Spiral wave instabilities
Snapshots of numerical simulations variants of the FitzHughNagumo
equation, based on the software EZSpiral.
This is joint work with Björn Sandstede.
Rigid Rotation
Rigidly rotating spirals in small domains, left and right picture. The
white circle illustrates the temporal path of a fixed point on the
spiral wave profile. The right pictures illustrates that the influence
of the boundary actually is negligible: the spiral wave can rotate
around almost any point in the domain. The middle picture of a large
domain shows the regular Archimedean structure in the farfield.
Meander  Core Region
Meandering, twofrequency motion of spirals is illustrated. Different
pictures correspond to different parameter values and illustrate a
typical final state after transients. The fixed point on the spiral profile
moves on a circle whose center rotates on a second, larger, circle in
the same (left picture) or the opposite direction (right picture). At
the transition, the center of the circle moves on a straight line.
See [1] (Postscript, PDF)
for a review of mathematical and experimental results.
Meander  Farfield
The effect of the twofrequency motion on the shape of the (now
almost) Archimedean spiral in the farfield is illustrated
below. Again, different pictures correspond to different parameter
values. The apparent change in shape results in a superimposed spiral
wave with large wavelength. In the left picture, the superimposed
spiral is oriented in the same direction as the primary spiral. In the
right picture, it takes the opposite orientation. At the transition,
the superimposed spiral degenerates to a sector.
The superpatterns consist of dark regions where the wavelength of the
primary spiral is shorter, and bright regions of larger
wavelength. The effect is enhanced by convolutiontype image processing.
See [2] (Postscript, PDF)
for reference.
Farfield breakup
A different instability mechanism caused by the essential spectrum of
the spiral wave. The wavetrains emitted by the spiral wave have become
unstable such that local variations in the wavenumber are
amplified. Super patterns again resemble superspirals. In the first
picture, the instability is of a convective nature. The super pattern
grows in amplitude but is, at the same time, advected towards the
boundary. After a long transient, the instability will disappear: the
primary Archimedean spiral is stable in any large but finite
domain. In the middle picture, the parameter driving the instability
is further increased until the threshold of absolute instability,
where the absolute spectrum [4] (Postscript, PDF) of the spiral wave has
crossed the imaginary axis. Perturbations now grow at fixed points of
the domain. Still, the instability is more pronounced at the
boundary. The subcritical nature of the instability amplifies the
compression and expansion of the wavetrains until they collide and
breakup. A final state in this parameter regime is depicted in the
right picture.
See [3] (Postscript, PDF)
for reference.
Core breakup
Similarly to farfield breakup, the instability here is caused by the
wave trains. However, perturbations are now advected towards the
center of the spiral wave. The final state of this subcritical
instability is an extremely incoherent pattern consisting of many
small spiral cores.
See [3] (Postscript, PDF)
for reference.
 [1] B. Sandstede, A. Scheel, C. Wulff

Dynamical behavior of patterns with euclidean symmetry
In ``Pattern formation in continuous and coupled systems'',
M. Golubitsky, D. Luss and S. Strogatz (eds.), SpringerVerlag.
IMA Volumes in Mathematics and its Applications 115 (1999),
249264.
 [2] B. Sandstede, A. Scheel

Superspiral structures of meandering and drifting spiral waves
Phys. Rev. Lett. 86 (2001), 171174.
 [3] B. Sandstede, A. Scheel

Absolute versus convective instability of spiral waves
Phys. Rev. E. 62 (2000), 77087714.
 [4] B. Sandstede, A. Scheel

Absolute and convective instabilities of waves on unbounded and large bounded domains
Physica D 145 (2000), 233277.
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