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We generalize our recent approach to reconstruction of phase dynamics of coupled oscillators from data [B. Kralemann et al., Phys. Rev. E, 77, 066205 (2008)] to cover the case of small networks of coupled periodic units. Starting from the multivariate time series, we first reconstruct genuine phases and then obtain the coupling functions in terms of these phases. The partial norms of these coupling functions quantify directed coupling between oscillators. We illustrate the method by different network motifs for three coupled oscillators and for random networks of five and nine units. We also discuss nonlinear effects in coupling., Comment: 6 pages, 5 figures, 27 references

In this work, we consider a ring of coupled electronic (Wien-bridge) oscillators from a perspective combining modeling, simulation, and experimental observation. Following up on earlier work characterizing the pairwise interaction of Wien-bridge oscillators by Kuramoto-Sakaguchi phase dynamics, we develop a lattice model for a chain thereof, featuring an exponentially decaying spatial kernel. We find that for certain values of the Sakaguchi parameter α, states of traveling phase-domain fronts involving the coexistence of two clearly separated regions of distinct dynamical behavior, can establish themselves in the ring lattice. Experiments and simulations show that stationary coexistence domains of synchronization only manifest themselves with the introduction of a local impurity; here an incoherent cluster of oscillators can arise reminiscent of the chimera states in a range of systems with homogeneous oscillators and suitable nonlocal interactions between them.

In this note we discuss the usage of the Dirac $\delta$ function in models of phase oscillators with pulsatile inputs. Many authors use a product of the delta function and the phase response curve in the right hand side of an ODE to describe the discontinuous phase dynamics in such systems. We point out that this notation has to be treated with care as it is ambiguous. We argue that the presumably most canonical interpretation does not lead to the intended behaviour in many cases., Comment: a serious flaw in the paper

We develop an approach for a fast experimental inference of synchronization properties of an oscillator. While the standard technique for determination of synchronization domains implies that the oscillator under study is forced with many different frequencies and amplitudes, our approach requires only several observations of a driven system. Reconstructing the phase dynamics from data, we successfully determine synchronization domains of noisy and chaotic oscillators. Our technique is especially important for experiments with living systems where an external action can be harmful and shall be minimized. Published by AIP Publishing.

A common observation is that large groups of oscillatory biological units often have the ability to synchronize. A paradigmatic model of such behavior is provided by the Kuramoto model, which achieves synchronization through coupling of the phase dynamics of individual oscillators, while each oscillator maintains a different constant inherent natural frequency. Here we consider the biologically likely possibility that the oscillatory units may be capable of enhancing their synchronization ability by adaptive frequency dynamics. We propose a simple augmentation of the Kuramoto model which does this. We also show that, by the use of a previously developed technique [Ott and Antonsen, Chaos 18, 037113 (2008)], it is possible to reduce the resulting dynamics to a lower dimensional system for the macroscopic evolution of the oscillator ensemble. By employing this reduction, we investigate the dynamics of our system, finding a characteristic hysteretic behavior and enhancement of the quality of the achieved synchronization.

We present results on experimental and theoretical studies of chaos in a solid-state ring laser with periodic pump modulation. We show that the synchronized chaos in the counter-propagating waves is observed for the values of pump modulation frequency fp satisfying the inequality f1 < fp < f2. The boundaries of this region, f1 and f2, depend on the pump-modulation depth. Inside the region of synchronized chaos we study not only dynamics of amplitudes of the counter-propagating waves but also the optical phases of them by mixing the fields of the counter-propagating waves and recording the intensity of the mixed signal. We demonstrate experimentally that in the regime of synchronized chaos the regular phase jumps appear during intervals between adjacent chaotic pulses. We improve the standard semi-classical model of a SSRL and consider an effect of spontaneous emission noise on the temporal evolution of intensities and phase dynamics in the regime of synchronized chaos. It is shown that at the parameters of the experimentally studied laser the noise strongly affects the temporal dependence of amplitudes of the counter-propagating waves.

We demonstrate that the direction of coupling of two interacting self-sustained electronic oscillators can be determined from the realizations of their signals. In our experiments, two electronic generators, operating in a periodic or a chaotic state, were subject to symmetrical or unidirectional coupling. In data processing, first the phases have been extracted from the observed signals and then the directionality of coupling was quantitatively estimated from the analysis of mutual dependence of the phase dynamics.

Close to a supercritical Hopf bifurcation, oscillatory media may be described, by the complex Ginzburg–Landau equation. The most important spatiotemporal behaviors associated with this dynamics are reviewed here. It is shown, on a few concrete examples, how real chemical oscillators may be described by this equation, and how its coefficients may be obtained from the experimental data. Furthermore, the effect of natural forcings, induced by the experimental realization of chemical oscillators in batch reactors, may also be studied in the framework of complex Ginzburg–Landau equations and its associated phase dynamics. We show, in particular, how such forcings may locally transform oscillatory media into excitable ones and trigger the formation of complex spatiotemporal patterns.

We analyze chaotic behavior found in numerical simulations of the transverse pattern dynamics of a laser demonstrating that in some cases chaos originates in phase dynamics and is of low dimension. Investigations of both a Ginzburg–Landau equation for the complex field amplitude of the laser output and a Kuramoto–Sivashinsky‐type equation for only the phase of that complex field equation find the same behavior. Both equations can be expanded in terms of spatial modes and in the chaotic regime the behavior of the modal amplitudes seems relatively independent. However, the fluctuations of the modal amplitudes are sufficiently correlated so that the spatiotemporal dynamics is a form of low dimensional chaos rather than a more complex turbulent behavior or even one that might merit the term spatiotemporal chaos.