Asymptotic Methods for Relaxation Oscillations and Applications

Grasman, J.

doi:10.1007/978-1-4612-1056-6

Cited by 236 publications

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“…For this topic, see [8,11] and [14]. Most rigorous analysis is carried out for two-dimensional autonomous and forced problems and it is not easy to extend this to more dimensions.…”

Section: Relaxation Oscillations and Quenchingmentioning

confidence: 99%

Singular perturbation methods for slow–fast dynamics

Verhulst

2007

Nonlinear Dyn

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Recently, geometric singular perturbation theory has been extended considerably while at the same time producing many new applications. We will review a number of aspects relevant to non-linear dynamics to apply this to periodic solutions within slow manifolds and to review a number of non-hyperbolic cases. The results are illustrated by examples.Keywords Singular perturbations . Slow manifolds . Periodic solutions . Nonhyperbolic This paper deals with slow-fast initial value problems that are of the forṁAs usual, ε is a small positive parameter, and an overdot denotes differentiation with respect to time. For a number of results, the vector fields f and g explicitly depending on time t present no obstruction. Part of the paper is a tutorial, but there are some new results. The Tikhonov theoremIn singular perturbations, certain attraction (or hyperbolicity) properties of the regular (outer) expansion play an essential part in the construction of the formal approximation. Remarkably enough, this hyperbolicity does not include the behaviour of the slow equation.In the constructions, the following theorem provides a basic boundary layer property of the solution. Theorem 1.1. (Tikhonov, 1952, see [15]) Consider the initial value probleṁ For f and g, we take sufficiently smooth vector functions in x, y and t; the dots represent (smooth) higherorder terms in ε.a. We assume that a unique solution of the initial value problem exists and suppose this holds also for the reduced probleṁ x = f (x, y, t), x(0) = x 0 , = g(x, y, t), with solutionsx(t),ȳ(t).Springer

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“…For this topic, see [8,11] and [14]. Most rigorous analysis is carried out for two-dimensional autonomous and forced problems and it is not easy to extend this to more dimensions.…”

Section: Relaxation Oscillations and Quenchingmentioning

confidence: 99%

Singular perturbation methods for slow–fast dynamics

Verhulst

2007

Nonlinear Dyn

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“…They are described mathematically as the solution of two or more nonlinear ordinary differential equations that exhibit different time scales. Over the years, reliable asymptotic techniques such as the method of matched asymptotic expansions [1][2][3] have been developed and successfully used to determine rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 3 . The values of the parameters are ε = 0.02 and λ = 1.…”

Section: Introductionmentioning

confidence: 99%

Slow–fast dynamics of a time-delayed electro-optic oscillator

Weicker

Erneux

D'Huys

et al. 2013

Phil. Trans. R. Soc. A.

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International audienceSquare-wave oscillations exhibiting different plateau lengths have been observed experimentally by investigating an electro-optic oscillator. In a previous study, we analysed the model delay differential equations and determined an asymptotic approximation of the two plateaus. In this paper, we concentrate on the fast transition layers between plateaus and show how they contribute to the total period. We also investigate the bifurcation diagram of all possible stable solutions. We show that the square waves emerge from the first Hopf bifurcation of the basic steady state and that they may coexist with stable low-frequency periodic oscillations for the same value of the control parameter

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“…Adding the perturbations f, g can not destroy this torus but only deforms it. In this example the torus is two-dimensional but the timescales of rotation, if µ is large enough, are in both directions determined by the timescales of relaxation oscillation (see Grasman, 1987) and so are O(1/µ).…”

Section: Deforming a Normally Hyperbolic Manifoldmentioning

confidence: 99%

Invariant Manifolds in Dissipative Dynamical Systems

Verhulst

2005

Acta Appl Math

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Invariant manifolds like tori, spheres and cylinders play an important part in dynamical systems. In engineering, tori correspond with the important phenomenon of multi-frequency oscillations. Normal hyperbolicity guarantees the robustness of these manifolds but in many applications weaker forms of hyperbolicity present more realistic cases and interesting phenomena. We will review the theory and present a number of examples using normalization-averaging techniques.

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Asymptotic Methods for Relaxation Oscillations and Applications

Abstract: The Editors welcome aII inquiries regarding the submisslon of manuscripts for the series. Final preparation of aII manuscrlpts will take place in the editorial offices of the series in the

Cited by 236 publications

References 0 publications

Singular perturbation methods for slow–fast dynamics

Singular perturbation methods for slow–fast dynamics

Slow–fast dynamics of a time-delayed electro-optic oscillator

Invariant Manifolds in Dissipative Dynamical Systems

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