Chimera Death: Symmetry Breaking in Dynamical Networks

For a network of generic oscillators with nonlocal topology and symmetry-breaking coupling we establish novel partially coherent inhomogeneous spatial patterns, which combine the features of chimera states (coexisting incongruous coherent and incoherent domains) and oscillation death (oscillation suppression), which we call “chimera death”. We show that due to the interplay of nonlocality and breaking of rotational symmetry by the coupling, two distinct scenarios from oscillatory behavior to a stationary state regime are possible: a transition from an amplitude chimera to chimera death via in-phase synchronized oscillations and a direct abrupt transition for larger coupling strength.

Figure 1

Space-time plots for the variable $y_j(t)$ for coupling strength $\sigma =14$ and varying nearest neighbors number $P$. (a) $P=4$: amplitude chimera, (b) $P=5$: in-phase synchronization, (c) $P=33$: chimera death (coexistence of spatially coherent and incoherent oscillation death). Other parameters: $N=100$, $\lambda =1$, $\omega =2$. Transients of $t=1000$ were cut off. Initial conditions were chosen similar to [12].

Figure 2

Amplitude chimera: Snapshots at $t=1000$ of (a) the variable $y_j$ and (b) the amplitude $r_j$ (with respect to the origin). Center of mass (c.m.) averaged over one period of each oscillator for (c) the variable $y_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}$, (d) the amplitude $r_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}$ (i.e., the shift of the c.m. from the origin). (e) Phase portraits of all oscillators in the complex $z=x+iy$ plane. In (c)–(e) transients of $t=1000$ were cut off. Other parameters: $N=100$, $P=4$, $\sigma =14$, $\lambda =1$, $\omega =2$.

Figure 3

Direct transition from amplitude chimera to chimera death for $\sigma =26$ by changing the coupling range. (a)–(c): Oscillating state (amplitude chimera) for $P=4$. (d)–(f): Stable inhomogeneous steady state (chimera death) for $P=5$. (a),(d): space-time plot for the variable $y_j$, (b),(e) snapshot for the variable $y_j$, (c),(f) phase portraits of all oscillators in the complex $z=x+iy$ plane. Transients of $t=1000$ were cut off. Other parameters: $N=100$, $\lambda =1$, $\omega =2$.

Figure 4

Chimera death for fixed value of coupling strength $\sigma =20$ and different values of coupling range $P/N$. Snapshots for the variable $y$: (a) $P=40$, (b) $P=20$, (c) $P=11$. Transients of $t=1000$ were cut off. Other parameters: $N=100$, $\lambda =1$, $\omega =2$.

Figure 5

Map of dynamic regimes for $N=100$ in the plane of coupling range $P/N$ and coupling strength $\sigma$. 1-CD: 1-cluster chimera death; 3-CD: 3-cluster chimera death; N-CD: multicluster ($>3$) chimera death. AC: amplitude chimera; SYNC: in-phase synchronized oscillations. Other parameters: $\lambda =1$, $\omega =2$. Diamonds, circles, and triangles mark the parameter values chosen in Figs. 1, 3, and 4, respectively.