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Absolute instabilities of standing pulses (2004).
| Content Provider | CiteSeerX |
|---|---|
| Author | Scheel, Arnd Sandstede, Björn |
| Abstract | We analyse instabilities of standing pulses in reaction-diffusion systems that are caused by an absolute instability of the homogeneous background state. Specifically, we investigate the impact of pitchfork, Turing and oscillatory bifurcations of the rest state on the standing pulse. At a pitchfork bifurcation, the standing pulse continues through the bifurcation point where it selects precisely one of the two bifurcating equilibria. At a Turing instability, symmetric pulses emerge that are spatially asymptotic to the bifurcating spatially-periodic Turing patterns. These pulses exist for any wavenumber inside the Eckhaus stability band. Oscillatory instabilities of the background state lead to genuinely time-periodic pulses that emit small wave trains with a unique selected wavenumber. We analyse these three bifurcations by studying the standingwave and modulated-wave equations: In this setup, pulses correspond to homoclinic orbits to equilibria that undergo reversible bifurcations. We use blow-up techniques to show that the relevant stable and unstable manifolds can be continued across the bifurcation point and to investigate both existence and stability of the bifurcating waves. |
| File Format | |
| Publisher Date | 2004-01-01 |
| Access Restriction | Open |
| Subject Keyword | Bifurcating Equilibrium Bifurcating Wave Bifurcation Point Spatially-periodic Turing Pattern Reaction-diffusion System Symmetric Pulse Absolute Instability Oscillatory Instability Undergo Reversible Bifurcation Blow-up Technique Time-periodic Pulse Modulated-wave Equation Oscillatory Bifurcation Turing Instability Pitchfork Bifurcation Rest State Unstable Manifold Eckhaus Stability Band Homogeneous Background State Background State Lead Standing Pulse |
| Content Type | Text |
