We analyze a minimal model of a population of identical oscillators with a nonlinear coupling—a generalization of the popular Kuramoto model. In addition to well-known for the Kuramoto model regimes of full synchrony, full asynchrony, and integrable neutral quasiperiodic states, ensembles of nonlinearly coupled oscillators demonstrate two novel nontrivial types of partially synchronized dynamic...

We consider the Kuramoto-Sivashinsky (KS) equation in one dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko, Scheurer, and Temam [18], and later improved by Collet, Eckmann, Epstein and Stubbe[1], and Goodman [10] to prove that lim sup t→∞ ||u||2 ≤ CL 3 2 . This result is slightly weaker than that recently announ...

The Kuramoto model describes a large population of coupled limit-cycle oscillators whose natural frequencies are drawn from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, while others remain incoherent. The mathematical analysis of this bifurcation has proved both prob...

Large networks of coupled oscillators appear in many branches of science, so that the kinds of phenomena they exhibit are not only of intrinsic interest but also of very wide importance. In 1975, Kuramoto proposed an analytically tractable model to describe these systems, which has since been successfully applied in many contexts and remains a subject of intensive research. Some related problem...

We investigate the effect of coupling delays on the synchronization properties of several network motifs. In particular, we analyze the synchronization patterns of unidirectionally coupled rings, bidirectionally coupled rings, and open chains of Kuramoto oscillators. Our approach includes an analytical and semianalytical study of the existence and stability of different in-phase and out-of-phas...

We analyze a Crank–Nicolson–type finite difference scheme for the Kuramoto– Sivashinsky equation in one space dimension with periodic boundary conditions. We discuss linearizations of the scheme and derive second–order error estimates.

We describe a Lohner-type algorithm for rigorous integration of dissipative PDEs. Using it for the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions we give a computer assisted proof the existence of multiple periodic orbits.

In this work we study a system of Kuramoto oscillators with identical frequencies in a Cayley tree. Heterogeneity in the frequency distribution is introduced in the root of the tree, allowing for analytical calculations of the phases evolution.

A prediction scheme for spatio-temporal time series is presented that is based on reconstructed local states. As a numerical example the ev olution of a Kuramoto-Siv ashinsky equation is forecasted using previously sampled data.

We present asymptotic relaxation estimates to bi-cluster configurations for the ensemble of Kuramoto oscillators with two different natural frequencies which have been observed in numerical simulations. We provide a set of initial configurations with a positive Lebesgue measure in T leading to bi-(point) cluster configurations consisting of linear combinations of two Dirac measures in super-thr...