Essential instabilities of fronts


Space-time plots of numerical simulations of a three species reaction-diffusion system. Color represents concentration of the first species. Time is plotted upwards. In both pictures, a front moving to the right destabilizes due to the essential spectrum crossing the imaginary axis. The instability manifests itself in the creation of stationary, spatially periodic patterns at one of the asymptotic states of the front. In a comoving frame, this amounts to an oscillatory instability. If the instability occurs ahead of the front, the front invades the standing, spatially periodic structure and experiences a periodic modulation in a comoving frame. We find a bifurcating periodic orbit - as expected in a Hopf bifurcation. If the instability occurs behind the front, the resulting pattern is not spatially periodic. There is a gap opening between the primary, large front and the Turing patterns behind. In this Hopf bifurcation, no periodic solutions bifurcate. This is joint work with Björn Sandstede. See [1] (Postscript, PDF) for mathematical results.


Turing instability ahead of the front

Turing instability behind the front



[1] B. Sandstede, A. Scheel
Essential instabilities of fronts: bifurcation and bifurcation failure
Dynamical Systems 16 (2001), 1-28.


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