Role of time scales and topology on the dynamics of complex networks

Abstract

The interplay between time scales and structural properties of complex networks of nonlinear oscillators can generate many interesting phenomena, like amplitude death, cluster synchronization, frequency synchronization, etc. We study the emergence of such phenomena and their transitions by considering a complex network of dynamical systems in which a fraction of systems evolves on a slower time scale on the network. We report the transition to amplitude death for the whole network and the scaling near the transitions as the connectivity pattern changes. We also discuss the suppression and recovery of oscillations and the crossover behavior as the number of slow systems increases. By considering a scale free network of systems with multiple time scales, we study the role of heterogeneity in link structure on dynamical properties and the consequent critical behaviors. In this case with hubs made slow, our main results are the escape time statistics for loss of complete synchrony as the slowness spreads on the network and the self-organization of the whole network to a new frequency synchronized state. Our results have potential applications in biological, physical, and engineering networks consisting of heterogeneous oscillators. Complexities of real world systems are studied mainly in terms of the nonlinearity in the intrinsic dynamics of their subsystems and the complex interaction patterns among them. In such systems, the variability and heterogeneity of the interacting subsystems can add a further level of complexity. In this context, heterogeneity arising from differing dynamical time scales offers several challenges and has applications in diverse fields, ranging from biology, economy, and sociology to physics and engineering. In our work, we study the interesting cooperative dynamics in interacting nonlinear systems of differing time scales using the framework of complex networks.