## Spiral wave instabilities

Snapshots of numerical simulations variants of the FitzHugh-Nagumo 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, two-frequency 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 two-frequency 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 convolution-type 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.), Springer-Verlag. IMA Volumes in Mathematics and its Applications 115 (1999), 249-264.

[2] B. Sandstede, A. Scheel
Super-spiral structures of meandering and drifting spiral waves
Phys. Rev. Lett. 86 (2001), 171-174.

[3] B. Sandstede, A. Scheel
Absolute versus convective instability of spiral waves
Phys. Rev. E. 62 (2000), 7708-7714.

[4] B. Sandstede, A. Scheel
Absolute and convective instabilities of waves on unbounded and large bounded domains
Physica D 145 (2000), 233-277.