Mode locking in systems of globally coupled phase oscillators

We investigate the dynamics of a Kuramoto-type system of globally coupled phase oscillators with equidistant natural frequencies and a coupling strength below the synchronization threshold. It turns out that in such cases one can observe a stable regime of sharp pulses in the mean field amplitude with a pulsation frequency given by spacing of the natural frequencies. This resembles a process known as mode locking in lasers and relies on the emergence of a phase relation induced by the nonlinear coupling. We discuss the role of the first and second harmonics in the phase-interaction function for the stability of the pulsations and present various bifurcating dynamical regimes such as periodically and chaotically modulated mode locking, transitions to phase turbulence, and intermittency. Moreover, we study the role of the system size and show that in certain cases one can observe type II supertransients, where the system reaches the globally stable mode-locking solution only after an exponentially long transient of phase turbulence.

Figure 1

Numerically calculated solutions of system (1, 2). (a)–(d) Time evolution of mean field amplitude $R_1\left(t\right),R_2\left(t\right)$, and phase velocities ${\dot{\theta }}_k\left(t\right)$; (e)–(g) snapshots of the phases at the time moments indicated by vertical lines in (a) and (c). Parameter values: (a) $(K,\gamma )=(1.25,0.7)$, (b) and (c) $(K,\gamma )=(0.91,1.0)$, (d) $K=0$. The system size is $N=21$.

Figure 2

Lyapunov spectra of the turbulent solutions at $(K,\gamma )=(0.91,1)$ for different system sizes $N\in \{21,31,41\}$ show extensive chaos. The time span of the computation is $2\times {10}^5$ units.

Figure 3

Interaction between two oscillators with phase difference $\varphi$ given by $f\left(\varphi \right)=\gamma sin\left(\varphi \right)+(1-\gamma )sin\left(2\varphi \right)$ for three different values of $\gamma$. For $2/3\le \gamma \le 1$ the interaction is everywhere attractive, while for $\gamma <2/3$ it becomes repulsive for $\left|\varphi \right|\approx \pi$, attracting distant oscillators to an antiphase position.

Figure 4

Time traces of order parameters $R_{1,2}$ (upper rows) together with normalized expansion rate of phase space volume $\mathrm{\Lambda }\left(t\right)/N$ and maximal instantaneous eigenvalue $max{\lambda }_i\left(t\right)$. (a) MLS with first harmonic interaction $(K,\gamma )=(0.91,1.0)$ [here only $R_1\left(t\right)$ is relevant]; (b) MLS with second harmonic interaction $(K,\gamma )=(1.25,0.7)$.

Figure 5

Logarithmic plot of the average transient time from random initial conditions to the stable MLS, varying the system size $N$. Averages are taken for $N\le 53$ over 300 realizations of initial conditions. For larger $N$, their number had to be restricted to 50. Parameters: $(K,\gamma )=(1.2,0.7)$.

Figure 6

Average transient times in logarithmic plot for varying $\gamma$. Coupling strength $K=1.2,1.25,1.3$, panels (a), (b), (c), respectively. Symbols and colors indicate different system sizes $N\in \{21,25,31\}$. Each point represents an average over 200 random initial condition.

Figure 7

For a system of $N=21$ the average and maximal order parameter are shown in (a) and (b), respectively. As initial conditions we used synchronous solutions obtained for $K>K_C$. Marked points correspond to parameter values of examples shown in other figures: $⟐$ = globally stable MLS, see Fig. 4 (b); $⊡$ = stable MLS coexisting with turbulence, see Fig. 4 (a); $◬$ = modulated MLS (comb splitting), see Fig. 9; $\odot$ = torus breakup, see Fig. 10; $\times$ = intermittent solution, see Fig. 11.

Figure 8

(a) Sampled return times $T_{\nu }$ for varying parameter $K,\gamma =0.82$. Different colors correspond to increasing (blue) and decreasing (orange) coupling strength. Narrow periodic windows occur between regions of phase chaos. (b) Sampled distances $\chi$ from the symmetry-invariant subspace with respect to the symmetry action $\sigma$ at the crossings of the section.

Figure 9

Comb splitting MLS for $(K,\gamma )=(0.765,1.0)$ and $N=21$. (a) Time trace of the order parameter $R_1\left(t\right)$; (b) orbit of period 7 in a two-dimensional Poincaré section with return times $T_{\nu }$ of consecutive section crossings plotted against each other; (c) spacings $\mathrm{\Delta }\mathrm{\Omega }$ of effective frequencies.

Figure 10

(a) Sampled return times $T_{\nu }$ between crossings of the Poincaré section (11) for $\gamma =0.82$ and varying $K$; blue and orange correspond to sweeping $K$ in the increasing and decreasing directions. (b) and (c) Two-dimensional representation of return times $T_{\nu }$ for selected values of $K$ given by dashed lines in panel (a). Closed curves in (b) represent invariant tori. Low-dimensional chaos in (c).

Figure 11

Time trace of $R_1\left(t\right)$ at $(K,\gamma )=(1.12,0.82)$, showing intermittency between phase chaos and a MLS. The rolling average (15) can be used to distinguish between the two regimes (red line).

Figure 12

Distance from the critical parameter $K_b$ and average fraction of mode locking $〈{\tau }_{\text{ML}}〉$ in a double logarithmic plot.