Phase Transitions and Chaos in Long-Range Models of Coupled Oscillators

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-driven coupled pendula. We also show that, in the Kuramoto model, the shape of the phase transition and the largest Lyapunov exponent behavior are strongly dependent on the distribution of the natural frequencies. Introduction. – Long-range interacting systems have been intensively studied in the last years and new methodologies have been developed in the attempt to understanding their intriguing features. One of the most promising directions is the combination of statistical mechanics tools and methods adopted in dynamical systems [1]. In particular, phase transitions have been extensively explored in both conservative and dissipative long-range systems. The Hamiltonian Mean Field (HMF) model [2] and the Kuramoto model [3–5] represent two paradigmatic toy models, the former conservative and the latter dissipative, for many real systems with long-range forces and have several applications. Both models share the same order parameter and display a spontaneous phase transition from an homogeneous/incoherent phase to a magnetized/synchronized one. In [6] we observed already that HMF and Kuramoto models can be considered as limiting cases (respectively conservative and overdamped) of a more general model of driven-damped coupled inertial oscillators. In this paper we present new numerical results which support a common scenario for the two models. More precisely, first we discuss the well known equilibrium features of the second order phase transition in the HMF model, then we study the stationary asymptotic behaviour of the Kuramoto model as a function of the coupling strength. We show that the shape of the dynamical phase transition in the Kuramoto model changes from a continuous to an abrupt one, depending on the distribution of the natural frequencies of the oscillators. We present numerical simulations which show that, as in the case of the HMF model, the largest Lyapunov exponent (LLE) and therefore the chaotic behavior exhibits a peak just before the critical value of the coupling, confirming a microscopic signature of the phase transition. Chaotic behavior in the Kuramoto model was partly discussed previously in ref. [7,8], but only for small sizes and peculiar initial conditions. In those papers the authors did not realize the strong dependence on the initial conditions, discussed in the following, and the fact that chaoticity seems to remain also in the thermodynamical limit. As far as we know, these results are reported for the first time and we think that they could provide with new insights for the study of dynamical phase transitions in systems displaying collective synchronization. Phase transition and chaos in the HMF model. – The Hamiltonian Mean Field model describes the dynamics ofN classical spins or rotators, characterized by the angles θi ∈ [−π, π[ and the coniugate momenta pi ∈]−∞,∞[, which can also be represented as particles moving on the unit circle. In its ferromagnetic version the Hamiltonian of the model is given by: