Frequency assortativity can induce chaos in oscillator networks

We investigate the effect of preferentially connecting oscillators with similar frequency to each other in networks of coupled phase oscillators (i.e., frequency assortativity). Using the network Kuramoto model as an example, we find that frequency assortativity can induce chaos in the macroscopic dynamics. By applying a mean-field approximation in combination with the dimension reduction method of Ott and Antonsen, we show that the dynamics can be described by a low dimensional system of equations. We use the reduced system to characterize the macroscopic chaos using Lyapunov exponents, bifurcation diagrams, and time-delay embeddings. Finally, we show that the emergence of chaos stems from the formation of multiple groups of synchronized oscillators, i.e., meta-oscillators.

Figure 1

Synchronization in nonassortative and assortative networks. Synchronization profiles $R$ (solid blue) and $B$ (dashed red) vs $K$ for examples of (a) nonassortative and (b) assortative networks of size $N=1000$ with constant degree $k=50$. For the reduced description on which our determination of $B$ is based we use $\tilde{N}=20$ and ${\omega }_{\text{max}},-{\omega }_{\text{min}}=3.126$.

Figure 2

Bifurcation diagrams and Lyapunov exponent. Bifurcation diagrams of the (a) full and (b) reduced dynamics calculated by plotting the values of $R\left(t\right)$ [for (a)] and $B\left(t\right)$ [for (b)] evaluated at the times of surfaces of section piercing. (c) The largest Lyapunov exponent ${\lambda }_{\text{LLE}}$ as a function of $K$ calculated using the reduced system.

Figure 3

Chaos vs periodicity. Time delay embeddings of the full system $\left[R\right(t),R(t-\tau \left)\right]$ for (a) $K=0.09$ and (b) $0.105$, and the reduced system $\left[B\right(t),B(t-\tau \left)\right]$ for (c) $K=0.09$ and (d) $0.105$ for $\tau =1$. Also for $K=0.09$ and $0.105$, the dynamic correlations $c_{ij}^{\omega }$ [(e) and (f), respectively] as calculated from the full system.