29 citations · 60 across the 14 of their papers we have counts for
16 papers
Different Singular Limits in a Gene Regulatory Network with Multiple Small Parameters
Lukas Baumgartner, Samuel Jelbart
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,\varepsil…
Switching, Multiple Time-Scales and Geometric Blow-Up in a Low-Dimensional Gene Regulatory Network
Samuel Jelbart, Kristian Uldall Kristiansen, Peter Szmolyan
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 t…
Characterising exchange of stability in scalar reaction-diffusion equations via geometric blow-up
Samuel Jelbart, Christian Kuehn, Alejandro Martínez Sánchez
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,…
Travelling Waves and Exponential Nonlinearities in the Zeldovich-Frank-Kamenetskii Equation
Samuel Jelbart, Kristian Uldall Kristiansen, Peter Szmolyan
We prove the existence of a family of travelling wave solutions in a variant of the , a reaction-diffusion equation which model…
Understanding broad-spike oscillations in a model of intracellular calcium dynamics
Behnaz Rahmani, Samuel Jelbart, Vivien Kirk +1
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 interes…
Rate and Bifurcation Induced Transitions in Asymptotically Slow-Fast Systems
Samuel Jelbart
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…
Extending Discrete Geometric Singular Perturbation Theory to Non-Hyperbolic Points
Samuel Jelbart, Christian Kuehn
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 prov…
A Formal Geometric Blow-up Method for Pattern Forming Systems
Samuel Jelbart, Christian Kuehn
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 o…
Geometric Blow-Up for Folded Limit Cycle Manifolds in Three Time-Scale Systems
Samuel Jelbart, Christian Kuehn, Sara-Viola Kuntz
Geometric singular perturbation theory provides a powerful mathematical framework for the analysis of 'stationary' multiple time-scale systems which possess a critical manifold, i.…
Geometric blow-up of a dynamic Turing instability in the Swift-Hohenberg equation
Felix Hummel, Samuel Jelbart, Christian Kuehn
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…
Discrete Geometric Singular Perturbation Theory
Samuel Jelbart, Christian Kuehn
We propose a mathematical formalism for discrete multi-scale dynamical systems induced by maps which parallels the established geometric singular perturbation theory for continuous…
Process-Oriented Geometric Singular Perturbation Theory and Calcium Dynamics
Samuel Jelbart, Nathan Pages, Vivien Kirk +2
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…
Singularly Perturbed Boundary-Equilibrium Bifurcations
Samuel Jelbart, Kristian Uldall Kristiansen, Martin Wechselberger
Boundary equilibria bifurcation (BEB) arises in piecewise-smooth systems when an equilibrium collides with a discontinuity set under parameter variation. Singularly perturbed BEB r…
Singularly Perturbed Boundary-Focus Bifurcations
Samuel Jelbart, Kristian Uldall Kristiansen, Martin Wechselberger
We consider smooth systems limiting as to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the…
Singularly Perturbed Oscillators with Exponential Nonlinearities
Samuel Jelbart, Kristian Uldall Kristiansen, Peter Szmolyan +1
Singular exponential nonlinearities of the form with small occur in many different applications. These terms have essential singularities for leading t…
Two-Stroke Relaxation Oscillators
Samuel Jelbart, Martin Wechselberger
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' re…