Coupled limit cycle oscillators with instantaneous mutual coupling offer a useful but idealized mathematical paradigm for the study of collective behavior in a wide variety of biological, physical and chemical systems. In most real-life systems however the interaction is not instantaneous but is delayed due to finite propagation times of signals, reaction times of chemicals, individual neuron firing periods in neural networks etc. We present a brief overview of the effect of time-delayed coupling on the collective dynamics of such coupled systems. Simple model equations describing two oscillators with a discrete time-delayed coupling as well as those describing linear arrays of a large number of oscillators with time-delayed global or local couplings are studied. Analytic and numerical results pertaining to time delay induced changes in the onset and stability of amplitude death and phase-locked states are discussed. A number of recent experimental and theoretical studies reveal interesting new directions of research in this field and suggest exciting future areas of exploration and applications.

Flavour equilibration for a thermally equilibrated but chemically non-equilibrated quark-gluon plasma is presented. Flavour equilibration is studied enforcing baryon number conservation. In addition to the usual processes like single additional gluon production $gg \rightleftharpoons ggg$ and its reverse and quark–antiquark pair production by gluon pair fusion $gg \rightleftharpoons q_{i}q_{i}^{-}$ and reverse thereof, processes like quark-flavour interchanging $q_{i}q_{i}^{-} \rightleftharpoons q_{j}q_{j}^{-}$ is also considered. The degree of equilibration is studied comparatively for various reactions/constraints that are being considered.

We investigate phase-locked solutions of a continuum field of nonlocally coupled identical phase oscillators with distance-dependent propagation delays. Equilibrium relations for both synchronous and travelling wave solutions in the parameter space characterizing the nonlocality and time delay are delineated. For the synchronous states a comprehensive stability diagram is presented that provides a heuristic synchronization condition as well as an analytic relation for the marginal stability curve. The relation yields simple stability expressions in the limiting cases of local and global coupling of phase oscillators.