Kuramoto model with time-varying parameters

We analyze the Kuramoto model generalized by explicit consideration of deterministically time-varying parameters. The oscillators' natural frequencies and/or couplings are influenced by external forces with constant or distributed strengths. A dynamics of the collective rhythms is observed, consisting of the external system superimposed on the autonomous one, a characteristic feature of many thermodynamically open systems. This deterministic, stable, continuously time-dependent, collective behavior is fully described, and the external impact to the original system is defined in both the adiabatic and the nonadiabatic limits.

Figure 1

The time-varying mean field for model A, Eq. (1), resembles the externally applied cosine (a)–(c) or chaotic forcing (d). Numerical simulations of the full system, Eq. (1) [thick (light blue) curve], are in agreement with the low-dimensional dynamics [dashed (red) curve], Eqs. (14, 15, 16, 17) (see text for details). Adiabatic [dotted (brown) curve] and nonadiabatic evolutions [dashed-dotted (green) curve], Eqs. (23, 24, 25, 26, 27), confirm the reduced dynamics in its limits (see text for details). The distribution $h\left(I\right)$ is (a), (b) the same as $g\left(\omega \right)$, $K=3.5$, $\epsilon =0.6$, $\Omega =5$, and $\Omega =0.05$, respectively; (c) independent Lorentzian, $K=4.5$, ${\gamma }_I=0.6$ and $\Omega =1$; and (d) bimodal $\delta$, $K=8$, $\gamma =1$, ${\gamma }_I=1$, and $\widehat{I}=1$.

Figure 2

Time-varying mean field for model B, Eq. (2), follows the external cosine (a) or the chaotic (b) forcing. The numerical simulation of Eq. (2) [thick (light blue) curve] coincides with the low-dimensional evolution [dashed (red) curve], Eq. (18), shown in (a), and Eq. (19), shown in (b). (a) Adiabatic [dotted (brown) curve], Eq. (26), and nonadiabatic evolution [dashed-dotted (green) curve], Eq. (25), for constant forcing with $K=3$, $\Omega =0.1$, and $\epsilon =0.33$. (b) Bimodal $\delta$-distributed strengths with $K=5$, ${\gamma }_I=1$, and $\widehat{I}=0$.

Figure 3

Magnitude of the response, $\Delta (\epsilon ,\Omega )$, of the NA model A to cosine forcing, Eq. (1). External forcing strengths follow the distribution of frequencies, $K=4.25$ and $\Omega \in [{10}^{-2},{10}^2]$. (a) Results from Eq. (14) for $\epsilon \in [0.05,0.99]$. (b) Nonadiabatic [dotted (black) lines], Eq. (21), and adiabatic [dashed (black) lines], Eq. (24), evolution for $\epsilon \in \{0.05,0.1055,0.2225,0.4693,0.99\}$, compared with the real dynamics [thick (light blue) lines], Eq. (14).