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We formulate a theory for the collective phase description of globally coupled noisy limit-cycle oscillators exhibiting macroscopic rhythms. Collective phase equations describing such macroscopic rhythms are derived by means of a two-step phase reduction. The collective phase sensitivity and collective phase coupling functions, which quantitatively characterize the macroscopic rhythms, are illustrated using three representative models of limit-cycle oscillators. As an important result of the theory, we demonstrate noise-induced anti-phase synchronization between macroscopic rhythms by direct numerical simulations of the three models., Comment: 19 pages, 11 figures

Using the previous approach outlined in Ratliff and Bridges (2016, 2015), a novel method is presented to derive the fifth order Kadomtsev–Petviashvili(KP) equation from periodic wavetrains. As a result, the coefficients and criterion for the fifth order KP to emerge take a universal form that can be determined a-priori, relating to the system’s conservation laws and the termination of a Jordan chain. Moreover, the analysis reveals that generically a mixed dispersive term q X X X Y appears within the final phase equation. The theory presented here is complimented by an example from the context of flexural gravity waves in shallow water and a higher order Nonlinear Schrodinger model relevant in plasma physics, demonstrating how the coefficients in this model are determined via elementary calculations.

Synchronization in the system of coupled non-identical non-isochronous van der Pol–Duffing oscillators with inertial and dissipative coupling is discussed. Generalized Adler’s equation is obtained and investigated in the presence of all relevant factors affecting the synchronization (non-isochronism of the oscillators, their non-identity, coupling of the dissipative and inertial types). Characteristic symmetries are revealed for the Adler’s equation responsible for equivalence of some of the factors. Numerical study of the parameters space of the initial differential equations is carried out using the method of charts of dynamic regimes in the parameter planes. Results obtained by both these approaches are compared and discussed.

The description of patterns in nonequilibrium systems is a fascinating subject. It becomes somewhat untractable if one insists to keep the original equations in their complete form, as the Oberbeck–Boussinesq equations for instance in Rayleigh–Benard convection. Various reduction scheme have been imagined, the ultimate one being the phase picture. I examine a simple version of the phase equation, relevant for systems of travelling rolls or for distributed self-oscillations. The boundary effects limit the ability of this phase to drift freely and yields a well-defined space structure, which can be analyzed in two different limits.

The Euclidean invariant phase dynamics for propagating spatially modulated structures is considered. The phase dynamics equation of motion is obtained in a general way for sufficiently smooth deformation of a striped pattern. The amplitude equation for a propagating pattern is also derived near a bifurcation threshold where the external parameter can contain a component oscillating in time. The theory is applied to the oscillatory patterns described by the amplitude equation. The new stability criterions for a propagating wave and a nonlinear standing wave are also presented.

Covariant long-scale equations of phase dynamics are formulated in a coordinate frame aligned with the phase gradient. The Landau-Ginzburg equation is used as a representative model for the analysis of stationary and wave patterns, as well as of weakly unstable phase dynamics of inhomogeneous oscillations; these results are also reproduced for a general reaction-diffusion system. The covariant amplitude equation at symmetry-breaking bifurcation is derived taking as a starting point a simple model of long-scale thermal convection, and extended to other systems with identical symmetry.

We consider a three-component model of an active medium where local oscillations are coupled to a slowly propagating field. Using the Kuramoto approximation of phase dynamics, we show that this uniform medium can support steady autonomous pacemakers. These were patterns spontaneously emerge from the background of uniform bulk oscillations and are stable with respect to small perturbations.

We use a combination of group theoretic and perturbation methods to analyze the stochastic wandering of bump solutions in a neural field model on the sphere S 2 . We first construct an explicit bump solution in the absence of external inputs and noise, by taking the synaptic weight distribution to be the sum of first-order spherical harmonics. The corresponding neural field equation is equivariant under the action of the special orthogonal group S O ( 3 ) , which implies that the bump is marginally stable with respect to rotations of the sphere. We then carry out an amplitude–phase decomposition of the solution in the presence of a weakly biased external input and weak noise, and use this to derive a pair of stochastic differential equations for the wandering of the bump, expressed in terms of angular coordinates on the sphere. The stochastic dynamics is a non-trivial generalization of the corresponding phase dynamics describing the wandering of a bump on a ring network with S O ( 2 ) symmetry, since S O ( 3 ) is non-abelian and S 2 is a curved manifold.

In the data analysis of oscillatory systems, methods based on phase reconstruction are widely used to characterize phase-locking properties and inferring the phase dynamics. The main component in these studies is an extraction of the phase from a time series of an oscillating scalar observable. We discuss a practical procedure of phase reconstruction by virtue of a recently proposed method termed iterated Hilbert transform embeddings. We exemplify the potential benefits and limitations of the approach by applying it to a generic observable of a forced Stuart–Landau oscillator. Although in many cases, unavoidable amplitude modulation of the observed signal does not allow for perfect phase reconstruction, in cases of strong stability of oscillations and a high frequency of the forcing, iterated Hilbert transform embeddings significantly improve the quality of the reconstructed phase. We also demonstrate that for significant amplitude modulation, iterated embeddings do not provide any improvement.

Properties of target patterns created by pacemakers, representing local regions with the modified oscillation frequency, are studied for twodimensional oscillatory reaction‐diffusion systems described by the complex Ginzburg‐Landau equation. An approximate analytical solution, based on the phase dynamics approximation, is constructed for a circular core and compared with numerical results for circular and square cores. The dependence of the wavenumber and frequency of generated waves on the size and frequency shift of the pacemaker is discussed. Instabilities of target patterns, involving repeated creations of ring-shaped amplitude defects, are further considered. c 2006 Elsevier B.V. All rights reserved.

The dynamics of the phase variable in a laser with injected signal are analyzed as the injected amplitude is modulated by a small periodic term. This term induces quasiperiodic oscillations in the real and imaginary parts of the field variable which make the phase dynamics much more complex than the constant injected signal case in which the real and imaginary parts of the field are periodic. Two-timing analysis, in which the frequency of the periodic component of the injected signal is represented as a small deviation from the fast-time frequency, yields slow-time evolution equations of a periodically forced Toda oscillator. The dynamics near resonance are discussed. A map determining whether the phase jumps up or down by π units is developed based on the slow-time evolution equations.

We extend the notion of phase locking to the case of chaotic oscillators. Different definitions of the phase are discussed. and the phase dynamics of a single self-sustanined chaotic oscillator subjected to external force is investigated. We describe regimes where the amplitude of the oscillator remains chaotic and the phase is synchronized by the external force. This effect is demonstrated for periodic and noisy driving. This phase synchronization is characterized via direct calculation of the phase, as well as by implicit indications, such as the resonant growth of the discrete component in the power spectrum and the appearance of a macroscopic average field in an ensemble of driven oscillators. The Rossler and the Lorenz systems are shown to provide examples of different phase coherence properties, with different response to the external force. A relation between the phase synchronization and the properties of the Lyapunov spectrum is discussed.

We present recent results obtained in the study of the instabilities induced by gravity on liquid layers hanging below solid surfaces, in situation of perfect wetting (without contact lines). Two cases are discussed: two-dimensional films of constant volume, and films continuously supplied with “fresh” liquid by an external source. In the first case, we show experimental evidences suggesting that the formation of a two-dimensional structure within the film arises by means of front propagation mechanisms. In the simplest situation, propagation of a one-dimensional structure (“roll” formation), the measured front speed is close to that deduced from the linear marginal stability theory. In the second case, the liquid film is hanging below an horizontal overflowing half-cylinder. Depending on the rate of supply, different regimes are observed (dripping, arrays of parallel jets, triangular sheets). The arrays of drops and jets exhibit interesting spatio-temporal phase dynamics: oscillations, pairing or nucleation of cells, forced tilt waves.

The competition between external and internal length scales is studied theoretically on a two-dimensional amplitude evolution model of a fluid layer heated from below. A spatially-periodic excitation is applied to the stress-free, isothermal boundaries at a wavenumber close to the critical wavenumber for the onset of the instability. Near resonance, the large-scale phase dynamics give rise to stationary phase solitons at high Prandtl numbers and propagating sine-Gordon solitons at low Prandtl numbers. The propagating solitons are associated with the external breaking of the Galilean invariance of the problem.