Folded Saddles and Faux Canards

Mitry, John; Wechselberger, Martin

doi:10.1137/15m1045065

Cited by 21 publications

(43 citation statements)

References 20 publications

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“…We note that the cycle shown in Fig. 9, panel 5 passes close to the faux canard of the folded saddle, and small oscillations seen as it does so are oscillations of the orbit around the faux canard; see [24] for further details about this kind of dynamics.…”

Section: Spike Adding Utilizing Two Time Scalesmentioning

confidence: 86%

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Spike-Adding in a Canonical Three-Time-Scale Model: Superslow Explosion and Folded-Saddle Canards

Desroches¹,

Kirk²

2018

SIAM J. Appl. Dyn. Syst.

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We examine the origin of complex bursting oscillations in a phenomenological ordinary differential equation model with three time scales. We show that bursting solutions in this model arise from a Hopf bifurcation followed by a sequence of spike-adding transitions, in a manner reminiscent of spike-adding transitions previously observed in systems with two time scales. However, the details of the process can be much more complex in this three-timescale context than in two-timescale systems. In particular, we find that spike-adding can involve canard explosions occurring on two different time scales and is associated with passage near a foldedsaddle singularity. We show that the form of spike-adding transition that occurs depends on the geometry of certain singular limit systems, specifically the relative positions of the critical and superslow manifolds. We also show that, unlike the case of spike-adding in two-timescale systems, the onset of a new spike in our model is not typically associated with a local maximum in the period of the bursting oscillation.

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Section: Spike Adding Utilizing Two Time Scalesmentioning

confidence: 86%

“…Note that the amplitude of such oscillations depends on the relative magnitude of the eigenvalues of the folded saddle in the desingularised slow reduced system. More detail about the dynamics near faux canards of folded saddles can be found in [24].…”

Section: Spike Adding Utilizing Three Time Scalesmentioning

confidence: 99%

Spike-Adding in a Canonical Three-Time-Scale Model: Superslow Explosion and Folded-Saddle Canards

Desroches¹,

Kirk²

2018

SIAM J. Appl. Dyn. Syst.

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“…Moreover, for each η > 0, no matter how small, there exists a solution of the perturbed BVP with z 2 (0) > 0 defined on the interval µ ∈ (0, µ η ). These type I solutions ( Figure 6(c)) contain segments that follow N r but ultimately connect to an attracting fast fiber and so resemble the faux canards familiar from the ODE setting, a term used to refer to solutions that start on a repelling slow manifold and connect to an attracting slow manifold [28]. Other notable solutions in slow-fast ODEs are jumpon canards, which start along an attracting fast fiber and connect to a repelling slow manifold [16].…”

Section: Stochastic Simulationsmentioning

confidence: 98%

Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system

et al. 2017

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A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow-fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for non-zero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary conditions. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition, and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow-fast spatial dynamical system.

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“…Our more detailed analysis of the slow flow (Section 3.1) reveals novel features of the dynamics, mainly due to the true canard that exists near the folded saddle. The family of faux canards existing by a result of [58] (see also [41]), also plays an important role. We fix two choices of Fenichel slow manifolds S r,ε and S a,ε , that extend to the neighbourhood of the folded saddle.…”

Section: Dynamics Of the Full Systemmentioning

confidence: 99%

“…Faux canards have recently been studied by Mitry and Wechselberger [41] who discovered that they can lead to rotational behaviour of the trajectories. Since the trajectories returning from a parabolic burst must pass close to the family of faux canards, some of this oscillatory behaviour may be a part of parabolic bursting; see e.g.…”

Section: Dynamics Of the Full Systemmentioning

confidence: 99%

Spike-adding in parabolic bursters: The role of folded-saddle canards

Desroches

Krupa

Rodrigues

2016

Physica D: Nonlinear Phenomena

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The present work develops a new approach to studying parabolic bursting, and also proposes a novel four-dimensional canonical and polynomial-based parabolic burster. In addition to this new polynomial system, we also consider the conductance-based model of the Aplysia R15 neuron known as the Plant model, and a reduction of this prototypical biophysical parabolic burster to three variables, including one phase variable, namely the Baer-Rinzel-Carillo (BRC) phase model. Revisiting these models from the perspective of slow-fast dynamics reveals that the number of spikes per burst may vary upon parameter changes, however the spike-adding process occurs in an explosive fashion that involves special solutions called canards. This spike-adding canard explosion phenomenon is analysed by using tools from geometric singular perturbation theory in tandem with numerical bifurcation techniques. We find that the bifurcation structure persists across all considered systems, that is, spikes within the burst are incremented via the crossing of an excitability threshold given by a particular type of canard orbit, namely the true canard of a folded-saddle singularity. However there can be a difference in the spike-adding transitions in parameter space from one case to another, according to whether the process is continuous or discontinuous, which depends upon the geometry of the folded-saddle canard. Using these findings, we construct a new polynomial approximation of the Plant model, which retains all the key elements for parabolic bursting, including the spike-adding transitions mediated by folded-saddle canards. Finally, we briefly investigate the presence of spike-adding via canards in planar phase models of parabolic bursting, namely the theta model by Ermentrout and Kopell.

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Folded Saddles and Faux Canards

Cited by 21 publications

References 20 publications

Spike-Adding in a Canonical Three-Time-Scale Model: Superslow Explosion and Folded-Saddle Canards

Spike-Adding in a Canonical Three-Time-Scale Model: Superslow Explosion and Folded-Saddle Canards

Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system

Spike-adding in parabolic bursters: The role of folded-saddle canards

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