Chimera States in Populations of Nonlocally Coupled Chemical Oscillators

Chimera states occur spontaneously in populations of coupled photosensitive chemical oscillators. Experiments and simulations are carried out on nonlocally coupled oscillators, with the coupling strength decreasing exponentially with distance. Chimera states with synchronized oscillators, phase waves, and phase clusters coexisting with unsynchronized oscillators are analyzed. Irregular motion of the cores of asynchronous oscillators is found in spiral-wave chimeras.

Figure 1

Experimental behavior of 40 oscillators coupled according to Eq. (1) in a ring configuration, with $n=10$, $\kappa =0.5$, $K^{\prime }=1$, and $\tau =30\mathrm{s}$. The experiment was started with a quasirandom initial phase distribution of the oscillators. (a) Snapshot showing the phase of each oscillator at $t=1220\mathrm{s}$. (b) Phase of each oscillator as a function of time. (c) Period of each oscillator at $t=900\mathrm{s}$ ($\mathrm{\text{blue}}+$) and 1500 s (red bullet). (d) Local order parameter $R$ according to Eq. (2) as a function of time, with $m=3$. See the Supplemental Material [25] for a video of the phase of each oscillator in (a) as a function of time.

Figure 2

Model simulations of 40 coupled BZ oscillators in a ring configuration, with $n=10$, $\kappa =0.4$, $K^{\prime }=6.3\times {10}^{-5}$, $\tau =35.0$, and ${\varphi }_0=1.6\times {10}^{-4}$. The simulation was started with a random initial phase distribution. (a) Snapshot showing the phase of each oscillator at $t=1.90\times {10}^4$. (b) Phase of each oscillator as a function of time. (c) Period of each oscillator at $t=1.85\times {10}^4$ ($\mathrm{\text{blue}}+$), $1.90\times {10}^4$ (red bullet). (d) Local order parameter $R$ as a function of time, with $m=3$. See the Supplemental Material [25] for a video of the phase of each oscillator in (a) as a function of time.

Figure 3

Simulations of phase-wave chimera states (top) and experimental measurements of phase-cluster chimera states (bottom). (a) Model simulations of 40 coupled BZ oscillators, with $n=10$, $\kappa =0.4$, $K^{\prime }=6.3\times {10}^{-5}$, $\tau =35.0$, and ${\varphi }_0=1.6\times {10}^{-4}$. The snapshot shows the phase of each oscillator at $t=1.93\times {10}^4$, where the red lines have been added to guide the eye. The simulation was started with a random initial phase distribution. (b) Phase of each oscillator in (a) as a function of time. (c) Measurements showing experimental phase-cluster chimera in 40 coupled BZ oscillators, with $n=10$, $\kappa =0.8$, $K^{\prime }=1.0$, and $\tau =30.0\mathrm{s}$. The experiment was started with a quasirandom initial phase distribution; the snapshot shows the phase of each oscillator at $t=1206\mathrm{s}$. (d) Local order parameter $R$ of oscillators in (c) as a function of time, with $m=3$. See the Supplemental Material [25] for videos of the phase of each oscillator in (a) and (c) as a function of time.

Figure 4

Simulations of spiral chimera states in populations of BZ oscillators. The system is composed of $50\times 50$ oscillators in a square-lattice configuration, with a coupling radius of $n=4$ and fixed boundary conditions. The top images show the phase of each oscillator in the lattice at $t=3500$ for values of delay $\tau =4.0$ (a) and 3.4 (b). Each simulation is initiated with a pair of symmetric counterrotating spirals, with $\tau =0$. The delay is switched on at $t=500$, and the simulation is continued to $t=3500$. (c), (d) Shown is the local order parameter $R$ at $t=3500$. The dark blue line shows the trajectory of the minimum in $R$ between $t=700$ and 3500. Parameters: $\kappa =0.3$, $K^{\prime }=1.4\times {10}^{-3}$, and ${\varphi }_0=1.1\times {10}^{-4}$. See the Supplemental Material [25] for videos of the evolution of each spiral pair in (a), (b) and the spiral-core trajectory as a function of time in (c), (d).