39], the method of multiple time scales [15,18,29,39], the method of averaging [3,41], and the WKBJ method [35] which were each developed for specific types of problems. In numerous examples the results obtained using the RG method are shown to agree [5,6,30,37,49] with those obtained from classical methods.…”

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

confidence: 99%

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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%

“…The notion of KBM vector fields was introduced in [3] and developed in [27,41], where KBM 0 was used. We follow [41] in calling this the Krylov-Bogolyubov-Mitropolskii average. We will assume that all f α,i in (6.1) are KBM Λ α,i .…”

Section: Rg Applied To Nonautonomous Equations (12)mentioning

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

“…In the particular case that k = n the expression of f 2 was found in [6], while when F 0 is identically zero (in particular this implies k = n), the expressions of the second order and third order bifurcation functions are known (see for instance [4] and the references therein). We remark that, in this particular case F 0 ≡ 0, the construction of these bifurcation functions are part of the averaging theory (for general results on averaging theory see for instance the books of Sanders, Verhulst and Murdock [16,17]). Sometimes, also in the general case F 0 ̸ ≡ 0 it is said that results like Theorem 1 describe the averaging method.…”

Section: Introductionmentioning

confidence: 99%

A second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter

Buică

Giné

Llibre

2012

Physica D: Nonlinear Phenomena

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Abstract. We deal with nonlinear T -periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation. In the end two applications are done. The key tool for proving the main result is the Lyapunov-Schmidt reduction method applied to the T -Poincaré-Andronov mapping.

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“…Then m = 1, Ω(u, φ) = 1 recovers the periodic case. Results on averaging in this situation, under certain assumptions on Ω, can be found in [BoMi61,SV85]. Exponential estimates are due to Neishtadt [Nei84] for m = 1 and to Simó [Sim94] for m > 1 and Ω(u, φ) = ω ∈ R m with Diophantine conditions on ω |( , ω)| ≥ γ| | −τ for some γ > 0, τ > m − 1 and all ∈ Z m (2.3) Partial differential equations with such a structure can be found in the context of systems of reaction-diffusion equations…”

Section: Examplesmentioning

confidence: 99%

“…The method proved to be very successful in the analysis of finite dimensional dynamical systems [LM88,SV85]. Now we consider problems, that can be written as an evolution equation with external forcing like (2.4,2.8) and under further assumption also (2.5) (see [Mat04]).…”

Section: Finite Order Estimates and Normal Formsmentioning

confidence: 99%

Exponential Estimates in Averaging and Homogenisation

Matthies

2006

Analysis, Modeling and Simulation of Multiscale Problems

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Abstract. Many partial differential equations with rapid spatial or temporal scales have effective descriptions which can be derived by homogenisation or averaging. In this article we deal with examples, where quantitative estimates of the error is possible for higher order homogenisation and averaging. In particular, we provide theorems, which allow homogenisation and averaging beyond all orders by giving exponential estimates of appropriately averaged and homogenised descriptions.Methods include iterated averaging transformations, optimal truncation of asymptotic expansions and highly regular solutions (Gevrey regularity). Prototypical examples are reaction-diffusion equations with heterogeneous reaction terms or rapid external forcing, nonlinear Schrödinger equations describing dispersion management, and second-order linear elliptic equations.

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Averaging Methods in Nonlinear Dynamical Systems

Cited by 1,237 publications

References 0 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

A second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter

Exponential Estimates in Averaging and Homogenisation

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