Methods and Applications of Singular Perturbations

Verhulst, Ferdinand

doi:10.1007/0-387-28313-7

Cited by 257 publications

(172 citation statements)

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“…Put simply, the goal of a singular expansion is to expand only part of a function's ε-dependence, while holding fixed some specific ε-dependence that captures one or more of the function's essential features. Further details of such expansions, and their role in singular perturbation theory, can be found in numerous textbooks [38][39][40][41]; see the text by Eckhaus [42] for a rigorous treatment of some aspects. See Refs.…”

Section: The Retarded Solution To the Wave Equation Ismentioning

confidence: 99%

Self-consistent gravitational self-force

2010

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I review the problem of motion for small bodies in General Relativity, with an emphasis on developing a self-consistent treatment of the gravitational self-force. An analysis of the various derivations extant in the literature leads me to formulate an asymptotic expansion in which the metric is expanded while a representative worldline is held fixed; I discuss the utility of this expansion for both exact point particles and asymptotically small bodies, contrasting it with a regular expansion in which both the metric and the worldline are expanded. Based on these preliminary analyses, I present a general method of deriving self-consistent equations of motion for arbitrarily structured (sufficiently compact) small bodies. My method utilizes two expansions: an inner expansion that keeps the size of the body fixed, and an outer expansion that lets the body shrink while holding its worldline fixed. By imposing the Lorenz gauge, I express the global solution to the Einstein equation in the outer expansion in terms of an integral over a worldtube of small radius surrounding the body. Appropriate boundary data on the tube are determined from a local-in-space expansion in a buffer region where both the inner and outer expansions are valid. This buffer-region expansion also results in an expression for the self-force in terms of irreducible pieces of the metric perturbation on the worldline. Based on the global solution, these pieces of the perturbation can be written in terms of a tail integral over the body's past history. This approach can be applied at any order to obtain a self-consistent approximation that is valid on long timescales, both near and far from the small body. I conclude by discussing possible extensions of my method and comparing it to alternative approaches.

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Section: The Retarded Solution To the Wave Equation Ismentioning

confidence: 99%

Self-consistent gravitational self-force

2010

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“…To put system (16) in the standard form of quasi-harmonic equations we have to apply the symplectic transformation p = K(0)y, q = L(0)x in (12). This leads with (15) and Table 2 to the transformed Hamiltonian…”

Section: Case 0: the Fpu Chain With Well-balanced Massesmentioning

confidence: 99%

“…); as u increases through the interval (0, u 1 ) the masses will differ more and more, producing generic Hamiltonians. To put system (16) in the standard form of perturbed harmonic equations we have to apply again a symplectic transformation, i.e. (12) and…”

Section: The Hamiltonian-hopf Bifurcationmentioning

confidence: 99%

The Inhomogeneous Fermi-Pasta-Ulam Chain, a Case Study of the 1 : 2 : 3 $1:2:3$ Resonance

Bruggeman

Verhulst

2017

Acta Appl Math

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A 4-particles chain with different masses represents a natural generalization of the classical Fermi-Pasta-Ulam chain. It is studied by identifying the mass ratios that produce prominent resonances. This is a technically complicated problem as we have to solve an inverse problem for the spectrum of the corresponding linearized equations of motion. In the case of such an inhomogeneous periodic chain with four particles each mass ratio determines a frequency ratio for the quadratic part of the Hamiltonian. Most prominent frequency ratios occur but not all. In general we find a one-dimensional variety of mass ratios for a given frequency ratio.A detailed study is presented of the resonance 1 : 2 : 3. A small cubic term added to the Hamiltonian leads to a dynamical behaviour that shows a difference between the case that two opposite masses are equal and a striking difference with the classical case of four equal masses. For two equal masses and two different ones the normalized system is integrable and chaotic behaviour is small-scale. In the transition to four different masses we find a Hamiltonian-Hopf bifurcation of one of the normal modes leading to complex instability and Shilnikov-Devaney bifurcation. The other families of short-periodic solutions can be localized from the normal forms together with their stability characteristics. For illustration we use action simplices and examples of behaviour with time.

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“…Introductions to the multiple timescale method can be found in [2], [16], [20], and [23]. A comparison of averaging and multiple timing by a number of important examples can be found in [10].…”

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

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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Methods and Applications of Singular Perturbations

Cited by 257 publications

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Self-consistent gravitational self-force

Self-consistent gravitational self-force

The Inhomogeneous Fermi-Pasta-Ulam Chain, a Case Study of the 1 : 2 : 3 $1:2:3$ Resonance

Profits and Pitfalls of Timescales in Asymptotics

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