Extending slow manifolds near transcritical and pitchfork singularities

Krupa, Martin; Szmolyan, Peter

doi:10.1088/0951-7715/14/6/304

Cited by 134 publications

(150 citation statements)

References 15 publications

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“…We remark that S may have singularities [142], but we assume here that this does not happen so that S is a smooth manifold. The points of S are equilibrium points for the layer equations (2.3).…”

Section: The Critical Manifold and The Slow Flowmentioning

confidence: 99%

Mixed-Mode Oscillations with Multiple Time Scales

Desroches¹,

Guckenheimer

Krauskopf³

et al. 2012

SIAM Rev.

486

619

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Abstract. Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale "mechanism." Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

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Section: The Critical Manifold and The Slow Flowmentioning

confidence: 99%

Mixed-Mode Oscillations with Multiple Time Scales

Desroches¹,

Guckenheimer

Krauskopf³

et al. 2012

SIAM Rev.

486

619

View full text Add to dashboard Cite

show abstract

“…For the TMDD model, the critical set reduces to two intersecting one dimensional manifolds. Using the results in [3], it can be shown that the slow manifold connects the two attracting branches of the critical set. For the dimerisation model, the critical set reduces to an incoming one dimensional manifold and an outgoing two dimensional manifold that intersect.…”

Section: Motivationmentioning

confidence: 99%

“…We will investigate what happens to the incoming manifold, denoted by S − a,ε , as it passes by a neighbourhood of the origin, the extension of S − a,ε is denoted byS − a,ε . We do this using the blow up method and building on the work of Krupa and Szmolyan [3].…”

Section: The General Problemmentioning

confidence: 99%

Extending Slow Manifolds Near a Degenerate Transcritical Intersection in Three Dimensions

Gavin

Aston

Derks

2017

Trends in Mathematics

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Motivated by a problem from pharmacology, we consider a general two parameter slow-fast system in which the critical set consists of a one dimensional manifold and a two dimensional manifold, intersecting transversally at the origin. Using geometric desingularisation, we show that for a subset of the parameter set there is an exchange of stabilities between the attracting components of the critical set and the direction of the continuation can be expressed in terms of the parameters. MotivationWe consider the pharmacological model of dimerisation where a receptor binds to two ligand molecules. The dimerisation model is an adaptation of the well studied target mediated drug disposition model (TMDD) in which the receptor binds to one ligand molecule, see [5] for more details. In both models, it is assumed that the binding is the fastest process. This gives a separation of time scales, which allows us to use geometric singular perturbation theory to analyse these models. For the TMDD model, the critical set reduces to two intersecting one dimensional manifolds. Using the results in [3], it can be shown that the slow manifold connects the two attracting branches of the critical set. For the dimerisation model, the critical set reduces to an incoming one dimensional manifold and an outgoing two dimensional manifold that intersect. By analysing this type of intersection using geometric desingularisation, we will show the existence of a transfer to the two dimensional manifold and determine the direction of the orbit on this manifold away from the intersection, in terms of the model parameters. The General ProblemThe canonical form for the problem of interest is given bẏThe critical set is the union of the one dimensional manifold {x 2 = y, x 1 = 0 : y ∈ R} and the two dimensional manifold {x 2 = −y : x 1 , y ∈ R}. The incoming manifold S − a = {x 2 = y, x 1 = 0 : y < 0} is attracting.

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“…It has also been used by Krupa and Szmolyan in e.g. [KS1], [KS2], [KS3] and [KS4]. The key element is the family blow up-a technique of rescaling variables in a geometrical way.…”

Section: Introductionmentioning

confidence: 99%

Canard solutions at non-generic turning points

Maesschalck

Dumortier

2005

Trans. Amer. Math. Soc.

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Abstract. This paper deals with singular perturbation problems for vector fields on 2-dimensional manifolds. "Canard solutions" are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a "turning point" and follow for a while a normally repelling branch of the singular curve. Following the geometric ideas developed by Dumortier and Roussarie in 1996 for the study of canard solutions near a generic turning point, we study canard solutions near non-generic turning points. Characterization of manifolds of canard solutions is given in terms of boundary conditions, their regularity properties are studied and the relation is described with the more traditional asymptotic approach. It reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work. Since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve, we also study their contact and its relation to the "control curve".

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Extending slow manifolds near transcritical and pitchfork singularities

Cited by 134 publications

References 15 publications

Mixed-Mode Oscillations with Multiple Time Scales

Mixed-Mode Oscillations with Multiple Time Scales

Extending Slow Manifolds Near a Degenerate Transcritical Intersection in Three Dimensions

Canard solutions at non-generic turning points

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