Coordinate-independent singular perturbation reduction for systems with three time scales

Kruff, Niclas; Walcher, Sebastian

doi:10.3934/mbe.2019255

Cited by 14 publications

(17 citation statements)

References 19 publications

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“…Our first result generalizes [40, Proposition 1 (ii)]. The proof is straightforward by induction using the argument in [40,Lemma 3] and its proof.…”

Section: Verification Of Hyperbolic Attractivitysupporting

confidence: 58%

“…Our next result allows a suitable first-order formulation without quantifier alternation. Its proof combines [40,Lemma 3] with our Lemma 3. . .…”

Section: Verification Of Hyperbolic Attractivitymentioning

confidence: 94%

“…Our definition of hyperbolic attractivity M k−1 M k refers to the eigenvalues of the Jacobians of the f * k , which cannot be directly obtained from the Jacobians of the f k [11,12]. Generalizing work on systems with three time scales [40], we take in this section a linear algebra approach to obtain the relevant eigenvalues without computing the f * k . To start with, recall the well-known Hurwitz criterion [36]:…”

Section: Verification Of Hyperbolic Attractivitymentioning

confidence: 99%

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Algorithmic Reduction of Biological Networks with Multiple Time Scales

et al. 2021

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We present a symbolic algorithmic approach that allows to compute invariant manifolds and corresponding reduced systems for differential equations modeling biological networks which comprise chemical reaction networks for cellular biochemistry, and compartmental models for pharmacology, epidemiology and ecology. Multiple time scales of a given network are obtained by scaling, based on tropical geometry. Our reduction is mathematically justified within a singular perturbation setting. The existence of invariant manifolds is subject to hyperbolicity conditions, for which we propose an algorithmic test based on Hurwitz criteria. We finally obtain a sequence of nested invariant manifolds and respective reduced systems on those manifolds. Our theoretical results are generally accompanied by rigorous algorithmic descriptions suitable for direct implementation based on existing off-the-shelf software systems, specifically symbolic computation libraries and Satisfiability Modulo Theories solvers. We present computational examples taken from the well-known BioModels database using our own prototypical implementations.

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“…Our first result generalizes [40, Proposition 1 (ii)]. The proof is straightforward by induction using the argument in [40,Lemma 3] and its proof.…”

Section: Verification Of Hyperbolic Attractivitysupporting

confidence: 58%

“…Our next result allows a suitable first-order formulation without quantifier alternation. Its proof combines [40,Lemma 3] with our Lemma 3. . .…”

Section: Verification Of Hyperbolic Attractivitymentioning

confidence: 94%

Section: Verification Of Hyperbolic Attractivitymentioning

confidence: 99%

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Algorithmic Reduction of Biological Networks with Multiple Time Scales

et al. 2021

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“…We emphasise that here, the presence of two singular perturbation parameters indicates the presence of three timescales, however, which distinguishes the relaxation oscillations observed for 0 < v 0 < v ss from vdP-type relaxation oscillations, despite the 'four strokes'. It is also important to note that in the presence of multiple small parameters, the number of timescales in general systems (2.9) can exceed the number of variables [70]. This is exemplified by system (3.43); see also [59], where the authors identify a nonstandard relaxation oscillation in a three timescale model for glycolitic oscillations in the plane.…”

Section: Three Timescales Stick-slip and Ducksmentioning

confidence: 99%

“…70) has the entire plane { = 0} as a manifold of equilibria, containing the degenerate subset Σ × {0} = {(x, 0, 0) : x ∈ R} ⊂ R 3 (i.e. the switching manifold) along which the system is non-smooth.…”

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confidence: 99%

Beyond Slow-Fast: Relaxation Oscillations in Singularly Perturbed Nonsmooth Systems

Jelbart

2021

Bull. Aust. Math. Soc.

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Though it is surely a cliché to say that a PhD thesis is more than the work of an individual, I feel compelled to reiterate the point here. It is true that this thesis is a product of countless hours of work on my part, however, it is also the result of close supervision by my advisor and co-authors, discussions with experts and other students, and the unwavering support of my friends and family.First and foremost, I would like to express my gratitude to my PhD supervisor Prof. Martin Wechselberger. Martin, thank you for the countless hours devoted to me and my work over the past four and a half years (let's not forget the honours year!). Thank you for your patience, and for keeping me on track, which cannot always have been easy. Among other things, you have taught me the importance of keeping 'the big picture' in mind. Above all, thank you for instilling and nurturing my passion for mathematics, which has only grown in these last few years. For this I am forever grateful.Secondly, I would like thank A/Prof. Kristian Uldall Kristiansen, who has in many ways been like a second, unofficial supervisor for me. Thank you for your time and support during my visit to DTU, as well as the countless emails and Skype calls made in an effort to guide me through the details. You have taught me the importance of rigour in mathematics, and inspired me to take on challenges I would not have otherwise thought possible. Truly, this thesis would not have been possible without your guidance and support.I would also like to thank Prof. Peter Szmolyan and Dr. Ilona Kosiuk. Firstly, I thank Peter for valuable feedback and contribution as a co-author on some of the material presented in this thesis. Secondly, I thank both Peter and Ilona for their earlier work which inspired much of this thesis.There are many who deserve thanks for contributing to my mathematical education more generally. I would like to thank A/Prof. Vivien Kirk, Prof. James Sneyd and Dr. Nathan Pages in particular for their hospitality during my visit to Auckland University. This visit (and the following months) taught me the value of collaborative work in mathematics. I would also like to thank several people 'from the office' for helpful discussions, DIY reading groups and the basic support required to get through a PhD.

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Geometric Blow-Up for Folded Limit Cycle Manifolds in Three Time-Scale Systems

Jelbart,

Kuehn,

Kuntz

2023

J Nonlinear Sci

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Geometric singular perturbation theory provides a powerful mathematical framework for the analysis of ‘stationary’ multiple time-scale systems which possess a critical manifold, i.e. a smooth manifold of steady states for the limiting fast subsystem, particularly when combined with a method of desingularisation known as blow-up. The theory for ‘oscillatory’ multiple time-scale systems which possess a limit cycle manifold instead of (or in addition to) a critical manifold is less developed, particularly in the non-normally hyperbolic regime. We use the blow-up method to analyse the global oscillatory transition near a regular folded limit cycle manifold in a class of three time-scale ‘semi-oscillatory’ systems with two small parameters. The systems considered behave like oscillatory systems as the smallest perturbation parameter tends to zero, and stationary systems as both perturbation parameters tend to zero. The additional time-scale structure is crucial for the applicability of the blow-up method, which cannot be applied directly to the two time-scale oscillatory counterpart of the problem. Our methods allow us to describe the asymptotics and strong contractivity of all solutions which traverse a neighbourhood of the global singularity. Our main results cover a range of different cases with respect to the relative time-scale of the angular dynamics and the parameter drift. We demonstrate the applicability of our results for systems with periodic forcing in the slow equation, in particular for a class of Liénard equations. Finally, we consider a toy model used to study tipping phenomena in climate systems with periodic forcing in the fast equation, which violates the conditions of our main results, in order to demonstrate the applicability of classical (two time-scale) theory for problems of this kind.

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Coordinate-independent singular perturbation reduction for systems with three time scales

Cited by 14 publications

References 19 publications

Algorithmic Reduction of Biological Networks with Multiple Time Scales

Algorithmic Reduction of Biological Networks with Multiple Time Scales

Beyond Slow-Fast: Relaxation Oscillations in Singularly Perturbed Nonsmooth Systems

Geometric Blow-Up for Folded Limit Cycle Manifolds in Three Time-Scale Systems

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