Dynamical systems with time-dependent coupling: Clustering and critical behaviour

We study the collective behaviour of an ensemble of coupled motile elements whose interactions depend on time and are alternatively attractive or repulsive. The evolution of interactions is driven by individual internal variables with autonomous dynamics. The system exhibits different dynamical regimes, with various forms of collective organization, controlled by the range of interactions and the dispersion of time scales in the evolution of the internal variables. In the limit of large interaction ranges, it reduces to an ensemble of coupled identical phase oscillators and, to some extent, admits to be treated analytically. We find and characterize a transition between ordered and disordered states, mediated by a regime of dynamical clustering.Comment: to appear in Physica

Dynamics of multi-frequency oscillator ensembles with resonant coupling

We study dynamics of populations of resonantly coupled oscillators having different frequencies. Starting from the coupled van der Pol equations we derive the Kuramoto-type phase model for the situation, where the natural frequencies of two interacting subpopulations are in relation 2:1. Depending on the parameter of coupling, ensembles can demonstrate fully synchronous clusters, partial synchrony (only one subpopulation synchronizes), or asynchrony in both subpopulations. Theoretical description of the dynamics based on the Watanabe-Strogatz approach is developed.Comment: 12 page

Scaling and singularities in the entrainment of globally-coupled oscillators

The onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling. The population is described by a Fokker-Planck equation for the distribution of phases which includes the diffusive effect of noise in the oscillator frequencies. The bifurcation from the phase-incoherent state is analyzed using amplitude equations for the unstable modes with particular attention to the dependence of the nonlinearly saturated mode $|\alpha_\infty|$ on the linear growth rate $\gamma$. In general we find $|\alpha_\infty|\sim \sqrt{\gamma(\gamma+l^2D)}$ where $D$ is the diffusion coefficient and $l$ is the mode number of the unstable mode. The unusual $(\gamma+l^2D)$ factor arises from a singularity in the cubic term of the amplitude equation.Comment: 11 pages (Revtex); paper submitted to Phys. Rev. Let

Spontaneous phase oscillation induced by inertia and time delay

We consider a system of coupled oscillators with finite inertia and time-delayed interaction, and investigate the interplay between inertia and delay both analytically and numerically. The phase velocity of the system is examined; revealed in numerical simulations is emergence of spontaneous phase oscillation without external driving, which turns out to be in good agreement with analytical results derived in the strong-coupling limit. Such self-oscillation is found to suppress synchronization and its frequency is observed to decrease with inertia and delay. We obtain the phase diagram, which displays oscillatory and stationary phases in the appropriate regions of the parameters.Comment: 5 pages, 6 figures, to pe published in PR

N-tree approximation for the largest Lyapunov exponent of a coupled-map lattice

The N-tree approximation scheme, introduced in the context of random directed polymers, is here applied to the computation of the maximum Lyapunov exponent in a coupled map lattice. We discuss both an exact implementation for small tree-depth $n$ and a numerical implementation for larger $n$s. We find that the phase-transition predicted by the mean field approach shifts towards larger values of the coupling parameter when the depth $n$ is increased. We conjecture that the transition eventually disappears.Comment: RevTeX, 15 pages,5 figure

A statistical mechanics of an oscillator associative memory with scattered natural frequencies

Analytic treatment of a non-equilibrium random system with large degrees of freedoms is one of most important problems of physics. However, little research has been done on this problem as far as we know. In this paper, we propose a new mean field theory that can treat a general class of a non-equilibrium random system. We apply the present theory to an analysis for an associative memory with oscillatory elements, which is a well-known typical random system with large degrees of freedoms.Comment: 8 pages, 4 figure

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 like the existence of an effective Hamiltonian for the dynamics. We also show how this approach can be used to numerically investigate the dynamical behavior of the model without finite size effects.Comment: 6 pages, 5 figures, Revtex file, to appear in J. Phys.

Partially and Fully Frustrated Coupled Oscillators With Random Pinning Fields

We have studied two specific models of frustrated and disordered coupled Kuramoto oscillators, all driven with the same natural frequency, in the presence of random external pinning fields. Our models are structurally similar, but differ in their degree of bond frustration and in their finite size ground state properties (one has random ferro- and anti-ferromagnetic interactions; the other has random chiral interactions). We have calculated the equilibrium properties of both models in the thermodynamic limit using the replica method, with emphasis on the role played by symmetries of the pinning field distribution, leading to explicit predictions for observables, transitions, and phase diagrams. For absent pinning fields our two models are found to behave identically, but pinning fields (provided with appropriate statistical properties) break this symmetry. Simulation data lend satisfactory support to our theoretical predictions.Comment: 37 pages, 7 postscript figure