Solvable Model of Spiral Wave Chimeras

Spiral waves are ubiquitous in two-dimensional systems of chemical or biological oscillators coupled locally by diffusion. At the center of such spirals is a phase singularity, a topological defect where the oscillator amplitude drops to zero. But if the coupling is nonlocal, a new kind of spiral can occur, with a circular core consisting of desynchronized oscillators running at full amplitude. Here, we provide the first analytical description of such a spiral wave chimera and use perturbation theory to calculate its rotation speed and the size of its incoherent core.

Figure 1

Snapshot of a spiral wave chimera on a square domain of side length 15, using an array of $151\times 151$ oscillators. Parameter values: $\alpha =0.15\pi$, $\omega =0$. Colors schematically indicate oscillation phases in panels (a)–(c), whereas in panel (d) they depict the amplitude $R(\mathbf{x})$ of the local order parameter according to the numerical values shown next to the color bar. (a) Spatial variation of $\sin \varphi$. (b) Close up of the incoherent core in panel (a). (c) $\sin [\Theta (\mathbf{x})]$ (phase of order parameter) (d) $R(\mathbf{x})$ (amplitude of order parameter). The amplitude drops to zero at the center of the core, consistent with the randomized phases of the oscillators there.

Figure 2

Comparison of results from simulation and perturbation theory. (a) Values of $\Delta$ from simulation (circles) compared with predicted values $\Delta =\alpha$ (solid line). (b) Comparison of measured values of $\rho$ (circles) with $\rho =(2/\sqrt{\pi })\alpha$ (solid line). The simulation was run on a square domain of side length 10, with $201\times 201$ oscillators.

Figure 3

Numerical solutions of Eq. (8). (a) $A(r)$ for two different values of $\alpha$, and $A_0(r)$. (b) $\Psi (r)/\alpha$ for three different values of $\alpha$. (c) Calculated values of $\Delta$ (circles), compared with the predicted result of $\Delta =\alpha$ (solid line). (d) Radius of incoherent core $\rho$: solutions of $A(\rho )=\Delta$ (circles), compared with the predicted value $\rho =(2/\sqrt{\pi })\alpha$ (solid line).

Figure 4

Effects of different kernels. Panels (a) and (b) show $\Delta$ and $\rho$, respectively, for kernels $2\pi G(r)$ given by: (A) $K_0(r)$; (B) $2e^{-r^2}$; (C) $e^{-r}$; (D) $2H(r)H(1-r)$, where $H$ is the Heaviside step function.