Relaxation oscillation and canard explosion

From MaRDI portal
Publication:5945081

DOI10.1006/jdeq.2000.3929zbMath0994.34032OpenAlexW2089995439MaRDI QIDQ5945081

Peter Szmolyan, Martin Krupa

Publication date: 13 October 2002

Published in: Journal of Differential Equations (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1006/jdeq.2000.3929



Related Items

Slow-fast dynamics in the truss core sandwich plate under excitations with high and low frequencies, Box dimension and cyclicity of Canard cycles, Neural excitability and singular bifurcations, Canard solutions in neural mass models: consequences on critical regimes, Viscous singular shock profiles for a system of conservation laws modeling two-phase flow, Chemical oscillators synchronized via an active oscillating medium: dynamics and phase approximation model, Spiking resonances in models with the same slow resonant and fast amplifying currents but different subthreshold dynamic properties, Canards and mixed-mode oscillations in a forest pest model, Multimodal regimes in a compartmental model of the dopamine neuron, Canard explosion in delay differential equations, Fractal analysis of canard cycles with two breaking parameters and applications, Singular perturbation methods for slow-fast dynamics, Bifurcations of mixed-mode oscillations in a stellate cell model, Abelian integrals and non-generic turning points, Modelling, singular perturbation and bifurcation analyses of bitrophic food chains, Phase transition of oscillators and travelling waves in a class of relaxation systems, Nonlinear sliding of discontinuous vector fields and singular perturbation, Controlling Canard cycles, A singular approach to discontinuous vector fields on the plane, Regular and slow-fast codimension 4 saddle-node bifurcations, A Lin's method approach for detecting all canard orbits arising from a folded node, Geometric desingularization of degenerate singularities in the presence of fast rotation: A new proof of known results for slow passage through Hopf bifurcations, Stability of traveling pulses with oscillatory tails in the FitzHugh-Nagumo system, Circuits with oscillatory hierarchical Farey sequences and fractal properties, The asymptotic critical wave speed in a family of scalar reaction-diffusion equations, Singularly perturbed oscillators with exponential nonlinearities, Coexisting two canards and their breakdown into chaos in the Van der pol oscillator under weak periodic perturbation, Linearly implicit Liénard systems, Inflection, canards and excitability threshold in neuronal models, Mixed-mode oscillations in a modified Chua's circuit, Asymptotic expansions for a family of non-generic canards using parametric representation, Multimodal oscillations in systems with strong contraction, Canard-induced complex oscillations in an excitatory network, Synchronization of weakly coupled canard oscillators, Canard phenomenon for an SIS epidemic model with nonlinear incidence, Slow-fast Bogdanov-Takens bifurcations, Extending slow manifold near generic transcritical canard point, Hunting French ducks in a noisy environment, Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales, Dynamical analysis and simulation of a 2-dimensional disease model with convex incidence, The role of slow system dynamics in predicting the degeneracy of slow invariant manifolds: the case of vdP relaxation-oscillations, Slow passage through canard explosion and mixed-mode oscillations in the forced Van der Pol's equation, The dynamics of ensemble of neuron-like elements with excitatory couplings, Dynamics and geometric desingularization of the multiple time scale FitzHugh Nagumo Rinzel model with fold singularity, Relaxation oscillations and dynamical properties in two time-delay slow-fast modified Leslie-Gower models, Stabilization of a class of slow-fast control systems at non-hyperbolic points, Multiple timescales, mixed mode oscillations and canards in models of intracellular calcium dynamics, Slow divergence integral and its application to classical Liénard equations of degree 5, Frequency control of singularly perturbed forced Duffing's oscillator, Relaxation oscillations in a slow-fast modified Leslie-Gower model, Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations, Slow divergence integral on a Möbius band, Asymptotic limit-cycle analysis of oscillating chemical reactions, Sector-delayed-Hopf-type mixed-mode oscillations in a prototypical three-time-scale model, Formal normal form of \(A_k\) slow-fast systems, Oscillations and pattern formation in a slow-fast prey-predator system, Relaxation oscillations and canard explosion in a predator-prey system of Holling and Leslie types, Regularization around a generic codimension one fold-fold singularity, Analytical insights on theta-gamma coupled neural oscillators, Excitable neurons, firing threshold manifolds and canards, Mixed-mode oscillations in a piecewise linear system with multiple time scale coupling, Canard solutions of two-dimensional singularly perturbed systems, Local analysis near a folded saddle-node singularity, Reduction of a model of an excitable cell to a one-dimensional map, The regularized visible fold revisited, Spike-adding canard explosion in a class of square-wave bursters, Inflection, canards and folded singularities in excitable systems: application to a 3D FitzHugh-Nagumo model, High-order study of the canard explosion in an aircraft ground dynamics model, Quasi-periodic states in coupled rings of cells, Mixed-mode oscillations in a stochastic, piecewise-linear system, Time-periodic perturbation of a Liénard equation with an unbounded homoclinic loop, Asymptotic behavior of periodic solutions in one-parameter families of Liénard equations, Relaxation oscillations and canards of a regulated two-gene model, Fractal dimensions and two-dimensional slow-fast systems, Effective reduction of a three-dimensional circadian oscillator model, Tracking particles in flows near invariant manifolds via balance functions, Singularly perturbed boundary-focus bifurcations, Coherent neural oscillations induced by weak synaptic noise, Reduced models of cardiomyocytes excitability: comparing Karma and FitzHugh-Nagumo, Limit cycles at oversteer vehicle, Analysis of a predator-prey model with specific time scales: a geometrical approach proving the occurrence of canard solutions, The entry-exit theorem and relaxation oscillations in slow-fast planar systems, Discretized fast-slow systems with canards in two dimensions, Canard limit cycle of the Holling-Tanner model, Dynamics of the Tyson-Hong-Thron-Novak circadian oscillator model, Asymptotic expansions for a degenerate canard explosion, Traveling pulses in a coupled FitzHugh-Nagumo equation, Relaxation oscillation and canard explosion for a SIRS model with nonlinear incidence rate, Analysis of the onset of a regime shift and detecting early warning signs of major population changes in a two-trophic three-species predator-prey model with long-term transients, Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple allee effect, Existence of canards under non-generic conditions, Trajectory-free approximation of phase space structures using the trajectory divergence rate, Fractal analysis of degenerate spiral trajectories of a class of ordinary differential equations, The number of limit cycles for regularized piecewise polynomial systems is unbounded, Mixed mode oscillations in the Bonhoeffer-van der Pol oscillator with weak periodic perturbation, Planar canards with transcritical intersections, Multiscale geometry of the Olsen model and non-classical relaxation oscillations, Canards, relaxation oscillations, and pattern formation in a slow-fast ratio-dependent predator-prey system, Analytical approximation of the canard explosion in a van der Pol system with the nonlinear time transformation method, Cyclicity of canard cycles with hyperbolic saddles located away from the critical curve, The dud canard: existence of strong canard cycles in \(\mathbb{R}^3\), Low-dimensional models of single neurons: a review, Existence of traveling wave solutions in a singularly perturbed predator-prey equation with spatial diffusion, Oscillatory and transient dynamics of a slow-fast predator-prey system with fear and its carry-over effect, Canard phenomena for a slow-fast predator-prey system with group defense of the prey, Revisiting the Kepler problem with linear drag using the blowup method and normal form theory, Unveiling the dynamics of canard cycles and global behaviour in a singularly perturbed predator-prey system with Allee effect in predator, Perturbation of a piecewise regular-singular Liénard system, Existence of traveling wave solutions for the perturbed modefied Gardner equation, Singular perturbations of generalized Holling type III predator-prey models with two canard points, Relaxation oscillation and canard explosion in a slow-fast predator-prey model with Beddington-DeAngelis functional response, Super-explosion and inverse canard explosion in a piecewise-smooth slow-fast Leslie-Gower model, Geometric blow-up for folded limit cycle manifolds in three time-scale systems, Relaxation oscillations in predator-prey systems, À propos de canards (Apropos canards), Relaxation Oscillations and Canard Explosion of a Leslie–Gower Predator–Prey System with Addictive Allee Effect, Canards for a reduction of the Hodgkin-Huxley equations, Dynamics in diffusive Leslie–Gower prey–predator model with weak diffusion, Regularizations of Two-Fold Bifurcations in Planar Piecewise Smooth Systems Using Blowup, Averaging, Folded Singularities, and Torus Canards: Explaining Transitions between Bursting and Spiking in a Coupled Neuron Model, Extending Melnikov theory to invariant manifolds on non-compact domains, Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations, Traveling Stripes in the Klausmeier Model of Vegetation Pattern Formation, Canard Cycles and Relaxation Oscillations in a Singularly Perturbed Leslie–Gower Predator–Prey Model with Allee Effect, Process-Oriented Geometric Singular Perturbation Theory and Calcium Dynamics, Stability Loss Delay and Smoothness of the Return Map in Slow-Fast Systems, Le Canard de Painlevé, A survey on the blow-up method for fast-slow systems, Global Stability and Canard Explosions of the Predator-Prey Model with the Sigmoid Functional Response, Multi-timescale systems and fast-slow analysis, Bifurcation Analysis of Wilson-Cowan Model with Singular Impulses, Geometric singular perturbation analysis to Camassa-Holm Kuramoto-Sivashinsky equation, Intrinsic determination of the criticality of a slow-fast Hopf bifurcation, Canards of Folded Saddle-Node Type I, Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases, Noise-induced Canard and Mixed-Mode Oscillations in Large-Scale Stochastic Networks, Travelling waves in diffusive Leslie-Gower prey-predator model, Fast Pulses with Oscillatory Tails in the FitzHugh--Nagumo System, Relaxation oscillations in a slow-fast predator-prey model with weak Allee effect and Holling-IV functional response, Study of a homoclinic canard explosion from a degenerate center, Wiggly canards: growth of traveling wave trains through a family of fast-subsystem foci, Birth, transition and maturation of canard cycles in a piecewise linear system with a flat slow manifold, Canard explosion near non-Liénard type slow-fast Hopf point, Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type III, Normal forms of Liénard type for analytic unfoldings of nilpotent singularities, Stochastic mixed-mode oscillations in the canards region of a cardiac action potential model, Canards in a minimal piecewise-linear square-wave burster, Attractors and long transients in a spatio-temporal slow-fast Bazykin's model, PERIODIC SOLUTIONS AND SLOW MANIFOLDS, CONTROL-BASED CONTINUATION OF PERIODIC ORBITS WITH A TIME-DELAYED DIFFERENCE SCHEME, Canards in Stiction: On Solutions of a Friction Oscillator by Regularization, Canard Phenomenon in an SIRS Epidemic Model with Nonlinear Incidence Rate, High-Order Analysis of Canard Explosion in the Brusselator Equations, Folded Saddles and Faux Canards, Fractal codimension of nilpotent contact points in two-dimensional slow-fast systems, On the approximation of the canard explosion point in singularly perturbed systems without an explicit small parameter, Unpeeling a Homoclinic Banana in the FitzHugh--Nagumo System, Computing the Stable Manifold of a Saddle Slow Manifold, Canard-cycle transition at a fast-fast passage through a jump point, Geometric desingularization of a cusp singularity in slow-fast systems with applications to Zeeman's examples, Multiple-S-Shaped Critical Manifold and Jump Phenomena in Low Frequency Forced Vibration with Amplitude Modulation, Discretized fast-slow systems near transcritical singularities, Anticipation via canards in excitable systems, Canard cycle transition at a slow-fast passage through a jump point, Local Theory for Spatio-Temporal Canards and Delayed Bifurcations, Polynomial normal forms of constrained differential equations with three parameters, A Stiction Oscillator with Canards: On Piecewise Smooth Nonuniqueness and Its Resolution by Regularization Using Geometric Singular Perturbation Theory, Food chain chaos with canard explosion, On maximum bifurcation delay in real planar singularly perturbed vector fields, Wave train selection by invasion fronts in the FitzHugh–Nagumo equation, DELAYED BIFURCATION IN FIRST-ORDER SINGULARLY PERTURBED PROBLEMS WITH A NONGENERIC TURNING POINT, Singularly perturbed boundary-equilibrium bifurcations, Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system, Tolerance and threshold in the extrinsic coagulation system, Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron, Localized and asynchronous patterns via canards in coupled calcium oscillators, FROM FIRST LYAPUNOV COEFFICIENTS TO MAXIMAL CANARDS, Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures, Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster, Chaos in homeostatically regulated neural systems, Canards in \(\mathbb{R}^3\), Canard Explosion and Relaxation Oscillation in Planar, Piecewise-Smooth, Continuous Systems, Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations, Canard solutions at non-generic turning points, Canard solutions in planar piecewise linear systems with three zones, Canard-like phenomena in piecewise-smooth Van der Pol systems, Delay induced canards in a model of high speed machining, Parabolic bursting, spike-adding, dips and slices in a minimal model, Extended and Symmetric Loss of Stability for Canards in Planar Fast-Slow Maps, Canards in piecewise-linear systems: explosions and super-explosions, Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator–prey systems, Canards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems, Canards and curvature: the ‘smallness of ε ’ in slow–fast dynamics, Regularization and Geometry of Piecewise Smooth Systems with Intersecting Discontinuity Sets, Duck traps: two-dimensional critical manifolds in planar systems, Delay-induced switched states in a slow–fast system, Constructive role of noise and diffusion in an excitable slow–fast population system, EXOTIC DYNAMICS IN NETWORKS OF COUPLED RINGS OF CELLS, Strange Dynamics in a Fractional Derivative of Complex-Order Network of Chaotic Oscillators, Canonical models for torus canards in elliptic bursters, Pulse Replication and Accumulation of Eigenvalues, Relaxation Oscillations and the Entry-Exit Function in Multidimensional Slow-Fast Systems, Understanding anomalous delays in a model of intracellular calcium dynamics, Relaxation oscillations in substrate-depletion oscillators close to the nonsmooth limit, Canard Explosion and Position Curves, Uniqueness of relaxation oscillations: A classical approach, A Stiction Oscillator under Slowly Varying Forcing: Uncovering Small Scale Phenomena using Blowup, Two-stroke relaxation oscillators, An elementary model of torus canards, A new type of relaxation oscillation in a model with rate-and-state friction, Slow-Fast Dynamics and Its Application to a Biological Model, Wave speeds for the FKPP equation with enhancements of the reaction function, Mixed mode oscillations in a conceptual climate model



Cites Work