Arnold tongues in the forced Kuramoto model with matrix coupling

Abstract

We consider a generalization of the Kuramoto model in which phase oscillators are represented by unit vectors coupled by a matrix of constant coefficients. We show that, when the oscillators are driven by an external periodic force, several resonances appear, giving rise to Arnold tongues that can be observed as the intensity and frequency of the external force are varied. Applying the Ott-Antonsen ansatz we obtain equations for the module and phase of order parameter. As these equations are explicitly time-dependent, we resort to extensive numerical simulations to uncover the resonant modes and their associated Arnold tongues and devil's staircases. These results contrast with the original forced Kuramoto model, where only $1:1$ resonance is possible.

Figures (6)

• Figure 1: Examples of trajectories for the matrix coupled Kuramoto model with external forces with $F = 0.1$ and (a) $\xi = 0.01$ and (b) $\xi = 0.15$. Panel (c) shows the x component of $\vec{r}$ for cases (a) (black) and (b) (red) in a short time interval.
• Figure 2: (a)Time-averaged order parameter $\langle r \rangle$ (left axis in blue) and Winding number (right axis in red) for different values of frequency drift $\xi$ and $F = 0.1$. (b) Values of the $x$-component of $\vec{r}$ found in the recurrence maps.
• Figure 3: Heatmap of the time averaged order parameter $\left \langle r \right \rangle$ for the oscillatory case.
• Figure 4: (a) Arnold tongues representing different mode locks in the matrix coupled Kuramoto model in the oscillatory state. (b) Zoom in an region of (a) showing a large variety of smaller tongues with different mode locks. (Color online)
• Figure 5: (a)Time-averaged order parameter $\langle r \rangle$ (left axis in blue) and Winding number (right axis in red) for different values of frequency drift $\xi$ and $F = 9.0$ (b-c) Examples of trajectories for the matrix coupled Kuramoto model set in the phase tuned states, with external forces with $F = 9.0$ and (b) $\xi = 9.54$ and (c) $\xi = 9.73$.