Relaxation Oscillation and Canard Explosion

Krupa, Martin; Szmolyan, Peter

doi:10.1006/jdeq.2000.3929

Cited by 455 publications

(651 citation statements)

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“…These regions correspond to U 1 and U 3 . In between there is an exponentially thin region centered at the canard solution consisting of trajectories that follow S r ε in a similar way as in classical canard phenomenon (canard explosion), see for example [11]. This region corresponds to U 2 .…”

mentioning

confidence: 85%

“…In the neighborhood of this equilibrium, the distinction between fast and slow variables is lost, and the analysis is further complicated by the fact that the two fold lines of the cusp surface, which themselves are already not normally hyperbolic, meet at the cusp point. Nevertheless, the system dynamics may be analyzed using the method of geometric desingularization, also known as the blowup method [2,3,4,5,10,11]. Here, the origin is blown up into a hyper-sphere, and the induced equilibria are either hyperbolic or semi-hyperbolic.…”

Section: Analysis Of a Singularly Perturbed Cusp By Means Of Geometrimentioning

confidence: 99%

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Geometric Desingularization of a Cusp Singularity in Slow–Fast Systems with Applications to Zeeman’s Examples

2013

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The cusp singularity -a point at which two curves of fold points meet-is a prototypical example in Takens' classification of singularities in constrained equations, which also includes folds, folded saddles, folded nodes, among others. In this article, we study cusp singularities in singularly perturbed systems for sufficiently small values of the perturbation parameter, in the regime in which these systems exhibit fast and slow dynamics. Our main result is an analysis of the cusp point using the method of geometric desingularization, also known as the blow-up method, from the field of geometric singular perturbation theory. Our analysis of the cusp singularity was inspired by the nerve impulse example of Zeeman, and we also apply our main theorem to it. Finally, a brief review of geometric singular perturbation theory for the two elementary singularities from the Takens' classification occurring for the nerve impulse example -folds and folded saddles -is included to make this article self-contained.

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confidence: 85%

Section: Analysis Of a Singularly Perturbed Cusp By Means Of Geometrimentioning

confidence: 99%

Geometric Desingularization of a Cusp Singularity in Slow–Fast Systems with Applications to Zeeman’s Examples

2013

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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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“…The situation considered here is principally different from the one in [5], where two pieces of critical manifold are attracting and one is repelling. In this paper, two pieces S a and S b of the critical manifold are not normally hyperbolic ( [4]) and consequently the geometric singular perturbation theory developed by N. Fenichel ( [1]) is not applicable to our case.…”

Section: Introductionmentioning

confidence: 95%

Frequency control of singularly perturbed forced duffing’s oscillator

Vrabel

Abas

2011

J Dyn Control Syst

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Abstract. We analyze the dynamics of the forced singularly perturbed differential equation of Duffing's type. We explain the appearance of the large frequency nonlinear oscillations of the solutions. It is shown that the frequency can be controlled by a small parameter at the highest derivative. We give some generalizations of results obtained recently by B.S. Wu, W.P. Sun and C.W. Lim, Analytical approximations to the double-well Duffing oscillator in large amplitude oscillations, Journal of Sound and Vibration, Volume 307, Issues 3-5, (2007), pp. 953-960. The new method for an analysis of the nonlinear oscillations which is based on the dynamic change of coordinates is proposed.

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Relaxation Oscillation and Canard Explosion

Cited by 455 publications

References 17 publications

Geometric Desingularization of a Cusp Singularity in Slow–Fast Systems with Applications to Zeeman’s Examples

Geometric Desingularization of a Cusp Singularity in Slow–Fast Systems with Applications to Zeeman’s Examples

Singular perturbation methods for slow–fast dynamics

Frequency control of singularly perturbed forced duffing’s oscillator

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