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The stability (or instability) of synchronization is important in a number of real world systems, including the power grid, the human brain and biological cells. For identical synchronization, the synchronizability of a network, which can…
The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among them. It has been observed that if the natural frequencies of the oscillators…
We investigate the engineering scenario where the objective is to synchronize heterogeneous oscillators in a distributed fashion. The internal dynamics of each oscillator are general enough to capture their time-varying natural frequency as…
Synchronization in a lattice of a finite population of phase oscillators with algebraically decaying, non-normalized coupling is studied by numerical simulations. A critical level of decay is found, below which full locking takes place if…
We consider a Kuramoto model for the dynamics of an excitable system consisting of two coupled active rotators. Depending on both the coupling strength and the noise, the two rotators can be in a synchronized or desynchronized state. The…
Cortical regions without direct neuronal connections have been observed to exhibit synchronized dynamics. A recent empirical study has further revealed that such regions that share more common neighbors are more likely to behave coherently.…
Rigorous conditions for cluster synchronization of Kuramoto oscillators are presented. The method of averaging plays an important role in stability analysis, but the standard Lyapunov's second method is not applicable due to the lack of…
We investigate algebraic and topological signatures of networks of coupled oscillators. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many…
In this work we investigate the stability of synchronized states for the Kuramoto model on scale-free and random networks in the presence of white noise forcing. We show that for a fixed coupling constant, the robustness of the globally…
From the flashes of fireflies to Josephson junctions and power infrastructure, networks of coupled phase oscillators provide a powerful framework to describe synchronization phenomena in many natural and engineered systems. Most real-world…
In this paper, we propose an $N$ oscillators Kuramoto model with quaternions $\mathbb{H}$. In case the coupling strength is strong, a sufficient condition of synchronization is established for general $N\geqslant 2$. On the other hand, we…
Improving the frequency precision by synchronizing a lattice of oscillators is studied in the phase reduction limit. For the most commonly studied case of purely dissipative phase coupling (the Kuramoto model) I confirm that the frequency…
We explore the behaviour of chaotic oscillators in hierarchical networks coupled to an external chaotic system whose intrinsic dynamics is dissimilar to the other oscillators in the network. Specifically, each oscillator couples to the…
Complex networks often possess communities defined based on network connectivity. When dynamics undergo in a network, one can also consider dynamical communities; i.e., a group of nodes displaying a similar dynamical process. We have…
Many complex networks display strong heterogeneity in the degree (connectivity) distribution. Heterogeneity in the degree distribution often reduces the average distance between nodes but, paradoxically, may suppress synchronization in…
The theoretical description of synchronization phenomena often relies on coupled units of continuous time noisy Markov chains with a small number of states in each unit. It is frequently assumed, either explicitly or implicitly, that…
We examine the onset of synchronization transition in a star network of Kuramoto phase oscillators in the presence of inertia and a time delay in the coupling. A direct correlation between the natural frequencies of the oscillators and…
We consider networks of coupled stochastic oscillators. When coupled we find strong collective oscillations, while each unit remains stochastic. In the limit (N\to \infty) we derive a system of integro-delay equations and show analytically…
In previous work, empirical evidence indicated that a time-varying network could propagate sufficient information to allow synchronization of the sometimes coupled oscillators, despite an instantaneously disconnected topology. We prove here…
We study the synchronization transition of Kuramoto oscillators in scale-free networks that are characterized by tunable local properties. Specifically, we perform a detailed finite size scaling analysis and inspect how the critical…