We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equati...

The nonlinear Kuramoto equations for n coupled oscillators are derived and studied. The oscillators are defined to be synchronized when they oscillate at the same frequency and their phases are all equal. A control-theoretic viewpoint reveals that synchronized states of Kuramoto oscillators are locally asymptotically stable if every oscillator is coupled to all others. The problem of synchroniz...

where μ, ν are positive constants. This equation, in the case μ = 0, was derived independently by Sivashinsky [1] and Kuramoto [2] with the purpose to model amplitude and phase expansion of pattern formations in different physical situations, for example, in the theory of a flame propagation in turbulent flows of gaseous combustible mixtures, see Sivashinsky [1], and in the theory of turbulence...

We analyze coherent structures in nonlocal dispersive active-dissipative nonlinear systems, using as a prototype the Kuramoto–Sivashinsky (KS) equation with an additional nonlocal term that contains stabilizing/destabilizing and dispersive parts. As for the local generalized Kuramoto–Sivashinsky (gKS) equation (see, e.g., [T. Kawahara and S. Toh, Phys. Fluids, 31 (1988), pp. 2103–2111]), we sho...

We present a class of extended Kuramoto models describing a flocking motion of particles on the infinite cylinder and provide sufficient conditions for the asymptotic formation of locked solutions where the distance between particles remains constant. Our proposed model includes the complex Kuramoto model for synchronization. We also provide several numerical simulation results and compare them...

Considerable attention has been devoted to the different synchronization phenomena [1] in biology. Such as complete synchronization, cluster synchronization, anti-synchronization, and so on[2, 3, 4, 5, 6]. During recent twenty years, phase synchronization has attracted much interest. Many dynamical behaviors in biological systems can be explained by phase synchronization finally[8]. This synchr...

We define a subclass of timed automata, called oscillator timed automata, suitable to model biological oscillators. The semantics of their interactions, parametric w.r.t. a model of synchronization, is introduced. We apply it to the Kuramoto model. Then, we introduce a logic, Kuramoto Synchronization Logic (KSL), and a model checking algorithm in order to verify collective synchronization prope...

We study the phase diagram of a generalized Winfree model. The modification is such that the coupling depends on the fraction of synchronized oscillators, a situation which has been noted in some experiments on coupled Josephson junctions and mechanical systems. We let the global coupling k be a function of the Kuramoto order parameter r through an exponent z such that z=1 corresponds to the st...