Geometry of Mixed-Mode Oscillations in the 3-D Autocatalator

1998 ◽  
Vol 08 (03) ◽  
pp. 505-519 ◽  
Author(s):  
Alexandra Milik ◽  
Peter Szmolyan ◽  
Helwig Löffelmann ◽  
Eduard Gröller
Keyword(s):  
Mixed Mode ◽  
Basic Mechanism ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Small Oscillations ◽  
One Dimensional ◽  
Mixed Mode Oscillations ◽  
Geometric Explanation ◽  
Natural Way ◽  
Reduced Problem

We present a geometric explanation of a basic mechanism generating mixed-mode oscillations in a prototypical simple model of a chemical oscillator. Our approach is based on geometric singular perturbation theory and canard solutions. We explain how the small oscillations are generated near a special point, which is classified as a folded saddle-node for the reduced problem. The canard solution passing through this point separates small oscillations from large relaxation type oscillations. This allows to define a one-dimensional return map in a natural way. This bimodal map is capable of explaining the observed bifurcation sequence convincingly.

Mixed-mode oscillations for slow-fast perturbed systems

2021 ◽  
Author(s):  
Yaru Liu ◽  
Shenquan Liu ◽  
Bo Lu ◽  
Jürgen Kurths
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Mixed Mode ◽  
Dynamic Properties ◽  
Differential Algebraic Equations ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Mixed Mode Oscillations ◽  
Average Analysis ◽  
Perturbed Systems

Abstract This article concerns the dynamics of mixed-mode oscillations (MMOs) emerging from the calcium-based inner hair cells (IHCs) model in the auditory cortex. The paper captures the MMOs generation mechanism based on the geometric singular perturbation theory (GSPT) after exploiting the average analysis for reducing the full model. Our analysis also finds that the critical manifold and folded surface are central to the mechanism of the existence of MMOs at the folded saddle for the perturbed system. The system parameters, such like the maximal calcium channels conductance, controls the firing patterns, and many new oscillations occur for the IHCs model. Tentatively, we conduct dynamic analysis combined with dynamic method based on GSPT by giving slow-fast analysis for the singular perturbed models and bifurcation analysis. In particular, we explore the two-slow-two-fast and three-slow-one-fast IHCs perturbed systems with layer and reduced problems so that differential-algebraic equations are obtained. This paper reveals the underlying dynamic properties of perturbed systems under singular perturbation theory.

scholarly journals Controlling Canard Cycles

2021 ◽  
Author(s):  
Hildeberto Jardón-Kojakhmetov ◽  
Christian Kuehn
Keyword(s):  
Mixed Mode ◽  
Blow Up ◽  
Van Der Pol Oscillator ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Van Der Pol ◽  
Critical Manifold ◽  
Mixed Mode Oscillations ◽  
Special Solutions ◽  
And Control

AbstractCanard cycles are periodic orbits that appear as special solutions of fast-slow systems (or singularly perturbed ordinary differential equations). It is well known that canard cycles are difficult to detect, hard to reproduce numerically, and that they are sensible to exponentially small changes in parameters. In this paper, we combine techniques from geometric singular perturbation theory, the blow-up method, and control theory, to design controllers that stabilize canard cycles of planar fast-slow systems with a folded critical manifold. As an application, we propose a controller that produces stable mixed-mode oscillations in the van der Pol oscillator.

A Surface of Heteroclinic Connections Between Two Saddle Slow Manifolds in the Olsen Model

2020 ◽  
Vol 30 (16) ◽  
pp. 2030048
Author(s):  
Elle Musoke ◽  
Bernd Krauskopf ◽  
Hinke M. Osinga
Keyword(s):  
Periodic Orbits ◽  
Slow Variable ◽  
Connecting Orbits ◽  
Stable And Unstable Manifolds ◽  
Small Oscillations ◽  
One Dimensional ◽  
Mixed Mode Oscillations ◽  
Slow Manifolds ◽  
Connecting Orbit ◽  
Crucial Ingredient

The Olsen model for the biochemical peroxidase-oxidase reaction has a parameter regime where one of its four variables evolves much slower than the other three. It is characterized by the existence of periodic orbits along which a large oscillation is followed by many much smaller oscillations before the process repeats. We are concerned here with a crucial ingredient for such mixed-mode oscillations (MMOs) in the Olsen model: a surface of connecting orbits that is followed closely by the MMO periodic orbit during its global, large-amplitude transition back to another onset of small oscillations. Importantly, orbits on this surface connect two one-dimensional saddle slow manifolds, which exist near curves of equilibria of the limit where the slow variable is frozen and acts as a parameter of the so-called fast subsystem. We present a numerical method, based on formulating suitable boundary value problems, to compute such a surface of connecting orbits. It involves a number of steps to compute the slow manifolds, certain submanifolds of their stable and unstable manifolds and, finally, a first connecting orbit that is then used to sweep out the surface by continuation. If it exists, such a surface of connecting orbits between two one-dimensional saddle slow manifolds is robust under parameter variations. We compute and visualize it in the Olsen model and show how this surface organizes the global return mechanism of MMO periodic orbits from the end of small oscillations back to a region of phase space where they start again.

scholarly journals Geometric analysis of mixed-mode oscillations in a model of electrical activity in human beta-cells

2021 ◽  
Author(s):  
Simone Battaglin ◽  
Morten Gram Pedersen
Keyword(s):  
Hopf Bifurcation ◽  
Electrical Activity ◽  
Beta Cells ◽  
Pancreatic Beta Cells ◽  
Mixed Mode ◽  
Original Model ◽  
Biophysical Model ◽  
Reduced Model ◽  
Geometric Singular Perturbation Theory ◽  
Mixed Mode Oscillations

AbstractHuman pancreatic beta-cells may exhibit complex mixed-mode oscillatory electrical activity, which underlies insulin secretion. A recent biophysical model of human beta-cell electrophysiology can simulate such bursting behavior, but a mathematical understanding of the model’s dynamics is still lacking. Here we exploit time-scale separation to simplify the original model to a simpler three-dimensional model that retains the behavior of the original model and allows us to apply geometric singular perturbation theory to investigate the origin of mixed-mode oscillations. Changing a parameter modeling the maximal conductance of a potassium current, we find that the reduced model possesses a singular Hopf bifurcation that results in small-amplitude oscillations, which go through a period-doubling sequence and chaos until the birth of a large-scale return mechanism and bursting dynamics. The theory of folded node singularities provide insight into the bursting dynamics further away from the singular Hopf bifurcation and the eventual transition to simple spiking activity. Numerical simulations confirm that the insight obtained from the analysis of the reduced model can be lifted back to the original model.

scholarly journals Geometric Analysis of Mixed-mode Oscillations in a Model of Electrical Activity in Human Beta-cells

2021 ◽  
Author(s):  
Simone Battaglin ◽  
Morten Gram Pedersen
Keyword(s):  
Hopf Bifurcation ◽  
Electrical Activity ◽  
Beta Cells ◽  
Pancreatic Beta Cells ◽  
Mixed Mode ◽  
Original Model ◽  
Biophysical Model ◽  
Reduced Model ◽  
Geometric Singular Perturbation Theory ◽  
Mixed Mode Oscillations

Abstract Human pancreatic beta-cells may exhibit complex mixed-mode oscillatory electrical activity, which underlies insulin secretion. A recent biophysical model of human beta-cell electrophysiology can simulate such bursting behavior, but a mathematical understanding of the model's dynamics is still lacking. Here we exploit time-scale separation to simplify the original model to a simpler three-dimensional model that retains the behavior of the original model and allows us to apply geometric singular perturbation theory to investigate the origin of mixed-mode oscillations. Changing a parameter modeling the maximal conductance of a potassium current, we nd that the reduced model possesses a singular Hopf bifurcation that results in small-amplitude oscillations, which go through a period-doubling sequence and chaos until the birth of a large-scale return mechanism and bursting dynamics. The theory of folded node singularities provide insight into the bursting dynamics further away from the singular Hopf bifurcation and the eventual transition to simple spiking activity. Numerical simulations confirm that the insight obtained from the analysis of the reduced model can be lifted back to the original model.

scholarly journals Propagating fronts for a viscous Hamer-type system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Giada Cianfarani Carnevale ◽  
Corrado Lattanzio ◽  
Corrado Mascia
Keyword(s):  
Perturbation Theory ◽  
Elliptic Equation ◽  
Burgers Equation ◽  
Type System ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Radiation Hydrodynamics ◽  
One Dimensional ◽  
Small Viscosity ◽  
Viscous Conservation Law

<p style='text-indent:20px;'>Motivated by radiation hydrodynamics, we analyse a <inline-formula><tex-math id="M1">\begin{document}$ 2\times2 $\end{document}</tex-math></inline-formula> system consisting of a one-dimensional viscous conservation law with strictly convex flux –the viscous Burgers' equation being a paradigmatic example– coupled with an elliptic equation, named <b>viscous Hamer-type system</b>. In the regime of small viscosity and for large shocks, namely when the profile of the corresponding underlying inviscid model undergoes a discontinuity –usually called <i>sub-shock</i>– it is proved the existence of a smooth propagating front, regularising the jump of the corresponding inviscid equation. The proof is based on <i>Geometric Singular Perturbation Theory</i> (GSPT) as introduced in the pioneering work of Fenichel [<xref ref-type="bibr" rid="b5">5</xref>] and subsequently developed by Szmolyan [<xref ref-type="bibr" rid="b21">21</xref>]. In addition, the case of small shocks and large viscosity is also addressed via a standard bifurcation argument.</p>

scholarly journals Complexity of a Peroxidase-Oxidase Reaction Model

2021 ◽  
Author(s):  
Jason Gallas ◽  
Marcus Hauser ◽  
Lars Folke Olsen
Keyword(s):  
Chemical Reaction ◽  
Mixed Mode ◽  
Period Doubling ◽  
Reaction Model ◽  
Chaotic Behaviour ◽  
Mixed Mode Oscillations ◽  
Oscillating Reaction ◽  
Oxidase Reaction

The peroxidase-oxidase oscillating reaction was the first (bio)chemical reaction to show chaotic behaviour. The reaction is rich in bifurcation scenarios, from period-doubling to peak-adding mixed mode oscillations. Here, we study...

Bursting patterns and mixed-mode oscillations in reduced Purkinje model

2018 ◽  
Vol 32 (05) ◽  
pp. 1850043 ◽  
Author(s):  
Feibiao Zhan ◽  
Shenquan Liu ◽  
Jing Wang ◽  
Bo Lu
Keyword(s):  
Purkinje Cell ◽  
Mixed Mode ◽  
Cell Model ◽  
Dynamical Behavior ◽  
Characteristic Index ◽  
Mixed Mode Oscillations ◽  
Codimension One ◽  
Bifurcation Phenomenon ◽  
Lyapunov Coefficient ◽  
Potential Link

Bursting discharge is a ubiquitous behavior in neurons, and abundant bursting patterns imply many physiological information. There exists a closely potential link between bifurcation phenomenon and the number of spikes per burst as well as mixed-mode oscillations (MMOs). In this paper, we have mainly explored the dynamical behavior of the reduced Purkinje cell and the existence of MMOs. First, we adopted the codimension-one bifurcation to illustrate the generation mechanism of bursting in the reduced Purkinje cell model via slow–fast dynamics analysis and demonstrate the process of spike-adding. Furthermore, we have computed the first Lyapunov coefficient of Hopf bifurcation to determine whether it is subcritical or supercritical and depicted the diagrams of inter-spike intervals (ISIs) to examine the chaos. Moreover, the bifurcation diagram near the cusp point is obtained by making the codimension-two bifurcation analysis for the fast subsystem. Finally, we have a discussion on mixed-mode oscillations and it is further investigated using the characteristic index that is Devil’s staircase.

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