Related papers: Basin sizes depend on stable eigenvalues in the Ku…
Synchronization is a ubiquitous phenomenon in complex systems. The Kuramoto model serves as a paradigmatic framework for understanding how coupled oscillators achieve collective rhythm. Conventional approaches focus on pairwise…
In applied sciences, such as physics and biology, it is often required to model the evolution of populations via dynamical systems. In this paper, we focus on the problem of approximating the basins of attraction of such models in case of…
We consider a generalization of the Kuramoto model of coupled oscillators to the situation where communities of oscillators having essentially different natural frequencies interact. General equations describing possible resonances between…
We consider a Kuramoto model of coupled oscillators that includes quenched random interactions of the type used by van Hemmen in his model of spin glasses. The phase diagram is obtained analytically for the case of zero noise and a…
We consider a mean-field model of coupled phase oscillators with quenched disorder in the natural frequencies and coupling strengths. A fraction $p$ of oscillators are positively coupled, attracting all others, while the remaining fraction…
We consider the Kuramoto model of globally coupled phase oscillators in its continuum limit, with individual frequencies drawn from a distribution with density of class $C^n$ ($n\geq 4$). A criterion for linear stability of the uniform…
An interesting problem in synchronization is the study of coupled oscillators, wherein oscillators with different natural frequencies synchronize to a common frequency and equilibrium phase difference. In this paper, we investigate the…
In adaptive dynamical networks, the dynamics of the nodes and the edges influence each other. We show that we can treat such systems as a closed feedback loop between edge and node dynamics. Using recent advances on the stability of…
We discuss the {\it nonlinear stability} of phase-locked states for globally coupled nonlinear oscillators with finite inertia, namely the modified Kuramoto model, in the context of the robust $\ell^{\infty}$-norm. We show that some classes…
The Kuramoto model, which describes synchronization phenomena, is a system of ordinary differential equations on $N$-torus defined as coupled harmonic oscillators. The order parameter is often used to measure the degree of synchronization.…
When the natural frequencies are allocated symmetrically in the Kuramoto model there exists an invariant torus of dimension [N/2]+1 (N is the population size). A global phase shift invariance allows to reduce the model to $N-1$ dimensions…
Imagine a group of oscillators, each endowed with their own rhythm or frequency, be it the ticking of a biological clock, the swing of a pendulum, or the glowing of fireflies. While these individual oscillators may seem independent of one…
A general stability analysis is presented for the determination of the transition from incoherent to coherent behavior in an ensemble of globally coupled, heterogeneous, continuous-time dynamical systems. The formalism allows for the…
Collective oscillations and patterns of synchrony have long fascinated researchers in the applied sciences, particularly due to their far-reaching importance in chemistry, physics, and biology. The Kuramoto model has emerged as a…
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…
A small-world network (SW) of similar phase oscillators, interacting according to the Kuramoto model is studied numerically. It is shown that deterministic Kuramoto dynamics on the SW networks has various stable stationary states. This can…
Although stable solutions of dynamical systems are typically considered more important than unstable ones, unstable solutions have a critical role in the dynamical integrity of stable solutions. In fact, usually, basins of attraction…
We investigate the role of inertia in the asynchronous state of a disordered Kuramoto model. We extend an iterative simulation scheme to the case of the Kuramoto model with inertia in order to determine the self-consistent fluctuation…
The Kuramoto model is a classical mathematical model in the field of non-linear dynamical systems that describes the evolution of coupled oscillators in a network that may reach a synchronous state. The relationship between the network's…
We present a detailed analysis of the stability of synchronized solutions to the Kuramoto system of oscillators. We derive an analytical expression counting the dimension of the unstable manifold associated to a given stationary solution.…