We consider two and three phase-oscillators as in the Kuramoto model of coupled oscillators, replacing the sine wave interaction with a sawtooth wave. We show that for the case of non-uniform input-symmetric coupling strengths, the non-smooth, piecewise-linear dynamics synchronizes when the coupling strengths are large enough to overcome the differences in the natural frequencies of the oscilla...

We study the chaotic behavior of the synchronization phase transition in the Kuramoto model. We discuss the relationship with analogous features found in the Hamiltonian Mean Field (HMF) model. Our numerical results support the connection between the two models, which can be considered as limiting cases (dissipative and conservative, respectively) of a more general dynamical system of damped-dr...

Although Command and Control (C2) is a complex activity, useful lessons about C2 can be learnt from simple computational models. In this paper, we describe experiments with two such models. The Kuramoto Model, though with some serious limitations, provides a representation of information flow and self-synchronization in an organization. A second (agent-based) model, based on factorization, prov...

A challenge to both understanding and modeling biochemical networks is integrating the effects of diffusion and stochasticity. Here, we use the theory of branching processes to give exact analytical expressions for the mean and variance of protein numbers as a function of time and position in a spatial version of an established model of gene expression. We show that both the mean and the magnit...

subject to appropriate initial and boundary conditions has been introduced in [15],[16] and in [23],[24] in studying phase turbulence and the flame front propagation in combustion theory. In the absence of any a priori estimates for the solutions of the scalar equation (1), most authors find it more convenient, for the mathematical study, to consider the differential form of the equation for u ...

Bursting neurons are characterized by an alternation of active and inactive state, during the active state they exhibit train of spikes and quiescent behavior in the inactive state.When bursting neurons are coupled they can synchronize depending on the coupling strength. If the coupling strength is increased above a critical value, a transition from an asynchronous state to a partial synchroniz...

A self-consistent equation to derive a discreteness-induced stochastic steady state is presented for reactiondiffusion systems. For this formalism, we use the so-called Kuramoto length, a typical distance over which a molecule diffuses in its lifetime, as was originally introduced to determine if local fluctuations influence globally the whole system. We show that this Kuramoto length is also r...