Spatiotemporal dynamics of the Kuramoto-Sakaguchi model with time-dependent connectivity

We study the dynamics of the paradigmatic Kuramoto-Sakaguchi model of identical coupled phase oscillators with various kinds of time-dependent connectivity using Eulerian discretization. We explore the parameter spaces for various types of collective states using the phase plots of the two statistical quantities, namely, the strength of incoherence and the discontinuity measure. In the quasistatic limit of the changing of coupling range, we observe how the system relaxes from one state to another and identify a few interesting collective dynamical states along the way. Under a sinusoidal change of the coupling range, the global order parameter characterizing the degree of synchronization in the system is shown to undergo a hysteresis with the coupling range. We also study the low-dimensional spatiotemporal dynamics of the local order parameter in the continuum limit using the recently developed Ott-Antonsen ansatz and justify some of our numerical results. In particular, we identify an intrinsic time scale of the Kuramoto system and show that the simulations exhibit two distinct kinds of qualitative behavior in two cases when the time scale associated with the switching of the coupling radius is very large compared to the intrinsic time scale and when it is comparable to the intrinsic time scale.

Figure 1

Phase plot for (a) the strength of incoherence and (b) the discontinuity measure for $\epsilon =0.5$ and $N=100$ after 5000 iterations and $\delta =1.0$; note the sharp change near $\alpha =\frac{\pi }{2}$.

Figure 2

Phase plots for (a) and (b) the strength of incoherence and (c) and (d) the discontinuity measure for $N=100$ after 5000 iterations and $\delta =1.0$ with (a) and (c) $\alpha =1.4$ and (b) and (d) $\alpha =1.6$; note the qualitative changes across $\alpha =\frac{\pi }{2}$.

Figure 3

Various collective states: (a) asynchronous ($\alpha =1.4, \epsilon =0.01$, and $r=0.01$), (b) synchronous ($\alpha =1.4, \epsilon =1.0$, and $r=0.49$), (c) multichimera ($\alpha =1.46, \epsilon =0.5$, and $r=0.15$), (d) ordered ($\alpha =1.6, \epsilon =1.0$, and $r=0.46$), (e) twisted ($\alpha =1.0, \epsilon =1.0$, and $r=0.02$), and (f) chimera ($\alpha =1.46,\epsilon =0.9,r=0.2$), all self-organized from a random initial state.

Figure 4

Collective states: (a) short-range-order state created from the random initial condition after sequentially using $r=0.12$ and 0.02 each for 5000 steps of iteration with $\alpha =\pi +1.0$ and (b) periodic multichimera created from the random initial condition after sequentially using $r=0.33$ for $20000$ steps and $r=0.49$ for $30000$ steps of iteration with $\alpha =\frac{\pi }{2}+0.11$ both for $N=1000$. Note that none of these states are obtainable directly from random initial conditions with the set of parameter values used in the second step throughout.

Figure 5

Phase dynamics of collective states with short-range order and their spatial correlation profile at the last time step: $L=1000$ and $\alpha =\pi +1.0$ for all, with the coupling radius $r$ switched to 0.02 at the $5000\text{th}$ step. The coupling radius at the first step was (a) 0.02, (b) 0.08, (c) 0.10, and (d) 0.12. Note the initial twisted states in all cases except the first one.

Figure 6

(a) Phase dynamics of the oscillators with variable coupling radius and (b) coevolution of the coupling radius $r$ (sinusoidal line) with order parameter $\left|Z\right|$ (nonsinusoidal line). In both, the initial phases were random, $N=1000, \epsilon =0.07, \alpha =\frac{\pi }{2}-0.1$, and the coupling radius varied from the nearest neighbor to global coupling with a time period of 1000 steps.

Figure 7

Variation of the hysteresis loop area with (a) the coupling strength $\epsilon$ for $N=100, R_0=25, T_r=1000$, and $\alpha =\frac{\pi }{2}-1.0$ and (b) the Sakaguchi phase $\alpha$ for $N=100, R_0=15, T_r=1000$, and $\epsilon =0.04$.

Figure 8

(a) Time evolution of the modulus of the local order parameter $\left|Z\right|$ in space and time and coevolution of the coupling radius $r$ (sinusoidal line) and (b) $\left|Z\right(20,t\left)\right|$ (nonsinusoidal line) in time for $\alpha =\frac{\pi }{2}-0.1$ and $\epsilon =0.5$, with the couplings oscillating between nearest neighbors to global coupling within a time period of 1000 units.

Figure 9

(a) Time evolution of the global order parameter obtained from direct simulation of the Kuramoto model with $\alpha =\frac{\pi }{2}-0.1, \epsilon =0.5, L=100$, coupling radius $r=0.46$, and $dt=1.0$ and (b) position of its peaks fitted with the straight line $y=58+8(x-1)$ corresponding to a period of 8 time units. Here the expected intrinsic time scale is $\frac{2\pi }{\epsilon sin\alpha }\approx 13$ and the observed oscillation occurs with a period roughly equal to that time scale.

Figure 10

(a) Time evolution of the modulus of the local order parameter $\left|Z\right|$ in space and time, (b) coevolution of the coupling radius $r$ (sinusoidal line) and $\left|Z\right(20,t\left)\right|$ (nonsinusoidal line) in time for $\alpha =\frac{\pi }{2}-0.01$ and $\epsilon =0.25$, and (c) the hysteresis loop associated with the couplings oscillating between nearest neighbors to global coupling within a time period of 200 units.

Figure 11

(a) Phase plot for strength of incoherence and (b) discontinuity measure for $\epsilon =0.07, N=100$, and $R_0=15$ averaged over $15000$ iterations and $\delta =1.0$; note the sharp change near $\alpha =\frac{\pi }{2}$. The line is the curve $y=\frac{2\pi }{\epsilon sin\left(x\right)}$.