Homoclinic bifurcations at the onset of pulse self-replication

DOI10.1016/j.jde.2006.03.026zbMath1217.34074OpenAlexW2136303057MaRDI QIDQ859515

Lambertus A. Peletier, Arjen Doelman, Tasso J. Kaper

Publication date: 16 January 2007

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

Full work available at URL: https://doi.org/10.1016/j.jde.2006.03.026

zbMATH Keywords

reaction-diffusion equationsautocatalysishomoclinic bifurcationssymmetric homoclinic orbitsmulti-pulse orbitsself-replicating pulses

Mathematics Subject Classification ID

Reaction-diffusion equations (35K57) Bifurcation theory for ordinary differential equations (34C23) Singular perturbations for ordinary differential equations (34E15) Homoclinic and heteroclinic solutions to ordinary differential equations (34C37)

Related Items (6)

Existence of traveling waves of general Gray-Scott models ⋮ Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain ⋮ Qualitative analysis of singular solutions for nonlinear elliptic equations with potentials ⋮ Self-replication of mesa patterns in reaction-diffusion systems ⋮ Critical exponent of a simple model of spot replication ⋮ Pulse Replication and Accumulation of Eigenvalues

Uses Software

• XPPAUT

Cites Work

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• The existence and stability of spike equilibria in the one-dimensional Gray -- Scott model on a finite domain
• Geometric singular perturbation theory for ordinary differential equations
• Dynamics of self-replicating patterns in the one-dimensional Gray-Scott model
• The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the pulse-splitting regime
• Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach
• Stability and instability in the Gray-Scott model: The case of equal diffusivities
• A skeleton structure of self-replicating dynamics
• Static spike autosolitons in the Gray-Scott model
• Slowly Modulated Two-Pulse Solutions in the Gray--Scott Model II: Geometric Theory, Bifurcations, and Splitting Dynamics
• Large stable pulse solutions in reaction-diffusion equations
• THE GIERER & MEINHARDT SYSTEM: THE BREAKING OF HOMOCLINICS AND MULTI-BUMP GROUND STATES
• A stability index analysis of 1-D patterns of the Gray-Scott model
• Stability of the Static Spike Autosolitons in the Gray--Scott Model
• Simulating, Analyzing, and Animating Dynamical Systems
• Pulse‐Splitting for Some Reaction‐Diffusion Systems in One‐Space Dimension
• The Existence and Stability of Spike Equilibria in the One‐Dimensional Gray–Scott Model: The Low Feed‐Rate Regime
• Homoclinic explosions and implosions
• Pattern formation in the one-dimensional Gray - Scott model
• The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form travelling waves
• The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. II. An initial-value problem with an immobilized or nearly immobilized autocatalyst
• Exact Homoclinic and Heteroclinic Solutions of the Gray--Scott Model for Autocatalysis
• Semistrong Pulse Interactions in a Class of Coupled Reaction-Diffusion Equations
• Homoclinic Stripe Patterns
• A Geometric Proof of the Kwong-McLeod Uniqueness Result
• Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system
• Invariant manifolds in singular perturbation problems for ordinary differential equations
• Invariant manifolds
• The stability of spike solutions to the one-dimensional Gierer-Meinhardt model
• \(2^n\)-splitting or edge-splitting? A manner of splitting in dissipative systems
• Elements of applied bifurcation theory

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