# Hopf Bifurcation and Exchange of Stability in Diffusive Media

@article{Brand2004HopfBA, 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}, year={2004}, volume={171}, pages={263-296} }

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…

## 20 Citations

Hopf Bifurcation From Viscous Shock Waves

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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.

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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.

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