Kuramoto model in the presence of additional interactions that break rotational symmetry

The Kuramoto model serves as a paradigm to study the phenomenon of spontaneous collective synchronization. We study here a nontrivial generalization of the Kuramoto model by including an interaction that breaks explicitly the rotational symmetry of the model. In an inertial frame (e.g., the laboratory frame), the Kuramoto model does not allow for a stationary state, that is, a state with time-independent value of the so-called Kuramoto (complex) synchronization order parameter $z\equiv r{e}^{i\psi }$. Note that a time-independent $z$ implies $r$ and $\psi$ are both time independent, with the latter fact corresponding to a state in which $\psi$ rotates at zero frequency (no rotation). In this backdrop, we ask: Does the introduction of the symmetry-breaking term suffice to allow for the existence of a stationary state in the laboratory frame? Compared to the original model, we reveal a rather rich phase diagram of the resulting model, with the existence of both stationary and standing wave phases. While in the former the synchronization order parameter $r$ has a long-time value that is time independent, one has in the latter an oscillatory behavior of the order parameter as a function of time that nevertheless yields a nonzero and time-independent time average. Our results are based on numerical integration of the dynamical equations as well as an exact analysis of the dynamics by invoking the so-called Ott-Antonsen ansatz that allows to derive a reduced set of time-evolution equations for the order parameter.

Figure 1

For the model (4) and considering for ${\omega }_j$'s the Lorentzian distribution (7) with $\gamma =0.5$ and ${\omega }_0=1.0$, the figure shows the thresholds ${\epsilon }_{1c}^{\mathrm{ISS}}\left({\epsilon }_2\right)$ and ${\epsilon }_{1c}^{\mathrm{SSS}}\left({\epsilon }_2\right)$ as a function of ${\epsilon }_2$, where ${\epsilon }_{1c}^{\mathrm{ISS}}\left({\epsilon }_2\right)$ [respectively, ${\epsilon }_{1c}^{\mathrm{SSS}}\left({\epsilon }_2\right)$] defines the stability threshold of an incoherent stationary state (ISS) [respectively, the existence threshold of a synchronized stationary state (SSS)].

Figure 2

For the model (4) and considering for ${\omega }_j$'s the Lorentzian distribution (7) with $\gamma =0.5$ and ${\omega }_0=1.0$, the figure depicts the two-parameter bifurcation diagram in the $({\epsilon }_1,{\epsilon }_2)$ plane. We show here the various stable phases and the phase boundaries, with bifurcation behavior observed as one crosses the phase boundaries. (a) The shaded regions, representing stable existence of the ISS (Incoherent Stationary State), the SSS (Synchronized Stationary State), and the SWS (Standing Wave State), have been constructed by analyzing the long-time numerical solution of the dynamics (4) for $N={10}^5$. In numerics, we distinguish between the different regions by requiring that at long times, $r\left(t\right)$ as a function of $t$ behaves differently for the ISS, the SSS, and the SWS. Namely, for the ISS, both the order parameter $r\left(t\right)$ and its time average take the value zero in the $t\to \infty$ limit. In the SWS, $r\left(t\right)$ as $t\to \infty$ oscillates in time around a time-independent nonzero value, thereby yielding a nonzero time average. For the SSS, too, $r\left(t\right)$ as $t\to \infty$ has a nonzero time-independent time average, but it does not oscillate, instead remains equal to a nonzero constant in time. The regions R1 and R2 represent multistability (hysteresis) between ISS-SSS and SSS-SWS, respectively. The dot-dashed blue line is obtained by using Eq. (27), the solid black line is obtained by using Eq. (29), while the dashed green line is obtained from an analysis of Eq. (23) by using the numerical package XPPAUT [18] in which we study the stability of the SWS; note that all these lines are based on the Ott-Ansatz-reduced dynamics corresponding to the dynamics (4). (b) Enlarged view of the boxed region in (a).

Figure 3

The first four panels show the quantity $R$, namely, the time-averaged order parameter in the long-time limit, see Eq. (10), as a function of adiabatically tuned ${\epsilon }_1$ and for four different values of ${\epsilon }_2$, see text. (e) shows $r\left(t\right)$ as a function of $t$ for ${\epsilon }_2=1.4$ and for three representative ${\epsilon }_1$ values, namely, (i) ${\epsilon }_1=0.35$, when we have an ISS (see Fig. 2), (ii) ${\epsilon }_1=1.2$, when we have a SWS, and (iii) ${\epsilon }_1=1.7$, when we have a SSS. In all cases, the lines are obtained from numerical integration of the dynamics (4) for $N=5\times {10}^5$, while symbols correspond to predictions based on the OA-ansatz-reduced dynamics (23) discussed in Sec. 3. The frequency distribution is a Lorentzian, see Eq. (7), with $\gamma =0.5$ and $\omega =1.0$.

Figure 4

Corresponding to the insets in panel (e) of Fig. 3, the figure shows ${\theta }_j$ for different $j$ and as a function of time at long times (the figure depicts the results not for the full range of $j$ but for $j=1,2,...,1000$). The color coding given on the side of each panel denotes the intensity of $\theta$ values. The top panel corresponds to the ISS, the middle panel to the SWS and the bottom panel to the SSS. As may be seen from the figure, unlike the ISS and the SSS, the SWS exhibits a stationary wave pattern: at a fixed $j$, the value of $\theta$ changes periodically as a function of time. The data are obtained from numerical integration of the dynamics (4) with parameter values and other details same as in Fig. 3.

Figure 5

Corresponding to the panel (e) in Fig. 3, the figure shows as a function of ${\epsilon }_1$ the mean-ensemble frequency $f$ and the mean-field frequency $\mathrm{\Omega }$ at long times. Here, we have compared numerical integration results with those based on the OA ansatz; for details of computation, see Sec. 4.

Figure 6

The four panels show the quantity $R$, namely, the time-averaged order parameter in the long-time limit, see Eq. (10), as a function of adiabatically-tuned ${\epsilon }_1$ and for four different values of ${\epsilon }_2$, see text. In all cases, the lines are obtained from numerical integration of the dynamics (4) for $N=5\times {10}^5$, while symbols correspond to predictions based on the OA-ansatz-reduced dynamics (23) discussed in Sec. 3. The frequency distribution is a Lorentzian, see Eq. (7), with $\gamma =0.5$ and $\omega =1.0$.

Figure 7

The four panels show the quantity $R$, namely, the time-averaged order parameter $R\left(t\right)$ in the long-time limit, see Eq. (10), as a function of adiabatically tuned ${\epsilon }_1$ and for four different values of ${\epsilon }_2$, see text. In all cases, the lines are obtained from numerical integration of the dynamics (4) for $N=5\times {10}^5$. The frequency distribution is a Gaussian, see Eq. (8), with ${\omega }_0=1.0$ and $\sigma =0.5$.