Collective dynamics of time-delay-coupled phase oscillators in a frustrated geometry

We study the effect of time delay on the dynamics of a system of repulsively coupled nonlinear oscillators that are configured as a geometrically frustrated network. In the absence of time delay, frustrated systems are known to possess a high degree of multistability among a large number of coexisting collective states except for the fully synchronized state that is normally obtained for attractively coupled systems. Time delay in the coupling is found to remove this constraint and to lead to such a synchronized ground state over a range of parameter values. A quantitative study of the variation of frustration in a system with the amount of time delay has been made and a universal scaling behavior is found. The variation in frustration as a function of the product of time delay and the collective frequency of the system is seen to lie on a characteristic curve that is common for all natural frequencies of the identical oscillators and coupling strengths. Thus time delay can be used as a tuning parameter to control the amount of frustration in a system and thereby influence its collective behavior. Our results can be of potential use in a host of practical applications in physical and biological systems in which frustrated configurations and time delay are known to coexist.

Figure 1

Frustrated systems of oscillators. Circles on the vertices represent phase oscillators, and the edges represent repulsive coupling links between the oscillators.

Figure 2

Plot of the stable collective frequencies ${\mathrm{\Omega }}_{\text{in}}$ and ${\mathrm{\Omega }}_{\text{out}}$ of the oscillators in the in-phase state (black solid line) and the out-of-phase state (red dashed line) as functions of the delay parameter $\tau$.

Figure 3

Plots showing the variation of the frustration parameter $F$ as a function of the delay parameter. $F$ is calculated using Eq. (2) from numerical solutions of Eq. (1) for the same set of initial phases for (a) three values of intrinsic frequency $\omega$ at fixed $\kappa =-2$, and (b) three values of coupling strength $\kappa$ at fixed $\omega =1.2$. (c) In this figure, $F$ is plotted against the product of collective frequency $\mathrm{\Omega }$ and delay parameter $\tau$ for various values of $\omega$ and $\kappa$ and a given set of initial phases. Frustration values for different intrinsic frequencies as well as coupling strengths are seen to lie on a common curve given by $F_1$ and $F_2$, which are obtained from Eqs. (10) and (11), respectively.

Figure 4

Numerical data points and analytical fit of the variation of the saturation time $t_{\text{sat}}$ are plotted with time delay $\tau$ for a system of three repulsive oscillators for $\omega =40$ and $\kappa =-2$, for which the critical time delay ${\tau }_c=0.0389$.

Figure 5

The phase patterns corresponding to the two equilibria of the six-oscillator network shown in Fig. 1 in the absence of time delay. In this figure, the phases of all six oscillators are plotted together. Part (a) is obtained from an initial condition that settles to the equilibrium state corresponding to $F=0.5$. This equilibrium state corresponds to a three-cluster pattern such that there are two oscillators in each cluster, and the oscillators in the same cluster share the same phase. Part (b) is obtained from an initial condition whose equilibrium state corresponds to $F=0.5505$. This equilibrium state has a six-cluster pattern, i.e., the instantaneous values of individual phases of all six oscillators are different. We have taken $\omega =0.7$ and $\kappa =-2$ to get this figure.

Figure 6

In this figure, $F$ is plotted against the product of collective frequency $\mathrm{\Omega }$ and delay parameter $\tau$ for (a) the system of six oscillators and (b) the $4\times 4$ triangular lattice with free boundaries. These systems show the existence of multiple equilibrium states characterized by different values of the frustration parameter for some values of delay, and therefore frustration $F$ can have multiple values at some values of $\mathrm{\Omega }\tau$. Depending on the choice of initial phases, the system can settle down to any of these equilibrium states. We have taken $\kappa =-2$ to obtain all the curves.

Figure 7

The blue dashed line with squares shows the percentage of initial phases that lead to an equilibrium state with a lower value of $F$ denoted by $F_{\text{lower}}$. The red dashed line with diamonds shows the percentage of initial phases that lead to the equilibrium states with a higher value of $F$ denoted by $F_{\text{upper}}$. The black dashed line with triangles shows the percentage of initial phases that lead to the in-phase equilibrium state labeled by In-Phase, and the magenta dashed line with circles shows the percentage of initial conditions for which the system has not settled to any stable equilibrium state even for a very large time span ($t_{\text{span}}=2000$ in simulations), and it is labeled as Unsteady.

Figure 8

The variation in the cluster pattern exhibited by the $4\times 4$ lattice of repulsively coupled oscillators with a change in the time-delay parameter $\tau$ for a given set of initial phases. At a certain value of delay, the system starts exhibiting the single cluster in-phase state. The collective frequency of the in-phase state decreases with a further increase in delay. We have taken $\omega =0.7$ and $\kappa =-2$.