Singularly perturbed boundary-equilibrium bifurcations

Jelbart, Samuel; Kristiansen, Kristian; Wechselberger, Martin

doi:10.1088/1361-6544/ac23b8

Cited by 13 publications

(16 citation statements)

References 37 publications

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“…Under the assumption 2, it follows that (12) gives a smooth vector-field V on (x, r, (ȳ, ¯ )) ∈ R n × [0, ∞) × S 1 by pullback of the vector-field associated with (11). In fact, due to the multiplication by , V has ¯ as a common factor.…”

Section: 1mentioning

confidence: 99%

“…Firstly, we work in the extended space (x, y, ) by adding the trivial equation ˙ = 0. At the same time, to ensure that = 0 is well-defined, we consider this extended system in terms of a fast time: (11) x = X z, φ y , y = Y z, φ y , = 0.…”

Section: 1mentioning

confidence: 99%

“…The blowup approach has also been used to study bifurcations of systems of the form (3), including boundary equilibrium bifurcations where equilibria of either Z ± collide with Σ upon parameter variation [11,12]. For this, the blowup transformation ( 12) is combined with additional blowup transformations in order to deal with lack of hyperbolicity.…”

Section: 1mentioning

confidence: 99%

“…In light of this, Section 2.1 suggests that we should consider (14) with respect to an even faster time-scale, corresponding to multiplying the right hand side by α again. But notice, despite the similarities, there is also a fundamental difference between ( 14) and (11) insofar that the discontinuity set of ( 14) for , α → 0 is y = 0, x ∈ Σ, p ∈ R, but the discontinuity only enters the p-equation. To avoid too many multiplications and subsequent divisions by the same quantities, we will therefore proceed more ad hoc in the following; in fact, the analysis will show that it is only necessary to multiply the right hand side of ( 14) by in order gain smoothness.…”

Section: 2mentioning

confidence: 99%

“…3. These results, together with [10,11,12,17] working on similar systems, were obtained by adapting methods from Geometric Singular Perturbation Theory [13,7]. In particular, these references use a modification of the blowup method [6,18] to gain smoothness of systems of the form (3).…”

Section: Introductionmentioning

confidence: 96%

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Blowup analysis of a hysteresis model based upon singular perturbations

Kristiansen¹

2022

Preprint

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In this paper, we provide a geometric analysis of a new hysteresis model that is based upon singular perturbations. Here hysteresis refers to a type of regularization of piecewise smooth differential equations where the past of a trajectory, in a small neighborhood of the discontinuity set, determines the vector-field at present. In fact, in the limit where the neighborhood of the discontinuity becomes smaller, hysteresis converges in an appropriate sense to Filippov's sliding vector-field. Recently, however,in [2] it has been shown that hysteresis, in contrast to regularization through smoothing, leads to chaos in the regularization of grazing bifurcations, even in two dimensions. The hysteresis model we analyze in the present paper -which was developed by Bonet et al [3] as an attempt to unify different regularizations of piecewise smooth systems -involves two singular perturbation parameters and includes a combination of slow-fast and nonsmooth effects. The description of this model is therefore -from the perspective of singular perturbation theory -challenging, even in two dimensions. In this paper, using blowup as our main technical tool, we present a detailed geometric description of the dynamics. Specifically, this description allows us to prove existence of an invariant cylinder carrying fast dynamics in the azimuthal direction and a slow drift in the axial direction. We find that the slow drift is given by Filippov's sliding to leading order. In the case of grazing, we identify two important parameter regimes that relate the model to smoothing and hysteresis. In particular, in one of these regimes we prove existence of chaotic dynamics through a horseshoe obtained by a folded saddle and a novel return mechanism.

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Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

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Blowup analysis of a hysteresis model based upon singular perturbations

Kristiansen¹

2022

Preprint

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A Review of Multiple-Time-Scale Dynamics: Fundamental Phenomena and Mathematical Methods

Kristiansen

2023

Mathematics Online First Collections

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In this paper, we review the most fundamental phenomena in multipletime scale systems as well as the Geometric Singular Perturbation Theory (GSPT) that can be used to analyze such systems. We put special emphasis on loss of normal hyperbolicity and illustrate how it relates to the different phenomena. Moreover, by working on lower dimensional model problems, we review the main technical tool of GSPT, the blowup method, and demonstrate -in line with recent research in this area -how it may provide detailed insight into problems of this kind.

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Blowup Analysis of a Hysteresis Model Based Upon Singular Perturbations

Kristiansen

2023

J Nonlinear Sci

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In this paper, we provide a geometric analysis of a new hysteresis model that is based upon singular perturbations. Here hysteresis refers to a type of regularization of piecewise smooth differential equations where the past of a trajectory, in a small neighborhood of the discontinuity set, determines the vector-field at present. In fact, in the limit where the neighborhood of the discontinuity vanishes, hysteresis converges in an appropriate sense to Filippov’s sliding vector-field. Recently (2022), however, Bonet and Seara showed that hysteresis, in contrast to regularization through smoothing, leads to chaos in the regularization of grazing bifurcations, even in two dimensions. The hysteresis model we analyze in the present paper—which was developed by Bonet et al in a paper from 2017 as an attempt to unify different regularizations of piecewise smooth systems—involves two singular perturbation parameters and includes a combination of slow–fast and nonsmooth effects. The description of this model is therefore—from the perspective of singular perturbation theory—challenging, even in two dimensions. Using blowup as our main technical tool, we prove existence of an invariant cylinder carrying fast dynamics in the azimuthal direction and a slow drift in the axial direction. We find that the slow drift is given by Filippov’s sliding vector-field to leading order. Moreover, in the case of grazing, we identify two important parameter regimes that relate the model to smoothing (through a saddle-node bifurcation of limit cycles) and hysteresis (through chaotic dynamics, due to a folded saddle and a novel return mechanism).

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Singularly perturbed boundary-equilibrium bifurcations

Cited by 13 publications

References 37 publications

Blowup analysis of a hysteresis model based upon singular perturbations

Blowup analysis of a hysteresis model based upon singular perturbations

A Review of Multiple-Time-Scale Dynamics: Fundamental Phenomena and Mathematical Methods

Blowup Analysis of a Hysteresis Model Based Upon Singular Perturbations

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