Computing Two-Dimensional Global Invariant Manifolds in Slow–fast Systems

England, James P.; Krauskopf, Bernd; Osinga, Hinke M.

doi:10.1142/s0218127407017562

Cited by 22 publications

(24 citation statements)

References 13 publications

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“…Despite its simplicity, this sweeping method fails to produce satisfactory results in some cases. In particular, strong convergence or divergence of trajectories toward one another makes the choice of the initial mesh problematic and can produce very nonuniform "coverage" of the desired manifold; see [61,62]. In multiple-time-scale systems, the fast exponential instability of Fenichel manifolds that are not attracting makes initial value solvers incapable of tracking these manifolds by forward integration.…”

Section: Numerical Methods For Slow-fast Systemsmentioning

confidence: 99%

Mixed-Mode Oscillations with Multiple Time Scales

Desroches¹,

Guckenheimer

Krauskopf³

et al. 2012

SIAM Rev.

507

635

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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: Numerical Methods For Slow-fast Systemsmentioning

confidence: 99%

Mixed-Mode Oscillations with Multiple Time Scales

Desroches¹,

Guckenheimer

Krauskopf³

et al. 2012

SIAM Rev.

507

635

View full text Add to dashboard Cite

show abstract

“…We find that for the pituitary model the effects are only at a quantitative level and the qualitative structure remains intact. We computed the two-dimensional global manifolds with the specialized method GLOBALIZEBVP (England et al, 2007;Krauskopf and Osinga, 2003) that builds the surface up as a collection of geodesic level sets, that is, a collection of closed curves (topological circles) with the property that points on the same curve lie at the same geodesic distance from the equilibrium or periodic orbit. The geodesic distance is the arclength of the shortest path on the manifold that connects the two objects, which need not be a trajectory.…”

Section: Modeling and Methodsmentioning

confidence: 99%

“…These twopoint boundary value problems are solved by continuation using the collocation routines in AUTO, which makes the method particularly suitable for systems with multiple time scales. We refer to England et al (2007) for more details. Visualization of the manifolds was done in GEOMVIEW (Phillips et al, 1993).…”

Section: Modeling and Methodsmentioning

confidence: 99%

Resetting Behavior in a Model of Bursting in Secretory Pituitary Cells: Distinguishing Plateaus from Pseudo-Plateaus

et al. 2007

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We study a recently discovered class of models for plateau bursting, inspired by models for endocrine pituitary cells. In contrast to classical models for fold-homoclinic (square-wave) bursting, the spikes of the active phase are not supported by limit cycles of the frozen fast subsystem, but are transient oscillations generated by unstable limit cycles emanating from a subcritical Hopf bifurcation around a stable steady state. Experimental time courses are suggestive of such fold-subHopf models because the spikes tend to be small and variable in amplitude; we call this pseudo-plateau bursting. We show here that distinct properties of the response to attempted resets from the silent phase to the active phase provide a clearer, qualitative criterion for choosing between the two classes of models. The fold-homoclinic class is characterized by induced active phases that increase towards the duration of the unperturbed active phase as resets are delivered later in the silent phase. For the fold-subHopf class of pseudo-plateau bursting, resetting is difficult and succeeds only in limited windows of the silent phase but, paradoxically, can dramatically exceed the native active phase duration.

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“…If the manifold M, in addition to (14), is such that M s = M τ , ∀ s, τ ∈ I, we will call it a time-independent invariant manifold.…”

Section: Invariant Manifolds In Systems Of Non-autonomous Ode'smentioning

confidence: 99%

“…Section 2.2) the algorithms developed specifically in this context will offer a better computational efficiency (e.g. [14,15]). Secondly, if a dynamical system depends periodically on time, the invariant manifold computations (and the associated transport considerations) can be reduced to the study of invariant sets of appropriately constructed Poincaré maps (e.g.…”

Section: Introductionmentioning

confidence: 99%

An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

Branicki

Wiggins

2009

Physica D: Nonlinear Phenomena

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a b s t r a c tWe present a computational method for determining the geometry of a class of three-dimensional invariant manifolds in non-autonomous (aperiodically time-dependent) dynamical systems. The presented approach can be also applied to analyse the geometry of 3D invariant manifolds in threedimensional, time-dependent fluid flows. The invariance property of such manifolds requires that, at any fixed time, they are given by surfaces in R 3 . We focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in R 3 . The presented technique can be employed, in particular, to compute codimension one (invariant) stable and unstable manifolds of hyperbolic trajectories in 3D non-autonomous dynamical systems which are crucial in the Lagrangian transport analysis. The same approach can also be used to determine evolution of an orientable 'material surface' in a fluid flow. These developments represent the first step towards a non-trivial 3D extension of the so-called lobe dynamics -a geometric, invariant-manifold-based framework which has been very successful in the analysis of Lagrangian transport in unsteady, two-dimensional fluid flows. In the developed algorithm, the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise C 2 interpolating functions. The method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution. A variant of the advancing front technique is used for remeshing, whenever necessary. Such an approach allows for computationally efficient determination of highly convoluted, evolving geometry of codimension one invariant manifolds in unsteady threedimensional flows. We show that the developed method is capable of providing detailed information on the evolving Lagrangian flow structure in three dimensions over long periods of time, which is crucial for a meaningful 3D transport analysis.

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Computing Two-Dimensional Global Invariant Manifolds in Slow–fast Systems

Cited by 22 publications

References 13 publications

Mixed-Mode Oscillations with Multiple Time Scales

Mixed-Mode Oscillations with Multiple Time Scales

Resetting Behavior in a Model of Bursting in Secretory Pituitary Cells: Distinguishing Plateaus from Pseudo-Plateaus

An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

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