A Trilogy of Ginzburg-Landau Spiral Waves on Spherical Geometries

Jia-Yuan Dai

National Center for Theoretical Sciences

Spiral waves are important patterns observed in biological and chemical systems including the ventricles with disorganized electrical activity and the Belousov-Zhabotinsky reaction. However, even on the existence of spiral waves, only a few rigorous results are available. Among various models describing spiral waves, I focus on the significant complex Ginzburg-Landau equation, because its global gauge symmetry offers an advantage for mathematical analysis.

The framework of my research is a trilogy: existence, (in-)stability, and delay feedback stabilization. The existence of spiral waves results from a global bifurcation analysis. The (in-)stability of spiral waves follows by an explicit construction of shooting curves. Then I adopt a noninvasive delay feedback control to stabilize some unstable spiral waves.

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