arXiv subjects

Samuel Jelbart

Publications and source records attributed to Samuel Jelbart.

16 recordsLinked to original sources

arXiv2026

Different Singular Limits in a Gene Regulatory Network with Multiple Small Parameters

We consider a planar ODE system from an important class of models for gene regulatory dynamics. The system depends singularly on the steepness parameters $0<\varepsilon_1,\varepsilon_2 \ll 1$ and converges to a piecewise-smooth system as these parameters tend to zero. Unlike previous studies, we do not assume that $\varepsilon_1 = \varepsilon_2$. As a consequence, the dynamics when $(\varepsilon_1, \varepsilon_2) \to (0,0)$ depends upon how the limit is taken. Using a preliminary blow-up in parameter space, we identify three distinct singular limits. We perform a two-parameter bifurcation analysis in each case, and apply multiple geometric blow-ups in variable and parameter space to determine the bifurcation structure and the associated global dynamics. Bogdanov-Takens bifurcations are revealed in two of three cases, and in one case in particular, a regularised visible-invisible two-fold singularity is shown to organise the unfolding of singular bifurcations in the vicinity of canards. Our results show that the qualitative dynamics and overall sensitivity of the system to parameter variation depends on the relative size of the steepness parameters. More generally, the analytical framework developed herein provides a systematic approach to singular perturbation problems with multiple independent small parameters that should apply well beyond gene regulatory network models.

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arXiv2025

Switching, Multiple Time-Scales and Geometric Blow-Up in a Low-Dimensional Gene Regulatory Network

ODE-based models for gene regulatory networks (GRNs) can often be formulated as smooth singular perturbation problems with multiple small parameters, some of which are related to time-scale separation, whereas others are related to 'switching' (proximity to a non-smooth singular limit). This motivates the study of reduced models obtained after (i) quasi-steady state reduction (QSSR), which utilises the time-scale separation, and (ii) piecewise-smooth approximations, which reduce the nonlinearity of the model by viewing highly nonlinear sigmoidal terms as singular perturbations of step functions. We investigate the interplay between the reduction methods (i)-(ii), in the context of a 4-dimensional GRN which has been used as a low-dimensional representative of an important class of (generally high-dimensional) GRN models in the literature. We begin by identifying a region in the small parameter plane for which this problem can be formulated as a smooth singularly perturbed system on a blown-up space, uniformly in the switching parameter. This allows us to apply Fenichel's coordinate-free theorems and obtain a rigorous reduction to a 2-dimensional system, that is a perturbation of the QSSR. Finally, we show that the reduced system features a Hopf bifurcation which does not appear in the QSSR system, due to the influence of higher order terms. Taken together, our findings suggest that the relative size of the small parameters is important for the validity of QSS reductions and the determination of qualitative dynamics in GRN models more generally. Although the focus is on the 4-dimensional GRN, our approach is applicable to higher dimensions.

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arXiv2024

Characterising exchange of stability in scalar reaction-diffusion equations via geometric blow-up

We study the exchange of stability in scalar reaction-diffusion equations which feature a slow passage through transcritical and pitchfork type singularities in the reaction term, using a novel adaptation of the geometric blow-up method. Our results are consistent with known results on bounded spatial domains which were obtained by Butuzov, Nefedov & Schneider using comparison principles like upper and lower solutions in [7], however, from a methodological point of view, the approach is motivated by the analysis of closely related ODE problems using geometric blow-up presented by Krupa & Szmolyan in [34]. After applying the blow-up transformation, we obtain a system of PDEs which can be studied in local coordinate charts. Importantly, the blow-up procedure resolves a spectral degeneracy in which continuous spectrum along the entire negative real axis is 'pushed back' so as to create a spectral gap in the linearisation about particular steady states which arise within the so-called entry and exit charts. This makes it possible to extend slow-type invariant manifolds into and out of a neighbourhood of the singular point using center manifold theory, in a manner which is conceptually analogous to the established approach in the ODE setting. We expect that the approach can be adapted and applied to the study of dynamic bifurcations in PDEs in a wide variety of different contexts.

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arXiv2024

Travelling Waves and Exponential Nonlinearities in the Zeldovich-Frank-Kamenetskii Equation

We prove the existence of a family of travelling wave solutions in a variant of the $\textit{Zeldovich-Frank-Kamenetskii (ZFK) equation}$, a reaction-diffusion equation which models the propagation of planar laminar premixed flames in combustion theory. Our results are valid in an asymptotic regime which corresponds to a reaction with high activation energy, and provide a rigorous and geometrically informative counterpart to formal asymptotic results that have been obtained for similar problems using $\textit{high activation energy asymptotics}$. We also go beyond the existing results by (i) proving smoothness of the minimum wave speed function $\overline c(\epsilon)$, where $0< \epsilon \ll 1$ is the small parameter, and (ii) providing an asymptotic series for a flat slow manifold which plays a role in the construction of travelling wave solutions for non-minimal wave speeds $c > \overline c(\epsilon)$. The analysis is complicated by the presence of an exponential nonlinearity which leads to two different scaling regimes as $\epsilon \to 0$, which we refer to herein as the $\textit{convective-diffusive}$ and $\textit{diffusive-reactive}$ zones. The main idea of the proof is to use the geometric blow-up method to identify and characterise a $(c,\epsilon)$-family of heteroclinic orbits which traverse both of these regimes, and correspond to travelling waves in the original ZFK equation. More generally, our analysis contributes to a growing number of studies which demonstrate the utility of geometric blow-up approaches to the study dynamical systems with singular exponential nonlinearities.

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arXiv2024

Understanding broad-spike oscillations in a model of intracellular calcium dynamics

Oscillations of free intracellular calcium concentration are thought to be important in the control of a wide variety of physiological phenomena, and there is long-standing interest in understanding these oscillations via the investigation of suitable mathematical models. Many of these models have the feature that different variables or terms in the model evolve on very different time-scales, which often results in the accompanying oscillations being temporally complex. Cloete et al [5] constructed an ordinary differential equation model of calcium oscillations in hepatocytes in an attempt to understand the origin of two distinct types of oscillation observed in experiments: narrow-spike oscillations in which rapid spikes of calcium concentration alternate with relatively long periods of quiescence, and broad-spike oscillations in which there is a fast rise in calcium levels followed by a slower decline then a period of quiescence. These two types of oscillation can be observed in the model if a single system parameter is varied but the mathematical mechanisms underlying the different types of oscillations were not explored in detail in [5]. We use ideas from geometric singular perturbation theory to investigate the origin of broad-spike solutions in this model. We find that the analysis is intractable in the full model, but are able to uncover structure in particular singular limits of a related model that point to the origin of the broad-spike solutions.

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arXiv2024

Rate and Bifurcation Induced Transitions in Asymptotically Slow-Fast Systems

This work provides a geometric approach to the study of bifurcation and rate induced transitions in a class of non-autonomous systems referred to herein as $\textit{asymptotically slow-fast systems}$, which may be viewed as 'intermediate' between the (smaller resp. larger) classes of asymptotically autonomous and non-autonomous systems. After showing that the relevant systems can be viewed as singular perturbations of a limiting system with a discontinuity in time, we develop an analytical framework for their analysis based on geometric blow-up techniques. We then provide sufficient conditions for the occurrence of bifurcation and rate induced transitions in low dimensions, as well as sufficient conditions for 'tracking' in arbitrary (finite) dimensions, i.e. the persistence of an attracting and normally hyperbolic manifold through the transitionary regime. The proofs rely on geometric blow-up, a variant of the Melnikov method which applies on non-compact domains, and general invariant manifold theory. The formalism is applicable in arbitrary (finite) dimensions, and for systems with forward and backward attractors characterised by non-trivial (i.e. non-constant) dependence on time. The results are demonstrated for low-dimensional applications.

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arXiv2023

Extending Discrete Geometric Singular Perturbation Theory to Non-Hyperbolic Points

We extend the recently developed discrete geometric singular perturbation theory to the non-normally hyperbolic regime. Our primary tool is the Takens embedding theorem, which provides a means of approximating the dynamics of particular maps with the time-1 map of a formal vector field. First, we show that the so-called reduced map, which governs the slow dynamics near slow manifolds in the normally hyperbolic regime, can be locally approximated by the time-one map of the reduced vector field which appears in continuous-time geometric singular perturbation theory. In the non-normally hyperbolic regime, we show that the dynamics of fast-slow maps with a unipotent linear part can be locally approximated by the time-1 map induced by a fast-slow vector field in the same dimension, which has a nilpotent singularity of the corresponding type. The latter result is used to describe (i) the local dynamics of two-dimensional fast-slow maps with non-normally singularities of regular fold, transcritical and pitchfork type, and (ii) dynamics on a (potentially high dimensional) local center manifold in $n$-dimensional fast-slow maps with regular contact or fold submanifolds of the critical manifold. In general, our results show that the dynamics near a large and important class of singularities in fast-slow maps can be described via the use of formal embedding theorems which allow for their approximation by the time-1 map of a fast-slow vector field featuring a loss of normal hyperbolicity.

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arXiv2023

A Formal Geometric Blow-up Method for Pattern Forming Systems

We extend and apply a recently developed approach to the study of dynamic bifurcations in PDEs based on the geometric blow-up method. We show that this approach, which has so far only been applied to study a dynamic Turing bifurcation in a cubic Swift-Hohenberg equation, can be coupled with a fast-slow extension of the method of multiple scales. This leads to a formal but systematic method, which can be viewed as a fast-slow generalisation of the formal part of classical modulation theory. We demonstrate the utility and versatility of this method by using it to derive modulation equations, i.e. simpler closed form equations which govern the dynamics of the formal approximations near the underlying bifurcation point, in the context of model equations with dynamic bifurcations of (i) Turing, (ii) Hopf, (iii) Turing-Hopf, and (iv) stationary long-wave type. The modulation equations have a familiar form: They are of real Ginzburg-Landau (GL), complex GL, coupled complex GL and Cahn-Hilliard type respectively. In contrast to the modulation equations derived in classical modulation theory, however, they have time-dependent coefficients induced by the slow parameter drift, they depend on spatial and temporal scales which scale in a dependent and non-trivial way, and the geometry of the space in which they are posed is non-trivial due to the blow-up transformation. The formal derivation of the modulation equations provides the first steps toward the rigorous treatment of these challenging problems, which remains for future work.

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arXiv2022

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

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 desingularization 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 Li\'enard 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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arXiv2022

Geometric blow-up of a dynamic Turing instability in the Swift-Hohenberg equation

We present a rigorous analysis of the slow passage through a Turing bifurcation in the Swift-Hohenberg equation using a novel approach based on geometric blow-up. We show that the formally derived multiple scales ansatz which is known from classical modulation theory can be adapted for use in the fast-slow setting, by reformulating it as a blow-up transformation. This leads to dynamically simpler modulation equations posed in the blown-up space, via a formal procedure which directly extends the established approach to the time-dependent setting. The modulation equations take the form of non-autonomous Ginzburg-Landau equations, which can be analysed within the blow-up. The asymptotics of solutions in weighted Sobelev spaces are given in two different cases: (i) A symmetric case featuring a delayed loss of stability, and (ii) A second case in which the symmetry is broken by a source term. In order to characterise the dynamics of the Swift-Hohenberg equation itself we derive rigorous estimates on the error of the dynamic modulation approximation. These estimates are obtained by bounding weak solutions to an evolution equation for the error which is also posed in the blown-up space. Using the error estimates obtained, we are able to infer the asymptotics of a large class of solutions to the dynamic Swift-Hohenberg equation. We provide rigorous asymptotics for solutions in both cases (i) and (ii). We also prove the existence of the delayed loss of stability in the symmetric case (i), and provide a lower bound for the delay time.

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arXiv2022

Discrete Geometric Singular Perturbation Theory

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 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 Poincar\'e maps, the geometry and dynamics of which can be described in detail using DGSPT.

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arXiv2021

Process-Oriented Geometric Singular Perturbation Theory and Calcium Dynamics

Phenomena in chemistry, biology and neuroscience are often modelled using ordinary differential equations (ODEs) in which the right-hand-side is comprised of terms which correspond to individual 'processes' or 'fluxes'. Frequently, these ODEs are characterised by multiple time-scale phenomena due to order of magnitude differences between contributing processes and the presence of switching, i.e., dominance or sub-dominance of particular terms as a function of state variables. We outline a heuristic procedure for the identification of small parameters in ODE models of this kind, with a particular emphasis on the identification of small parameters relating to switching behaviours. This procedure is outlined informally in generality, and applied in detail to a model for intracellular calcium dynamics characterised by switching and multiple (more than two) time-scale dynamics. A total of five small parameters are identified, and related to a single perturbation parameter by a polynomial scaling law based on order of magnitude comparisons. The resulting singular perturbation problem has a time-scale separation which depends on the region of state space. We prove the existence and uniqueness of stable relaxation oscillations with three distinct time-scales using a coordinate-independent formulation of GSPT in combination with the blow-up method. We also provide an estimate for the period of the oscillations, and consider a number of possibilities for their onset under parameter variation.

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arXiv2021

Singularly Perturbed Boundary-Equilibrium Bifurcations

Boundary equilibria bifurcation (BEB) arises in piecewise-smooth systems when an equilibrium collides with a discontinuity set under parameter variation. Singularly perturbed BEB refers to a bifurcation arising in singular perturbation problems which limit as some $\epsilon \to 0$ to piecewise-smooth (PWS) systems which undergo a BEB. This work completes a classification for codimension-1 singularly perturbed BEB in the plane initiated by the present authors in [19], using a combination of tools from PWS theory, geometric singular perturbation theory (GSPT) and a method of geometric desingularization known as blow-up. After deriving a local normal form capable of generating all 12 singularly perturbed BEBs, we describe the unfolding in each case. Detailed quantitative results on saddle-node, Andronov-Hopf, homoclinic and codimension-2 Bogdanov-Takens bifurcations involved in the unfoldings and classification are presented. Each bifurcation is singular in the sense that it occurs within a domain which shrinks to zero as $\epsilon \to 0$ at a rate determined by the rate at which the system loses smoothness. Detailed asymptotics for a distinguished homoclinic connection which forms the boundary between two singularly perturbed BEBs in parameter space are also given. Finally, we describe the explosive onset of oscillations arising in the unfolding of a particular singularly perturbed boundary-node (BN) bifurcation. We prove the existence of the oscillations as perturbations of PWS cycles, and derive a growth rate which is polynomial in $\epsilon$ and dependent on the rate at which the system loses smoothness. For all the results presented herein, corresponding results for regularized PWS systems are obtained via the limit $\epsilon \to 0$.

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arXiv2020

Singularly Perturbed Boundary-Focus Bifurcations

We consider smooth systems limiting as $\epsilon \to 0$ to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with $0 < \epsilon \ll 1$ using a combination of geometric singular perturbation theory and blow-up. We show that the type of BF bifurcation in the PWS system determines the bifurcation structure for the smooth system within an $\epsilon-$dependent domain which shrinks to zero as $\epsilon \to 0$, identifying a supercritical Andronov-Hopf bifurcation in one case, and a supercritical Bogdanov-Takens bifurcation in two other cases. We also show that PWS cycles associated with BF bifurcations persist as relaxation cycles in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation cycles to regular cycles within the $\epsilon-$dependent domain described above. Our results are applied to models for Gause predator-prey interaction and mechanical oscillation subject to friction.

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arXiv2019

Singularly Perturbed Oscillators with Exponential Nonlinearities

Singular exponential nonlinearities of the form $e^{h(x)\epsilon^{-1}}$ with $\epsilon>0$ small occur in many different applications. These terms have essential singularities for $\epsilon=0$ leading to very different behaviour depending on the sign of $h$. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $\epsilon\rightarrow 0$ limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in Kristiansen [20], and prove - for both systems - existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.

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arXiv2019

Two-Stroke Relaxation Oscillators

Two-stroke relaxation oscillations consist of two distinct phases per cycle - one slow and one fast - which distinguishes them from the well-known van der Pol-type 'four-stroke' relaxation oscillations. This type of oscillation can be found in singular perturbation problems in non-standard form, where the slow-fast timescale splitting is not necessarily reflected in a slow-fast variable splitting. The existing literature on such non-standard problems has developed primarily through applications - we compliment this by providing a general framework for the application of geometric singular perturbation theory in this non-standard setting and illustrate its applicability by proving existence and uniqueness results on a general class of two-stroke relaxation oscillators. We apply this non-standard geometric singular perturbation toolbox to a collection of examples arising in the dynamics of nonlinear transistors and models for mechanical oscillators with friction.

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