• Corpus ID: 119628372

Forced translational symmetry-breaking for abstract evolution equations: the organizing center for blocking of travelling waves

  title={Forced translational symmetry-breaking for abstract evolution equations: the organizing center for blocking of travelling waves},
  author={Victor G. LeBlanc and Christian Roy},
  journal={arXiv: Dynamical Systems},
We consider two parameter families of differential equations on a Banach space X, where the parameters c and $\epsilon$ are such that: (1) when $\epsilon=0$, the differential equations are symmetric under the action of the group of one-dimensional translations SE(1) acting on X, whereas when $\epsilon\neq 0$, this translation symmetry is broken, (2) when $\epsilon=0$, the symmetric differential equations admit a smooth family of relative equilibria (travelling waves) parametrized by the drift… 

Figures from this paper

The Origin of Wave Blocking for a Bistable Reaction-Diffusion Equation : A General Approach

Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in

Wave Blocking Phenomena and Ecological Applications

The growing flow of people and goods around the globe has allowed new, non-native species to establish and spread in already fragile ecosystems. The introduction of invasive species can have a

Invasion pinning in a periodically fragmented habitat

A geometric approach to studying pinning or blocking of a bistable travelling wave is presented, using ideas from the theory of symmetric dynamical systems to make quantitative predictions about how spatial heterogeneities in dispersal and/or reproduction rates contribute to halting biological invasion fronts in reaction–diffusion models with an Allee effect.

Invasion pinning in a periodically fragmented habitat

Biological invasions can cause great damage to existing ecosystems around the world. Most landscapes in which such invasions occur are heterogeneous. To evaluate possible management options, we need



Dynamics of Spiral Waves on Unbounded Domains Using Center-Manifold Reductions

Abstract An equivariant center-manifold reduction near relative equilibria ofG-equivariant semiflows on Banach spaces is presented. In contrast to previous results, the Lie groupGinduces a strongly

Translational Symmetry-Breaking for Spiral Waves

This paper investigates the effects on spiral wave dynamics of breaking the translation symmetry while keeping the rotation symmetry by introducing a small perturbation in the five-dimensional center bundle equations (describing Hopf bifurcation from one-armed spiral waves) which is SO(2)-equivariant but not equivariant under translations.

The non-local Fisher–KPP equation: travelling waves and steady states

We consider the Fisher–KPP equation with a non-local saturation effect defined through an interaction kernel ϕ(x) and investigate the possible differences with the standard Fisher–KPP equation. Our

Positive travelling fronts for reaction–diffusion systems with distributed delay

We give sufficient conditions for the existence of positive travelling wave solutions for multi-dimensional autonomous reaction–diffusion systems with distributed delay. To prove the existence of

Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms.

This work explains the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delay time increases and derives a nonlinear equation governing the motion of fronts, which includes a term with delay.

Bifurcations and traveling waves in a delayed partial differential equation.

The resulting mathematical model is a nonlinear first-order partial differential equation for the cell density u(t,x) in which there is retardation in both temporal (t) and maturation variables (x), and contains three parameters.

The approach of solutions of nonlinear diffusion equations to travelling front solutions

AbstractThe paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation ut—uxx—∞;(u)=O, x∈(—∞, ∞) , in the case ∞(0)=∞(1)=0, ∞′(0)<0, ∞′(1)<0. Commonly, a travelling

Wave-Block in Excitable Media Due to Regions of Depressed Excitability

A geometrical method is presented that allows one to easily compute the critical gap length above which a steady state solution, and thus block, first occurs and is used to show that block associated with any local inhomogeneity must be associated with a limit point bifurcation.

Wave-blocking phenomena in bistable reaction-diffusion systems

The following bistable reaction-variable diffusion systems including a parameter $\sigma $ is considered: \[ u_1 = ( d( x )u_x )_x + \frac{1}{\sigma }f( u,v ), \]\[ u_1 = ( {d( x )u_x } )_x + \sigma