Computing the Stable Manifold of a Saddle Slow Manifold

DOI10.1137/17M1132458zbMath1403.37036MaRDI QIDQ4608085

Vivien Kirk, Hinke M. Osinga, Saeed Farjami

Publication date: 15 March 2018

Published in: SIAM Journal on Applied Dynamical Systems (Search for Journal in Brave)

zbMATH Keywords

two-point boundary value problemgeometric singular perturbation theorytransient dynamicsseparatrixsaddle slow manifold

Mathematics Subject Classification ID

Dynamics induced by flows and semiflows (37C10) Singular perturbations of ordinary differential equations (34D15) Numerical solution of boundary value problems involving ordinary differential equations (65L10) Invariant manifold theory for dynamical systems (37D10) Invariant manifolds for ordinary differential equations (34C45) Computational methods for bifurcation problems in dynamical systems (37M20) Singular perturbations for ordinary differential equations (34E15) Numerical nonlinear stabilities in dynamical systems (65P40) Stability of manifolds of solutions to ordinary differential equations (34D35) Systems with slow and fast motions for nonlinear problems in mechanics (70K70)

Related Items

Computational Singular Perturbation Method for Nonstandard Slow-Fast Systems, On analysis of inputs triggering large nonlinear neural responses slow-fast dynamics in the wendling neural mass model, Switching in Cerebellar Stellate Cell Excitability in Response to a Pair of Inhibitory/Excitatory Presynaptic Inputs: A Dynamical System Perspective, A Surface of Heteroclinic Connections Between Two Saddle Slow Manifolds in the Olsen Model, Robust domain of attraction estimation for a tumor growth model, Saddle slow manifolds and canard orbits in \(\mathbb{R}^{4}\) and application to the full Hodgkin-Huxley model

Uses Software

• AUTO
• AUTO-07P

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Multiple time scale dynamics
• Attraction-based computation of hyperbolic Lagrangian coherent structures
• Mathematical foundations of neuroscience
• The geometry of mixed-mode oscillations in the Olsen model for the peroxidase-oxidase reaction
• Invariant manifolds of singularly perturbed ordinary differential equations
• Geometric singular perturbation theory for ordinary differential equations
• Singular perturbations and conditional stability
• The structure of a flow in the vicinity of an almost periodic motion
• Tracking invariant manifolds with differential forms in singularly perturbed systems
• Normally hyperbolic invariant manifolds in dynamical systems
• Analysis of bursting in a thalamic neuron model
• Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram
• Full system bifurcation analysis of endocrine bursting models
• Canards in a surface oxidation reaction
• A showcase of torus canards in neuronal bursters
• Mathematical physiology. I: Cellular physiology
• A Contour Algorithm for Computing Stable Fiber Bundles of Nonautonomous, Noninvertible Maps
• AN ANALYTICAL AND NUMERICAL STUDY OF A MODIFIED VAN DER POL OSCILLATOR
• On the Approximation of Stable and Unstable Fiber Bundles of (Non)Autonomous ODEs — A Contour Algorithm
• HYPERBOLICITY AND INVARIANT MANIFOLDS FOR PLANAR NONAUTONOMOUS SYSTEMS ON FINITE TIME INTERVALS
• Invariant manifolds and control of hyperbolic trajectories on infinite- or finite-time intervals
• The Geometry of Slow Manifolds near a Folded Node
• Introduction to Applied Nonlinear Dynamical Systems and Chaos
• Rigorous Enclosures of a Slow Manifold
• Geometry of Mixed-Mode Oscillations in the 3-D Autocatalator
• Computing Slow Manifolds of Saddle Type
• Computation of Saddle-Type Slow Manifolds Using Iterative Methods
• A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS
• Continuation-Based Numerical Detection of After-Depolarization and Spike-Adding Thresholds
• Parameter shifts for nonautonomous systems in low dimension: bifurcation- and rate-induced tipping
• Food chain chaos with canard explosion
• Invariant manifolds in singular perturbation problems for ordinary differential equations
• Invariant manifolds
• Relaxation oscillation and canard explosion
• Canards in \(\mathbb{R}^3\)