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  (Postscript, PDF) for mathematical results.
Turing instability ahead of the front
Turing instability behind the front
-  B. Sandstede, A. Scheel
Essential instabilities of fronts: bifurcation and bifurcation failure
Dynamical Systems 16 (2001), 1-28.
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