Wave Propagation in Nonlocally Coupled Oscillators With Noise

The onset of undamped wave propagation in noisy self-oscillatory media is identified with a Hopf bifurcation of the corresponding effective dynamical system obtained by properly renormalizing the effects of noise. We illustrate this fact on a dense array of nonlocally coupled phase oscillators for which a mean-field idea works exactly in deriving such effective dynamical equations.Comment: 4 page

Rotating Spiral Waves with Phase-Randomized Core in Non-locally Coupled Oscillators

Rotating spiral waves with a central core composed of phase-randomized oscillators can arise in reaction-diffusion systems if some of the chemical components involved are diffusion-free. This peculiar phenomenon is demonstrated for a paradigmatic three-component reaction-diffusion model. The origin of this anomalous spiral dynamics is the effective non-locality in coupling, whose effect is stronger for weaker coupling. There exists a critical coupling strength which is estimated from a simple argument. Detailed mathematical and numerical analyses are carried out in the extreme case of weak coupling for which the phase reduction method is applicable. Under the assumption that the mean field pattern keeps to rotate steadily as a result of a statistical cancellation of the incoherence, we derive a functional self-consistency equation to be satisfied by this space-time dependent quantity. Its solution and the resulting effective frequencies of the individual oscillators are found to agree excellently with the numerical simulation.Comment: 10 pages, 6 figure

Collective Phase Sensitivity

The collective phase response to a macroscopic external perturbation of a population of interacting nonlinear elements exhibiting collective oscillations is formulated for the case of globally-coupled oscillators. The macroscopic phase sensitivity is derived from the microscopic phase sensitivity of the constituent oscillators by a two-step phase reduction. We apply this result to quantify the stability of the macroscopic common-noise induced synchronization of two uncoupled populations of oscillators undergoing coherent collective oscillations.Comment: 6 pages, 3 figure

Collective dynamical response of coupled oscillators with any network structure

We formulate a reduction theory that describes the response of an oscillator network as a whole to external forcing applied nonuniformly to its constituent oscillators. The phase description of multiple oscillator networks coupled weakly is also developed. General formulae for the collective phase sensitivity and the effective phase coupling between the oscillator networks are found. Our theory is applicable to a wide variety of oscillator networks undergoing frequency synchronization. Any network structure can systematically be treated. A few examples are given to illustrate our theory.Comment: 4 pages, 2 figure

Onset of Collective Oscillation in Chemical Turbulence under Global Feedback

Preceding the complete suppression of chemical turbulence by means of global feedback, a different universal type of transition, which is characterized by the emergence of small-amplitude collective oscillation with strong turbulent background, is shown to occur at much weaker feedback intensity. We illustrate this fact numerically in combination with a phenomenological argument based on the complex Ginzburg-Landau equation with global feedback.Comment: 6 pages, 8 figures; to appear in Phys. Rev.

Noise-induced Turbulence in Nonlocally Coupled Oscillators

We demonstrate that nonlocally coupled limit-cycle oscillators subject to spatiotemporally white Gaussian noise can exhibit a noise-induced transition to turbulent states. After illustrating noise-induced turbulent states with numerical simulations using two representative models of limit-cycle oscillators, we develop a theory that clarifies the effective dynamical instabilities leading to the turbulent behavior using a hierarchy of dynamical reduction methods. We determine the parameter region where the system can exhibit noise-induced turbulent states, which is successfully confirmed by extensive numerical simulations at each level of the reduction.Comment: 23 pages, 17 figures, to appear in Phys. Rev.

Slow Switching in Globally Coupled Oscillators: Robustness and Occurrence through Delayed Coupling

The phenomenon of slow switching in populations of globally coupled oscillators is discussed. This characteristic collective dynamics, which was first discovered in a particular class of the phase oscillator model, is a result of the formation of a heteroclinic loop connecting a pair of clustered states of the population. We argue that the same behavior can arise in a wider class of oscillator models with the amplitude degree of freedom. We also argue how such heteroclinic loops arise inevitably and persist robustly in a homogeneous population of globally coupled oscillators. Although the heteroclinic loop might seem to arise only exceptionally, we find that it appears rather easily by introducing the time-delay in the population which would otherwise exhibit perfect phase synchrony. We argue that the appearance of the heteroclinic loop induced by the delayed coupling is then characterized by transcritical and saddle-node bifurcations. Slow switching arises when the system with a heteroclinic loop is weakly perturbed. This will be demonstrated with a vector model by applying weak noises. Other types of weak symmetry-breaking perturbations can also cause slow switching.Comment: 10 pages, 14 figures, RevTex, twocolumn, to appear in Phys. Rev.