Loss of Coherence in Dynamical Networks: Spatial Chaos and Chimera States

We discuss the breakdown of spatial coherence in networks of coupled oscillators with nonlocal interaction. By systematically analyzing the dependence of the spatiotemporal dynamics on the range and strength of coupling, we uncover a dynamical bifurcation scenario for the coherence-incoherence transition which starts with the appearance of narrow layers of incoherence occupying eventually the whole space. Our findings for coupled chaotic and periodic maps as well as for time-continuous Rössler systems reveal that intermediate, partially coherent states represent characteristic spatiotemporal patterns at the transition from coherence to incoherence.

Figure 1

Regions of coherence for system (1) in the $(r,\sigma )$ parameter plane with wave numbers $k=1$, 2, and 3. Snapshots of typical coherent states $z_i$ are shown in the insets. The color code inside the regions distinguishes different time periods of the states. The coherence-incoherence bifurcation (CIB) curve separates regions with coherent and incoherent dynamics. In the hatched and shaded (color) regions below CIB, two-cluster incoherent states exist. Completely synchronized chaotic states exist in the light hatched region bounded by the blowout bifurcation curve BB. Parameters: $a=3.8$ and $N=100$.

Figure 2

Coherence-incoherence bifurcation for coupled chaotic logistic maps for fixed coupling radius $r=0.32$ (black triangles in Fig. 1). For each value of the coupling parameter $\sigma$ (decreasing from top to bottom, $\sigma =0.43$, 0.4, 0.32, 0.3, 0.2, and 0.1, respectively), snapshots (left columns) and space-time plots (right columns) are shown. Other parameters are as in Fig. 1.

Figure 3

(a) Local order parameter for $r=0.32$ and different $\sigma$ as in Fig. 2 (approximated with $\delta =0.025$, $N=400$). (b) Regions of stability for the two-cluster solutions with different asymmetries ($n_1:n_2$) as a function of $\sigma$ for $r=0.35$. Other parameters are as in Fig. 1.

Figure 4

Snapshots with decreasing coupling strength $\sigma$: coherence-incoherence transition for a ring of (a) 100 coupled periodic logistic maps ($a=3.2$) for a coupling radius $r=0.1$ and (b) 100 nonlocally coupled Rössler systems ($a=0.42$, $b=2$, $c=4$) with $r=0.3$.