Hopf Bifurcation and Exchange of Stability in Diffusive Media

  title={Hopf Bifurcation and Exchange of Stability in Diffusive Media},
  author={Thomas Brand and Markus Kunze and Guido Schneider and Thorsten Seelbach},
  journal={Archive for Rational Mechanics and Analysis},
Abstract.We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses a continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a pair of imaginary eigenvalues crosses the imaginary axis. For a reaction-diffusion-convection system we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a Hopf… 
Hopf Bifurcation From Viscous Shock Waves
Using spatial dynamics, a Hopf bifurcation theorem is proved for viscous Lax shocks in viscous conservation laws that is unique, exponentially localized in space, periodic in time, and their speed satisfies the Rankine–Hugoniot condition.
The Existence of Bifurcating Invariant Tori in a Spatially Extended Reaction-Diffusion-Convection System with Spatially Localized Amplification
The bifurcation of a family of invariant tori which may contain quasiperiodic solutions is proved for this spatially extended reaction-diffusion-convection system with a marginally stable ground state and a spatially localized amplification.
A Hopf-Bifurcation Theorem for the Vorticity Formulation of the Navier–Stokes Equations in ℝ3
We prove a Hopf-bifurcation theorem for the vorticity formulation of the Navier–Stokes equations in ℝ3 in case of spatially localized external forcing. The difficulties are due to essential spectrum
Nonlinear convective instability of fronts: a case study
We consider a model system, consisting of two nonlinearly coupled partial differential equations, to investigate nonlinear convective instabilities of travelling waves. The system exhibits front
Stability of traveling waves in partly parabolic systems
We review recent results on stability of traveling waves in partly parabolic reaction-diffusion systems with stable or marginally stable equilibria. We explain how attention to what are apparently
Stability of a planar front in a multidimensional reaction-diffusion system
We study the planar front solution for a class of reaction diffusion equations in multidimensional space in the case when the essential spectrum of the linearization in the direction of the front
Hopf Bifurcation from Fronts in the Cahn–Hilliard Equation
We study Hopf bifurcation from traveling-front solutions in the Cahn–Hilliard equation. The primary front is induced by a moving source term. Models of this form have been used to study a variety of
Existence of periodic solutions of Boussinesq system
This paper is devoted to the study of the dynamical behavior of a Boussinesq system, which is a basic model in describing the flame propagation in a gravitationally stratified medium. This system
The Banach fixed point theorem application to Hopf bifurcation of a generalized Boussinesq system
This paper applies the Banach fixed point theorem to the study of the dynamical behavior of a three dimensional Boussinesq system with the temperature-dependent viscosity and thermal diffusivity
Stability of planar fronts for a class of reaction diffusion systems
The purpose of this thesis is to study stability of one-dimensional traveling waves and multidimensional planar fronts as well as space-independent steady states for a class of reaction diffusion


Exchange of stability and finite-dimensional dynamics in a bifurcation problem with marginally stable continuous spectrum
AbstractWe consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses continuous spectrum up to the imaginary axis, for
Bifurcation to spiral waves in reaction-diffusion systems
For a large class of reaction-diffusion systems on the plane, we show rigorously that m-armed spiral waves bifurcate from a homogeneous equilibrium when the latter undergoes a Hopf bifurcation. In
Bifurcation from the essential spectrum without sign condition on the nonlinearity
  • M. Kunze
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2001
We study a problem from nonlinear optics that leads to an integro-differential equation which can be written as the abstract bifurcation problem Au = λLu + ∇φ(u). For such a type of equation, a
Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations
We present a general method for studying long-time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It
On the infinitely many solutions of a semilinear elliptic equation
A dynamical systems approach is developed for studying the spherically symmetric solutions of $\Delta u + f(u) = 0$, where $f(u)$ grows like $| u |^\sigma u$ as $| u | \to \infty $ . Various scalings
Geometric Theory of Semilinear Parabolic Equations
Preliminaries.- Examples of nonlinear parabolic equations in physical, biological and engineering problems.- Existence, uniqueness and continuous dependence.- Dynamical systems and liapunov
Non-linear Stability of Modulated Fronts¶for the Swift–Hohenberg Equation
Abstract: We consider front solutions of the Swift–Hohenberg equation ∂tu= -(1+ ∂x2)2u + ɛ2u -u3. These are traveling waves which leave in their wake a periodic pattern in the laboratory frame. Using
Bifurcation into spectral gaps
Keywords: spectral gaps ; existence ; bifurcation ; unbounded ; selfadjoint operator ; real Hilbert space ; bounded ; selfadjoint operators ; bound states ; nonlinear ; Schrodinger equations ;