A moment-based approach to the dynamical solution of the Kuramoto model

We examine the dynamics of the Kuramoto model with a new analytical approach. By defining an appropriate set of moments the dynamical equations can be exactly closed. We discuss some applications of the formalism such as the existence of an effective Hamiltonian for the dynamics. We also show how this approach can be used to numerically investigate the dynamical behaviour of the model without finite-size effects. The study of the dynamical behaviour of systems with a very large number of mutual interacting units is a well debated subject. It is a topic with interest in many different interdisciplinar fields. The cooperation between the members of a population may lead to very rich dynamical situations ranging from chaos, periodicity, phase locking, synchronization to self-organized critical states, just to cite a few [1, 2]. In the presence of disorder such interaction can be frustrated and this yields new types of behaviour. In the realm of disordered systems much work has been devoted to the study of models with relaxational dynamics, for instance spin-glass models [3]. In those cases there exists a Hamiltonian function which governs the dynamics of the system. A large body of information can be obtained by using the tools of statistical mechanics. One of the main results at equilibrium is that the fluctuation–dissipation theorem is obeyed. But it is definitely interesting to study the dynamical behaviour of dissipative systems in the presence of external driving forces. A simple model of this type was proposed by Kuramoto to analyse synchronization phenomena in populations of weakly nonlinearly coupled oscillators [4]. It has recently become a subject of extensive studies due to its applications to biology, chemistry and physics [5]. The purpose of this paper is to present a new analytical approach to the Kuramoto model based on the definition of a suitable hierarchy of moments. It allows us to reproduce previous known results and, in addition, gives a new insight into the nature of the problem. Here, we will present the method and consider its potential applications leaving detailed analysis for future work. Through this formalism it is possible to analyse some aspects of the model that deserve special attention. As an example, it has been suggested that, under certain conditions, it is possible to define a suitable Hamiltonian function § E-mail address: [email protected] ‖ E-mail address: [email protected] 0305-4470/97/238095+09$19.50 c © 1997 IOP Publishing Ltd 8095 8096 C J Perez and F Ritort from which it is possible to compute stationary properties of the system within the usual thermodynamic formalism such as ground states and universality classes at zero temperature [6] and equilibrium Boltzmann distribution in the more general case at finite temperature [7]. Our method can answer this question in a simple way. We will also show how our approach can be used to numerically investigate the behaviour of the Kuramoto model free of finite-size effects. Particular results will be obtained for the bimodal distribution case. Our formalism complements other recent theories developed to analyse the Kuramoto model. In particular, it is worthwhile mentioning the order function approach [8] useful for studying properties of the stationary states of the system as well as the critical exponent of the order parameter at the onset of entrainment. Another interesting method was proposed in [9] based on kinetic theory and suitable to deal with questions related to the time dependence of the probability density of the system. The Kuramoto model is defined by a set of N oscillators whose state can be specified in terms of only one degree of freedom, the phase. Each phase {φi; 1 6 i 6 N} follows the dynamical equation