Stability, bifurcations and edge oscillations in standing pulse solutions to an inhomogeneous reaction-diffusion system

Rubin, Jonathan E.

doi:10.1017/s0308210500031073

Cited by 15 publications

(16 citation statements)

References 28 publications

(58 reference statements)

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“…In conjunction with these observations, we point to an earlier example in which geometric singular perturbation theory was used to establish the existence of standing wave solutions in a RD model of the Fabry-Perot interferometer, which involves spatially dependent coefficients. See [21].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Pulse Dynamics in a Three-Component System: Existence Analysis

2008

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In this article, we analyze the three-component reaction-diffusion system originally developed by Schenk et al. (PRL 78:3781-3784, 1997). The system consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a prototype three-component system that generates rich pulse dynamics and interactions, and this richness is the main motivation for the analysis we present. We demonstrate the existence of stationary one-pulse and two-pulse solutions, and travelling one-pulse solutions, on the real line, and we determine the parameter regimes in which they exist. Also, for one-pulse solutions, we analyze various bifurcations, including the saddle-node bifurcation in which they are created, as well as the bifurcation from a stationary to a travelling pulse, which we show can be either subcritical or supercritical. For two-pulse solutions, we show that the third component is essential, since the reduced bistable two-component system does not support them. We also analyze the saddle-node bifurcation in which two-pulse solutions are created. The analytical method used to construct all of these pulse solutions is geometric singular perturbation theory, which allows us to show that these solutions lie in the transverse intersections of invariant manifolds in the phase space of the associated six-dimensional travelling wave system. Finally, as we illustrate with numerical simulations, these solutions form the backbone of the rich pulse dynamics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among others.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Pulse Dynamics in a Three-Component System: Existence Analysis

2008

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“…The proof of Theorem 3 is a direct application of the Exchange Lemma, more precisely, of the theorem in [17,Section 4], which applies the Exchange Lemma to persistence of singular homoclinic orbits like the one in Figure 4. The presence of a third scale governed by ε us or, equivalently, byε us is solely exploited to enforce the transversality conditions required by the application of the exchange lemma to (8) via the singular limit (26)(27). Genericity inε us arises by imposing some of these transversality conditions.…”

Section: Standing Casementioning

confidence: 99%

“…The stability analysis is beyond the scope of the present paper but predictions can be made in accordance with what is known about the stability of the two-scale models with a cubic nonlinearity. Standing pulses in two-scale models are known to be stable if the space-scale separation δ s is much smaller than the time-scale separation [27,Theorem 4.2], namely δ l ε s Likewise, to the best of our knowledge, the stability of traveling pulses has been analyzed only in the absence of diffusion in the adaptation variable [16], which corresponds to the limit…”

Section: Stability Of Bursting Wavesmentioning

confidence: 99%

A Three-Scale Model of Spatio-Temporal Bursting

Franci¹,

Sepulchre²

2016

SIAM J. Appl. Dyn. Syst.

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Abstract. We study spatio-temporal bursting in a three-scale reaction diffusion equation organized by the winged cusp singularity. For large time-scale separation the model exhibits traveling bursts, whereas for large space-scale separation the model exhibits standing bursts. Both behaviors exhibit a common singular skeleton, whose geometry is fully determined by persistent bifurcation diagrams of the winged cusp. The modulation of spatio-temporal bursting in such a model naturally translates into paths in the universal unfolding of the winged-cusp.Key words. spatio-temporal bursting, reaction-diffusion, singularity theory, geometric singular perturbation theory AMS subject classifications. 37N25,37G99,34E13,35K57,35B361. Introduction. In the recent paper [12], we proposed that temporal bursting can be conveniently studied in a three-time-scale model with a single scalar winged-cusp nonlinearity, a basic organizing center in the language of singularity theory. Because of its particular structure, the model is readily interpreted as a three-scale generalization of the celebrated FitzHugh-Nagumo model, a two-time-scale model similarly organized by a single scalar hysteresis nonlinearity.With enough time-scale separation, the (temporal) patterns exhibited by the trajectories of those models are robust and modulable in the parameter space, because they are in oneto-one correspondence with the persistent bifurcation diagrams of the universal unfolding of the hysteresis and winged-cusp, respectively.We showed that this feature is of prime interest in neurophysiology, for instance, because it provides insight in the physiological parameters that can regulate the continuous transition between distinct temporal patterns as observed in experimental neuronal recordings.The present paper pursues this investigation by adding diffusion in the three-scale temporal model and exploring the resulting three-scale spatio-temporal patterns: traveling bursts and standing bursts.The close relationship of the model with its two-dimensional counterpart is of great help in the analysis because we extensively rely on the existing theory for two-scale models with a hysteresis nonlinearity. We study the existence of traveling bursts and standing bursts in our three-scale model by mimicking the analysis of traveling pulses and standing pulses in earlier two-scale models. This geometric analysis exploits the singular limit to construct the skeleton

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“…And the assumptions on (f, g) for stability hold for both competition cases (not covering case (1.2)) and predator-prey cases. The stability of traveling waves for some other systems were also considered in [8,9].…”

Section: Introductionmentioning

confidence: 99%