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A dead zone in the interaction between two dynamical systems is a region of their joint phase space where one system is insensitive to the changes in the other. These can arise in a number of contexts, and their presence in phase interaction functions has interesting dynamical consequences for the emergent dynamics. In this paper, we consider dead zones in the interaction of general coupled dynamical systems. For weakly coupled limit cycle oscillators, we investigate criteria that give rise to dead zones in the phase interaction functions. We give applications to coupled multiscale oscillators where coupling on only one branch of a relaxation oscillation can lead to the appearance of dead zones in a phase description of their interaction.
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The dynamics of networks of interacting dynamical systems depend on the nature of the coupling between individual units. We explore networks of oscillatory units with coupling functions that have dead zones, that is, the coupling functions are zero on sets with interior. For such networks, it is convenient to look at the effective interactions between units rather than the (fixed) structural connectivity to understand the network dynamics. For example, oscillators may effectively decouple in particular phase configurations. Along trajectories the effective interactions are not necessarily static, but the effective coupling may evolve in time. Here, we formalize the concepts of dead zones and effective interactions. We elucidate how the coupling function shapes the possible effective interaction schemes and how they evolve in time.
Spontaneous synchronization is a remarkable collective effect observed in nature, whereby a population of oscillating units, which have diverse natural frequencies and are in weak interaction with one another, evolves to spontaneously exhibit collective oscillations at a common frequency. The Kuramoto model provides the basic analytical framework to study spontaneous synchronization. The model comprises limit-cycle oscillators with distributed natural frequencies interacting through a mean-field coupling. Although more than forty years have passed since its introduction, the model continues to occupy the centre-stage of research in the field of non-linear dynamics, and is also widely applied to model diverse physical situations. In this brief review, starting with a derivation of the Kuramoto model and the synchronization phenomenon it exhibits, we summarize recent results on the study of a generalized Kuramoto model that includes inertial effects and stochastic noise. We describe the dynamics of the generalized model from a different yet a rather useful perspective, namely, that of long-range interacting systems driven out of equilibrium by quenched disordered external torques. A system is said to be long-range interacting if the inter-particle potential decays slowly as a function of distance. Using tools of statistical physics, we highlight the equilibrium and nonequilibrium aspects of the dynamics of the generalized Kuramoto model, and uncover a rather rich and complex phase diagram that it exhibits, which underlines the basic theme of intriguing emergent phenomena that are exhibited by many-body complex systems.
Suppose we are given a system of coupled oscillators on an arbitrary graph along with the trajectory of the system during some period. Can we predict whether the system will eventually synchronize? This is an important but analytically intractable question especially when the structure of the underlying graph is highly varied. In this work, we take an entirely different approach that we call learning to predict synchronization (L2PSync), by viewing it as a classification problem for sets of graphs paired with initial dynamics into two classes: `synchronizing or `non-synchronizing. Our conclusion is that, once trained on large enough datasets of synchronizing and non-synchronizing dynamics on heterogeneous sets of graphs, a number of binary classification algorithms can successfully predict the future of an unknown system with surprising accuracy. We also propose an ensemble prediction algorithm that scales up our method to large graphs by training on dynamics observed from multiple random subgraphs. We find that in many instances, the first few iterations of the dynamics are far more important than the static features of the graphs. We demonstrate our method on three models of continuous and discrete coupled oscillators -- The Kuramoto model, the Firefly Cellular Automata, and the Greenberg-Hastings model.
We study the effects of delayed coupling on timing and pattern formation in spatially extended systems of dynamic oscillators. Starting from a discrete lattice of coupled oscillators, we derive a generic continuum theory for collective modes of long wavelength. We use this approach to study spatial phase profiles of cellular oscillators in the segmentation clock, a dynamic patterning system of vertebrate embryos. Collective wave patterns result from the interplay of coupling delays and moving boundary conditions. We show that the phase profiles of collective modes depend on coupling delays.
We study synchronization dynamics of a population of pulse-coupled oscillators. In particular, we focus our attention in the interplay between networks topological disorder and its synchronization features. Firstly, we analyze synchronization time $T$ in random networks, and find a scaling law which relates $T$ to networks connectivity. Then, we carry on comparing synchronization time for several other topological configurations, characterized by a different degree of randomness. The analysis shows that regular lattices perform better than any other disordered network. The fact can be understood by considering the variability in the number of links between two adjacent neighbors. This phenomenon is equivalent to have a non-random topology with a distribution of interactions and it can be removed by an adequate local normalization of the couplings.
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