Slow Unfoldings of Contact Singularities in Singularly Perturbed Systems Beyond the Standard Form

Lizarraga, Ian; Marangell, Robert; Wechselberger, Martin

doi:10.1007/s00332-020-09647-4

Cited by 5 publications

(5 citation statements)

References 35 publications

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“…Corollary 4.9. Consider the C r -smooth fast-slow map (35) with a regular fold point (0, 0) ∈ S satisfying (38). For ρ > 0 sufficiently small there exists an ε 0 > 0 such that for all ε ∈ (0, ε 0 ) the extension of the attracting slow manifold S a ε leaves U at a point (x, y) = (ρ, Y out (ρ, ε)), where…”

Section: Regular Fold Dynamicsmentioning

confidence: 99%

“…Regular contact points can be defined analogously to regular contact points in continuous-time GSPT; see [38]…”

Section: Regular Contact Points In Arbitrary Dimensionsmentioning

confidence: 99%

“…A recent formulation of GSPT for fast-slow ODE systems in this class can be found in [49], see also [19,29,36,39,38]. We emphasise that similarly to the theory for ODEs, the class of fast-slow maps defined by (1) contains, but is not limited to, the well-known class of fast-slow maps in standard form…”

Section: Introductionmentioning

confidence: 99%

“…This allows us to show that the dynamics on the center manifold can be approximated by the time-1 map of a (k+1)-dimensional fast-slow ODE system with a regular contact point. This is advantageous, given that the dynamics of the latter have already been described in [48,49] (see also [38] for more on contact points in fast-slow ODEs).…”

Section: Introductionmentioning

confidence: 99%

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Discrete geometric singular perturbation theory

Jelbart¹,

Kuehn²

2023

DCDS

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We propose a mathematical formalism for discrete multi-scale dynamical systems induced by maps which parallels the established geometric singular perturbation theory for continuous-time fast-slow systems. We identify limiting maps corresponding to both 'fast' and 'slow' iteration under the map. A notion of normal hyperbolicity is defined by a spectral gap requirement for the multipliers of the fast limiting map along a critical fixed-point manifold <inline-formula><tex-math id="M1">\begin{document}$ S $\end{document}</tex-math></inline-formula>. We provide a set of Fenichel-like perturbation theorems by reformulating pre-existing results so that they apply near compact, normally hyperbolic submanifolds of <inline-formula><tex-math id="M2">\begin{document}$ S $\end{document}</tex-math></inline-formula>. The persistence of the critical manifold <inline-formula><tex-math id="M3">\begin{document}$ S $\end{document}</tex-math></inline-formula>, local stable/unstable manifolds <inline-formula><tex-math id="M4">\begin{document}$ W^{s/u}_{loc}(S) $\end{document}</tex-math></inline-formula> and foliations of <inline-formula><tex-math id="M5">\begin{document}$ W^{s/u}_{loc}(S) $\end{document}</tex-math></inline-formula> by stable/unstable fibers is described in detail. The practical utility of the resulting discrete geometric singular perturbation theory (DGSPT) is demonstrated in applications. First, we use DGSPT to identify singular geometry corresponding to excitability, relaxation, chaotic and non-chaotic bursting in a map-based neural model. Second, we derive results which relate the geometry and dynamics of fast-slow ODEs with non-trivial time-scale separation and their Euler-discretized counterpart. Finally, we show that fast-slow ODE systems with fast rotation give rise to fast-slow Poincaré maps, the geometry and dynamics of which can be described in detail using DGSPT.

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Section: Regular Fold Dynamicsmentioning

confidence: 99%

“…Regular contact points can be defined analogously to regular contact points in continuous-time GSPT; see [38]…”

Section: Regular Contact Points In Arbitrary Dimensionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Discrete geometric singular perturbation theory

Jelbart¹,

Kuehn²

2023

DCDS

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“…[21,34,36,42,65], and finally for the more general class (1) in [72] and related works, e.g. [8,20,40,47,48,49]. This theory provides a mathematical framework for identifying and analysing the geometry and dynamics of the so-called layer (or fast subsystem) and reduced (or slow subsystem) problems, obtained after taking ε → 0 in (1) on the fast and slow time-scales t and τ = εt respectively.…”

Section: Introductionmentioning

confidence: 99%

Discrete Geometric Singular Perturbation Theory

Jelbart¹,

Kuehn²

2022

Preprint

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We propose a mathematical formalism for discrete multi-scale dynamical systems induced by maps which parallels the established geometric singular perturbation theory for continuous-time fast-slow systems. We identify limiting maps corresponding to both 'fast' and 'slow' iteration under the map. A notion of normal hyperbolicity is defined by a spectral gap requirement for the multipliers of the fast limiting map along a critical fixed-point manifold S. We provide a set of Fenichel-like perturbation theorems by reformulating pre-existing results so that they apply near compact, normally hyperbolic submanifolds of S. The persistence of the critical manifold S, local stable/unstable manifolds W s/u loc (S) and foliations of W s/u loc (S) by stable/unstable fibers is described in detail. The practical utility of the resulting discrete geometric singular perturbation theory (DGSPT) is demonstrated in applications. First, we use DGSPT to identify singular geometry corresponding to excitability, relaxation, chaotic and non-chaotic bursting in a mapbased neural model. Second, we derive results which relate the geometry and dynamics of fastslow ODEs with non-trivial time-scale separation and their Euler-discretized counterpart. Finally, we show that fast-slow ODE systems with fast rotation give rise to fast-slow Poincaré maps, the geometry and dynamics of which can be described in detail using DGSPT.

show abstract