Two types of quasiperiodic partial synchrony in oscillator ensembles

We analyze quasiperiodic partially synchronous states in an ensemble of Stuart-Landau oscillators with global nonlinear coupling. We reveal two types of such dynamics: in the first case the time-averaged frequencies of oscillators and of the mean field differ, while in the second case they are equal, but the motion of oscillators is additionally modulated. We describe transitions from the synchronous state to both types of quasiperiodic dynamics, and a transition between two different quasiperiodic states. We present an example of a bifurcation diagram, where we show the borderlines for all these transitions, as well as domain of bistability.

Figure 1

Bifurcation diagram for ${\varepsilon }_1=3$. Blue solid line: here the largest real eigenvalue equals zero; red bold line: here real parts of complex roots are equal to zero. Red filled circle marks the Takens-Bogdanov point. Theoretical analysis is complemented by the numerical study, which reveals five domains: stable full synchrony (SYN), quasiperiodic partial synchrony (QPS-I and QPS-II), and two domains where bistability between synchrony and partial synchrony is observed (SYN/QPS-I and SYN/QPS-II). Red dashed and cyan dotted curves are obtained numerically; they denote emergence of the amplitude modulation and of the frequency difference, respectively; see text for details. The line ${\varepsilon }_2=0,{\eta }_1>3$ corresponds to the partially synchronous but not quasiperiodic bunch states. Four horizontal black dashed-dotted lines show the cuts of the diagram illustrated in Figs. 2, 4, 7, 8, and in the text.

Figure 2

Transition from synchrony to quasiperiodic partial synchrony of type I for small ${\varepsilon }_2=0.3$. (a) Mean field amplitude $\rho$. (b) Frequency difference is the essential feature of the QPS-I dynamics. Here we see that the mean field is faster than the oscillators, and that the frequency difference $\nu -\mathrm{\Omega }$ smoothly grows beyond the transition point. (c) Frequency modulation of the oscillators and of the mean field is quantified by the standard deviations of their instantaneous frequencies, denoted as ${\sigma }_o$ and ${\sigma }_f$; these quantities are shown by blue solid and red bold curves, respectively. (d) Here blue solid and red bold curves show the standard deviations of oscillator and mean field amplitudes, denoted as ${\delta }_o$ and ${\delta }_f$, respectively.

Figure 3

(a)–(c) Phase portraits, illustrating the synchrony-to-QPS-I transition at ${\varepsilon }_2=0.3$. Gray solid and blue bold curves show trajectories of an oscillator and of the mean field, respectively for ${\eta }_1=3.5$ (a), ${\eta }_1=7$ (b), and ${\eta }_1=10$ (c). Open circles (magenta) show a snapshot of the ensemble. (d)–(f) Probability distribution $w$ of instantaneous frequency of the oscillator, ${\omega }_o$ (bold black curve), and of the mean field, ${\omega }_f$ (red curve) for ${\eta }_1=3.5,{\eta }_1=7$, and ${\eta }_1=10$, respectively.

Figure 4

Transition from synchrony to QPS-II and then to QPS-I, for ${\varepsilon }_2=3$. The shown quantities are same as in Fig. 2. In (d) we additionally show the minimal value of the oscillator amplitude $r$ (dashed curve). The first transition, synchrony to QPS-II, takes place at ${\eta }_1=7$, as predicted by the stability analysis. At this point the amplitude and frequency modulation of oscillators emerge smoothly, but the average frequencies are still equal, $\nu =\mathrm{\Omega }$. At ${\eta }_1\approx 8.8$ the second transition takes place; here the frequency difference appears by jump.

Figure 5

Phase portraits, illustrating the transitions at ${\varepsilon }_2=3$ (the same curves as in Fig. 3). (a) QPS-II state, ${\eta }_1=7.1$. (b) Close to the transition from QPS-II to QPS-I, ${\eta }_1=8.5$. (c) QPS-I state at ${\eta }_1=10$.

Figure 6

Same as in Fig. 5, but for slightly nonidentical oscillators. Now it is easy to see that the trajectory (represented by a sequence of oscillators' states marked with circles) in (a) does not encircle the origin directly, but encircles the mean field, and therefore the frequency difference $\nu -\mathrm{\Omega }=0$; this is the QPS-I state. The trajectory in (c) directly encircles the origin and therefore the frequencies differ; this is the QPS-II state.

Figure 7

Hysteresis at the transition from synchrony to QPS-I for ${\varepsilon }_2=0.7$. The shown quantities are mean field amplitude $\rho$ (a), frequency difference $\nu -\mathrm{\Omega }$ (b), and standard deviations ${\sigma }_o$ (c) and ${\delta }_o$ (d) of the instantaneous frequency and of the amplitude of the oscillators. In each panel solid blue line shows the results obtained for nearly asynchronous initial conditions, while dashed-dotted red line corresponds to nearly synchronous initial conditions.

Figure 8

Hysteresis at transitions between synchrony, QPS-I and QPS-II for ${\varepsilon }_2=0.95$. The shown quantities are the same as in Fig. 7.

Figure 9

Chaotic dynamics of the model (11). Black solid line shows trajectory of an oscillator; blue bold line depicts the mean field. Open circles show a snapshot of the ensemble. (a) Phase portraits; (b) time dependence. Notice then when the oscillator amplitude decreases, the mean field makes an additional rotation with respect to the oscillator; thus, the time-averaged frequencies differ, as is typical for a QPS-I state.