Mechanism of Desynchronization in the Finite-Dimensional Kuramoto Model

We study how a decrease of the coupling strength causes a desynchronization in the Kuramoto model of $N$ globally coupled phase oscillators. We show that, if the natural frequencies are distributed uniformly or close to that, the synchronized state can robustly split into any number of phase clusters with different average frequencies, even culminating in complete desynchronization. In the simplest case of $N=3$ phase oscillators, the course of the splitting is controlled by a Cherry flow. The general $N$-dimensional desynchronization mechanism is numerically illustrated for $N=5$.

Figure 1

Frequency-splitting bifurcation diagrams for the Kuramoto model from Eq. (1) with $N=3$ [(a),(b)] and $N=5$ [(c),(d)] phase oscillators, with a nonuniform distribution of the natural frequencies ${\omega }_i$: (a) $\{-0.97,-0.4,-0.2\}$, (b) $\{-1.0,0.0,0.953\}$, (c) $\{-1.0022,-0.50446,0.00106,0.49869,0.99804\}$, and (d) $\{-1.00064,-0.50017,0.00347,0.50170,0.99852\}$. ${\overline{\omega }}_i=\langle {\dot{\psi }}_i\rangle$, where $\langle ·\rangle$ denotes averaging in time. $\Omega$ is the mean frequency.

Figure 2

Schematic representation of the phase portraits for Eq. (2) with ${\Delta }_1={\Delta }_2=\Delta$. (a) $\Delta =0$: The phase portrait contains a stable node $O$, saddles $S_1$, $S_2$, and $S_3$, and unstable nodes $N_1$ and $N_2$. (b) $\Delta >0$ and $1.70<K/\Delta <3$ (Cherry flow): The phase portrait contains a stable node $O$, a saddle $S$, and an unstable periodic orbit $P$, and is foliated by stable manifolds $W_1^S$ and $W_2^S$ of the saddle $S$, as indicated by the gray regions.

Figure 3

(a) Two-parameter bifurcation diagram for Eq. (1) with $N=3$. The Cherry flow region is hatched. The main resonant tongues $0:1$, $1:3$, $1:2$, $2:3$, $3:4$, and $1:1$ are shown. (b) Enlargement from (a). Parameter variations along the arrows in (a) and (b) correspond to the bifurcation diagrams in Figs. 1a, 1b, respectively.

Figure 4

Two-parameter bifurcation diagram for the Kuramoto model from Eq. (1) with $N=5$ and ${\Delta }_i=0.5$, $i=\overline{2,4}$. Four main resonant tongues are shown.