Asymmetry-induced synchronization in oscillator networks

A scenario has recently been reported in which in order to stabilize complete synchronization of an oscillator network—a symmetric state—the symmetry of the system itself has to be broken by making the oscillators nonidentical. But how often does such behavior—which we term asymmetry-induced synchronization (AISync)—occur in oscillator networks? Here we present the first general scheme for constructing AISync systems and demonstrate that this behavior is the norm rather than the exception in a wide class of physical systems that can be seen as multilayer networks. Since a symmetric network in complete synchrony is the basic building block of cluster synchronization in more general networks, AISync should be common also in facilitating cluster synchronization by breaking the symmetry of the cluster subnetworks.

Figure 1

Multilayer construction of AISync networks. (a) Example of a symmetric network of $N=4$ heterogeneous oscillators and $K=3$ types of (directed) links with associated coupling functions ${\mathit{H}}^{\left(1\right)}$, ${\mathit{H}}^{\left(2\right)}$, and ${\mathit{H}}^{\left(3\right)}$. (b) One of many possible multilayer networks corresponding to the network in (a), with $L=2$ layers and $n=LN=8$ identical subnodes. Subnodes are labeled with node indices, with prime and double prime indicating layers 1 and 2, respectively. (c) Flattened, monolayer representation of the multilayer network in (b). (d) Block structure of the adjacency matrix $\tilde{A}$ for the monolayer network in (c). Colors indicate different types of nodes ($F^{\left(i\right)}$, diagonal blocks) and links (${\tilde{A}}^{\left(i{i}^{\prime }\right)}$, off-diagonal blocks).

Figure 2

Example of consensus system showing AISync. (a) Symmetric network of $N=3$ homogeneous nodes, each with $L=2$ subnodes coupled by a directed link (from subnode $i^{\prime \prime }$ to $i^{\prime }$). (b) The same network but with heterogeneous nodes, in which the direction of the internal sublink in the (light) cyan node is the opposite of that in the (dark) green nodes. In both (a) and (b) we show the corresponding node-level visualization of the network at the top right. (c) Stability region (shaded gray) for the consensus dynamics. All the transverse modes for the homogeneous system in (a) are unstable (red squares), while those for the heterogeneous system in (b) are stable (blue dots).

Figure 3

Example of coupled Lorenz systems showing AISync. (a) Symmetric network of $N=3$ nodes, each with $L=2$ directionally coupled subnodes of Lorenz oscillators. Here we show an instance of a heterogeneous system in which the sublink direction in one node (cyan) is different from the other two (green). (b) Contour plots of ${\mathrm{\Psi }}_{=}(\mathrm{red})$ for the case of homogeneous nodes (all-green or all-cyan nodes) and ${\mathrm{\Psi }}_{\ne }\left(\mathrm{blue}\right)$ for heterogeneous nodes (one or two green nodes). The shaded region corresponds to AISync systems, for which ${\mathrm{\Psi }}_{=}>0$ and ${\mathrm{\Psi }}_{\ne }<0$. (c) Sample trajectory of the system for $a=8$ and $b=6$ [cross symbol in (b)], exhibiting AISync. The first component $x_1$ of the Lorenz oscillator state vector is shown for all $n=6$ subnodes.

Figure 4

Example of coupled electro-optic systems showing AISync. (a–c) Networks with heterogeneous (a) and homogeneous nodes (b, c). (d) Stability function $\psi \left(\lambda \right)$ for the electro-optic system. (e–g) Numerical results indicating where each system in (a–c) is synchronizable in the parameter space. Each pixel is categorized into three classes according to 24 independent simulations from random initial conditions (see text for details). (h) The AISync region (shaded purple), which is the union of yellow and green regions in (e) minus the analogous unions in (f) and (g). The red contours encode the MTLE for the heterogeneous system.

Figure 5

Statistics on the prevalence of AISync-favoring networks. Shown as functions of (a) external sublink density and (b) network size $N$. Both panels show the fraction of systems with AISync strength $r>0.05$ among those with circulant network structures, where the external sublink density is given by $D/\left[L^2(N-1)\right]$, and $D$ is the number of external sublinks received by a node (which is the same for all nodes).

Figure 6

Stability function $\psi \left(\lambda \right)$ for the AISync system in Fig. 3.