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.

Amplitude-mediated chimera states.

We investigate the possibility of obtaining chimera state solutions of the nonlocal complex Ginzburg-Landau equation (NLCGLE) in the strong coupling limit when it is important to retain amplitude variations. Our numerical studies reveal the existence of a variety of amplitude-mediated chimera states (including stationary and nonstationary two-cluster chimera states) that display intermittent emergence and decay of amplitude dips in their phase incoherent regions. The existence regions of the single-cluster chimera state and both types of two-cluster chimera states are mapped numerically in the parameter space of C(1) and C(2), the linear and nonlinear dispersion coefficients, respectively, of the NLCGLE. They represent a new domain of dynamical behavior in the well-explored rich phase diagram of this system. The amplitude-mediated chimera states may find useful applications in understanding spatiotemporal patterns found in fluid flow experiments and other strongly coupled systems.

Phase-locked patterns and amplitude death in a ring of delay-coupled limit cycle oscillators.

We study the existence and stability of phase-locked patterns and amplitude death states in a closed chain of delay coupled identical limit cycle oscillators that are near a supercritical Hopf bifurcation. The coupling is limited to nearest neighbors and is linear. We analyze a model set of discrete dynamical equations using the method of plane waves. The resultant dispersion relation, which is valid for any arbitrary number of oscillators, displays important differences from similar relations obtained from continuum models. We discuss the general characteristics of the equilibrium states including their dependencies on various system parameters. We next carry out a detailed linear stability investigation of these states in order to delineate their actual existence regions and to determine their parametric dependence on time delay. Time delay is found to expand the range of possible phase-locked patterns and to contribute favorably toward their stability. The amplitude death state is studied in the parameter space of time delay and coupling strength. It is shown that death island regions can exist for any number of oscillators N in the presence of finite time delay. A particularly interesting result is that the size of an island is independent of N when N is even but is a decreasing function of N when N is odd.