Many-Body Theory of Synchronization by Long-Range Interactions

Synchronization of coupled oscillators on a $d$-dimensional lattice with the power-law coupling $G(r) = g_0/r^\alpha$ and randomly distributed intrinsic frequency is analyzed. A systematic perturbation theory is developed to calculate the order parameter profile and correlation functions in powers of $\epsilon = \alpha/d-1$. For $\alpha \le d$, the system exhibits a sharp synchronization transition as described by the conventional mean-field theory. For $\alpha > d$, the transition is smeared by the quenched disorder, and the macroscopic order parameter \Av\psi decays slowly with $g_0$ as |\Av\psi| \propto g_0^2.Comment: 4 pages, 2 figure

Diffusion-induced instability and chaos in random oscillator networks

We demonstrate that diffusively coupled limit-cycle oscillators on random networks can exhibit various complex dynamical patterns. Reducing the system to a network analog of the complex Ginzburg-Landau equation, we argue that uniform oscillations can be linearly unstable with respect to spontaneous phase modulations due to diffusional coupling - the effect corresponding to the Benjamin-Feir instability in continuous media. Numerical investigations under this instability in random scale-free networks reveal a wealth of complex dynamical regimes, including partial amplitude death, clustering, and chaos. A dynamic mean-field theory explaining different kinds of nonlinear dynamics is constructed.Comment: 6 pages, 3 figure

Birhythmicity, Synchronization, and Turbulence in an Oscillatory System with Nonlocal Inertial Coupling

We consider a model where a population of diffusively coupled limit-cycle oscillators, described by the complex Ginzburg-Landau equation, interacts nonlocally via an inertial field. For sufficiently high intensity of nonlocal inertial coupling, the system exhibits birhythmicity with two oscillation modes at largely different frequencies. Stability of uniform oscillations in the birhythmic region is analyzed by means of the phase dynamics approximation. Numerical simulations show that, depending on its parameters, the system has irregular intermittent regimes with local bursts of synchronization or desynchronization.Comment: 21 pages, many figures. Paper accepted on Physica

Chimeras in networks of planar oscillators

Chimera states occur in networks of coupled oscillators, and are characterized by having some fraction of the oscillators perfectly synchronized, while the remainder are desynchronized. Most chimera states have been observed in networks of phase oscillators with coupling via a sinusoidal function of phase differences, and it is only for such networks that any analysis has been performed. Here we present the first analysis of chimera states in a network of planar oscillators, each of which is described by both an amplitude and a phase. We find that as the attractivity of the underlying periodic orbit is reduced chimeras are destroyed in saddle-node bifurcations, and supercritical Hopf and homoclinic bifurcations of chimeras also occur.Comment: To appear, Phys. Rev.

Self-Emerging and Turbulent Chimeras in Oscillator Chains

We report on a self-emerging chimera state in a homogeneous chain of nonlocally and nonlinearly coupled oscillators. This chimera, i.e. a state with coexisting regions of complete and partial synchrony, emerges via a supercritical bifurcation from a homogeneous state and thus does not require preparation of special initial conditions. We develop a theory of chimera basing on the equations for the local complex order parameter in the Ott-Antonsen approximation. Applying a numerical linear stability analysis, we also describe the instability of the chimera and transition to a phase turbulence with persistent patches of synchrony

Chimera Ising Walls in Forced Nonlocally Coupled Oscillators

Nonlocally coupled oscillator systems can exhibit an exotic spatiotemporal structure called chimera, where the system splits into two groups of oscillators with sharp boundaries, one of which is phase-locked and the other is phase-randomized. Two examples of the chimera states are known: the first one appears in a ring of phase oscillators, and the second one is associated with the two-dimensional rotating spiral waves. In this article, we report yet another example of the chimera state that is associated with the so-called Ising walls in one-dimensional spatially extended systems, which is exhibited by a nonlocally coupled complex Ginzburg-Landau equation with external forcing.Comment: 7 pages, 5 figures, to appear in Phys. Rev.

Synchronous solutions and their stability in nonlocally coupled phase oscillators with propagation delays

We study the existence and stability of synchronous solutions in a continuum field of non-locally coupled identical phase oscillators with distance-dependent propagation delays. We present a comprehensive stability diagram in the parameter space of the system. From the numerical results a heuristic synchronization condition is suggested, and an analytic relation for the marginal stability curve is obtained. We also provide an expression in the form of a scaling relation that closely follows the marginal stability curve over the complete range of the non-locality parameter.Comment: accepted in Phys. Rev. E (2010

Chimera states in heterogeneous networks

Chimera states in networks of coupled oscillators occur when some fraction of the oscillators synchronise with one another, while the remaining oscillators are incoherent. Several groups have studied chimerae in networks of identical oscillators, but here we study these states in a heterogeneous model for which the natural frequencies of the oscillators are chosen from a distribution. We obtain exact results by reduction to a finite set of differential equations. We find that heterogeneity can destroy chimerae, destroy all states except chimerae, or destabilise chimerae in Hopf bifurcations, depending on the form of the heterogeneity.Comment: Revised text. To appear, Chao

Shear diversity prevents collective synchronization

Large ensembles of heterogeneous oscillators often exhibit collective synchronization as a result of mutual interactions. If the oscillators have distributed natural frequencies and common shear (or nonisochronicity), the transition from incoherence to collective synchronization is known to occur at large enough values of the coupling strength. However, here we demonstrate that shear diversity cannot be counterbalanced by diffusive coupling leading to synchronization. We present the first analytical results for the Kuramoto model with distributed shear, and show that the onset of collective synchronization is impossible if the width of the shear distribution exceeds a precise threshold

The Kuramoto model with distributed shear

We uncover a solvable generalization of the Kuramoto model in which shears (or nonisochronicities) and natural frequencies are distributed and statistically dependent. We show that the strength and sign of this dependence greatly alter synchronization and yield qualitatively different phase diagrams. The Ott-Antonsen ansatz allows us to obtain analytical results for a specific family of joint distributions. We also derive, using linear stability analysis, general formulae for the stability border of incoherence.Comment: 6 page