Chimera states in purely local delay-coupled oscillators

We study the existence of chimera states in a network of locally coupled chaotic and limit-cycle oscillators. The necessary condition for chimera state in purely local coupled oscillators is discussed. At first, we numerically observe the existence of chimera or multichimera states in the locally coupled Hindmarsh–Rose neuron model. We find that delay time in the nonlinear local coupling reduces the domain of the coherent island in the parameter space of the synaptic coupling strength and time delay, and thus the coherent region can be completely eliminated once the time delay exceeds a certain threshold. We then consider another form of nonlinearity in the local coupling, and the existence of chimera states is observed in the time-delayed Mackey–Glass system and in a Van der Pol oscillator. We also discuss the effect of time delay in local coupling for the existence of chimera states in Mackey–Glass systems. The nonlinearity present in the coupling function plays a key role in the emergence of chimera or multichimera states. A phase diagram for the chimera state is identified over a wide parameter space.

Figure 1

(left panel) Snapshot of all neurons in locally coupled Hindmarsh–Rose oscillators without synaptic coupling delay, i.e., $\tau =0.0$ for (a) disordered state, $\epsilon =0.18$, (b) multichimera state, $\epsilon =0.63$, (c) chimera state, $\epsilon =0.67$, and (d) coherent state, $\epsilon =2.0$. (right panel) Corresponding mean phase velocities of disordered, multichimera, chimera, and coherent states.

Figure 2

Snapshot of all neurons in locally coupled Hindmarsh–Rose oscillators [Eq. (6)] for (a) disordered state, $\epsilon =0.4$, (b) multichimera state, $\epsilon =0.58$, (c) chimera state, $\epsilon =0.66$, and (d) coherent state, $\epsilon =2.0$. The value of synaptic coupling delay is $\tau =5$, and the number of oscillators is taken as $N=200$.

Figure 3

Variation of strength of incoherence (SI) and discontinuity measure (DM) for different values of synaptic coupling strength $\epsilon$ and fixed value of synaptic coupling delay $\tau$, for (a), (b) $\tau =5$, (c), (d) $\tau =10$, and (e), (f) $\tau =25$. Here the number $M=20$ of bins and $\delta =0.05$. The regions I, II, and III represent disordered, chimera or multichimera, and coherent states, respectively.

Figure 4

Two-parameter phase diagram in $\epsilon -\tau$ plane for 200 locally coupled Hindmarsh–Rose oscillators. (a) Strength of incoherence is used as color bar, the region of incoherence is represented by red, coherence by blue, and chimera or multichimera states by cyan or yellow, respectively. (b) Discontinuity measure is used to distinguish chimera and multichimera states. Green indicates incoherent states, black is for coherent states, red is for multichimera, and blue is for chimera states. Here $M=40$ and $\delta =0.05$.

Figure 5

Snapshots of locally coupled Mackey–Glass oscillators for different values of coupling strength. Left column shows incoherent state, $\epsilon =0.058$ (top panel); multichimera state, $\epsilon =0.09$ (middle panel); chimera state, $\epsilon =0.18$ (bottom panel). Right column shows corresponding spacetime plots. Here $N=150$ and $\tau =0.0$.

Figure 6

The time-averaged frequency of the oscillators of the locally coupled Mackey–Glass oscillators without coupling delay ($\tau =0$) for different values of coupling strength $\epsilon$. (a) Incoherent frequency for $\epsilon =0.058$, (b) coexisting of multiple domains of coherent and incoherent frequency for $\epsilon =0.09$, (c) coexisting of coherent and incoherent domains of frequency for $\epsilon =0.18$. All the parameters are the same as in Fig. 5.

Figure 7

Variation of (a) strength of incoherence and (b) discontinuity measure for different values of coupling strength in locally coupled Mackey–Glass time-delay oscillators. Here $N=150,M=30,\delta =0.05$, and $\tau =0.0$.

Figure 8

(a) Bifurcation diagram with respect to system delay time ${\tau }_0$ of an isolated Mackey–Glass oscillator. Two-parameter phase space in (b) $\epsilon -{\tau }_0$ plane for coupling delay time $\tau =0$, (c) $\epsilon -\tau$ plane for system delay time ${\tau }_0=1.0$ where individual oscillators are in a periodic state, and (d) $\epsilon -\tau$ plane for system delay time ${\tau }_0=2.0$ where each oscillator is in a chaotic state. These figures indicate regions of incoherence by green, chimera states by blue, and multichimera states by red. Here we have taken $N=150,M=30$, and $\delta =0.05$.

Figure 9

Locally coupled Mackey–Glass oscillators: (a) snapshots of the variable $x_i$, (b) corresponding profiles of mean phase velocity ${\omega }_i$, (c) spacetime plot showing the existence of chimera state for $\epsilon =0.1,\tau =0.0$. (d) Two-parameter phase diagram in $\epsilon -\tau$ plane. The incoherent, chimera, and multichimera states are represented by green, blue, and red dotted points, respectively. The other parameters are $\alpha =0.6$ and $\beta =0.06$.

Figure 10

Locally coupled Van der Pol oscillators: (a) Snapshots in the phase space $(x_i,y_i)$ where the limit cycle of the uncoupled oscillator is shown by a black line, (b) snapshots of the variable $x_i$, (c) corresponding mean phase velocity ${\omega }_i$, and (d) spacetime plot showing the existence of a chimera state for $\epsilon =0.04,\tau =1.0,\alpha =1.0,\beta =0.25$.