Spiral Wave Chimeras in Complex Oscillatory and Chaotic Systems

We demonstrate for the first time that spiral wave chimeras—spiral waves with spatially extended unsynchronzied cores—can exist in complex oscillatory and even locally chaotic homogeneous systems under nonlocal coupling. Using ideas from phase synchronization, we show in particular that the unsynchronized cores exhibit a distribution of different frequencies, thus generalizing the main concept of chimera states beyond simple oscillatory systems. In contrast to simple oscillatory systems, we find that spiral wave chimeras in complex oscillatory and locally chaotic systems are characterized by the presence of synchronization defect lines (SDLs), along which the dynamics follows a periodic behavior different from that of the bulk. Whereas this is similar to the case of local coupling, the type of the prevailing SDLs is very different.

Figure 1

Equivalent values of $\alpha$ computed by matching the numerically obtained phase response curves for the Rössler model to the sinusoidal phase response curve are shown in the left panel, for parameters $a=0.2$, $c=5.7$, and varying $b$ values. Several representative phase response curves are shown in the right panel.

Figure 2

(a) Snapshot of the scalar field $X(\mathbf{r},t)$ for the nonlocal Rössler model, exhibiting a spiral wave chimera. Parameters used are $(a,b,c)=(0.2,1.7,5.7)$ and $K=0.05$, in the period-1 (simple oscillatory) regime of the system. Simulations are performed on a square lattice of size $N=512$ with $\Delta x=0.0977$ and no-flux boundary conditions using an Euler scheme with $\Delta t=0.01$ and a spatial convolution for the coupling [8]. (b) Time-averaged frequencies over $\sim 1700$ oscillations, corresponding to (a), recorded at the cross section $y=27.125$.

Figure 3

(a)–(c) Snapshots of $X(\mathbf{r},t)$ in the period-2, period-4, and chaotic regimes of the nonlocal Rössler system. Parameters $a=0.2$, $b=1.25$ are fixed whereas $c=5.7$, 6.7, and 7.0. (d)–(f) SDLs in the system corresponding to (a)–(c). Period-1 and period-2 SDLs are indicated as gray and black in (e) and (f), respectively.

Figure 4

(a) Local time series corresponding to Fig. 3 showing different oscillatory behaviors far from the spiral core: solid line $(a,b,c)=(0.2,1.7,5.7)$, dashed line $(a,b,c)=(0.2,1.25,5.7)$, dash-dotted line $(a,b,c)=(0.2,1.25,6.7)$, dash-dash-dotted line $(a,b,c)=(0.2,1.25,7.0)$. (b) Local phase space trajectory in the core [$(x,y)=(27.54,27.64)$] and in the bulk [$(x,y)=(37.30,34.86)$] of the spiral wave chimera, in the chaotic regime $(a,b,c)=(0.2,1.25,7.0)$.

Figure 5

Snapshots of local rotation periods $n(\mathbf{r})$ corresponding to Fig. 3c shown at two different times $t=1000$ (left) and $t=7500$ (right). Note that values of $n(\mathbf{r},t)$ measured in the spiral wave can differ by up to one period, since the wave front has not reached the entire system. In the right panel, an elongated “trail” of oscillators indicates a moving, unsynchronized spiral core. The oscillators at the current core location are unsynchronized. However, the oscillators that were at the previous core location are now synchronized but with a large phase lag.