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We demonstrate in numerical experiments that estimators of strength and directionality of coupling between oscillators based on modeling of their phase dynamics [D. A. Smirnov and B. P. Bezruchko, Phys. Rev. E 68, 046209 (2003)] are widely applicable. Namely, although the expressions for the estimators and their confidence bands are derived for linear uncoupled oscillators under the influence of independent sources of Gaussian white noise, they turn out to allow reliable characterization of coupling from relatively short time series for different properties of noise, significant phase nonlinearity of the oscillators, and nonvanishing coupling between them. We apply the estimators to analyze a two-channel human intracranial epileptic electroencephalogram (EEG) recording with the purpose of epileptic focus localization. [ABSTRACT FROM AUTHOR]

We consider a system of coupled swarmalators moving in two dimensional space and explore its collective behavior. Here the swarmalators represent the oscillators that can sync and swarm in space, following the previous study [O'Keeffe et al., Nat. Commun., 8, 1504 (2017)]. The internal state of each swarmalator is represented by its phase angle, and the swarmalators are free to move in the plane according to an equation of motion where the phase and spatial dynamics are coupled with each other. In particular, the phase coupling between the swarmalators is attractive (positive) one, so the coupling makes the swarmalators have their phase difference minimized. The collective behavior of the system is found to be different depending on the extent of the interplay between the phase and spatial dynamics: Specifically, when the extent of the interplay between the phase and spatial dynamics is so weak as to be negligible, the phase dynamics of our system recovers that of the conventional mean-field X Y model. On the other hand, when a certain extent of the interplay is present, the system is found to exhibit the correlated phase where the overall order does not occur. Interestingly, it is found that the correlated phase is the same as the active phase wave found in the system of swarmalators with repulsive phase coupling [O'Keeffe et al., Nat. Commun., 8, 1504 (2017)]. We also find that the system exhibits two different phase transitions: One is the transition from the sync state to the active phase wave state, and the other one is the transition from the active phase wave state to the async state. We perform the finite-size scaling analysis and investigate the transition nature. [ABSTRACT FROM AUTHOR]

We experimentally address the long-time dynamics of an artificially curved growth interface in directional solidification. Repetitive cell nucleations are found to appear in a disordered way but to eventually organize themselves coherently, at long times. This behavior is recovered by simulation of a nonlinear advection-diffusion model for the phase dynamics. The existence of a periodic attractor is shown by deriving a Liapunov functional for the cellular pattern organization on time ranges that include the singular events of cell nucleation. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We study numerically forced synchronization of a heterogeneous multilayer network in the regime of a complex spatiotemporal pattern. Retranslating the master chimera structure in a driving layer along subsequent layers is considered, and the peculiarities of forced synchronization are studied depending on the nature and degree of heterogeneity of the network, as well as on the degree of asymmetry of the inter-layer coupling. We also analyze the possibility of synchronizing all the network layers with a given accuracy when the coupling parameters are varied. [ABSTRACT FROM AUTHOR]

We consider the Kuramoto-Sakaguchi model of identical coupled phase oscillators with a common noisy forcing. While common noise always tends to synchronize the oscillators, a strong repulsive coupling prevents the fully synchronous state and leads to a nontrivial distribution of oscillator phases. In previous numerical simulations, the formation of stable multicluster states has been observed in this regime. However, we argue here that because identical phase oscillators in the Kuramoto-Sakaguchi model form a partially integrable system according to the Watanabe-Strogatz theory, the formation of clusters is impossible. Integrating with various time steps reveals that clustering is a numerical artifact, explained by the existence of higher order Fourier terms in the errors of the employed numerical integration schemes. By monitoring the induced change in certain integrals of motion, we quantify these errors. We support these observations by showing, on the basis of the analysis of the corresponding Fokker-Planck equation, that two-cluster states are non-attractive. On the other hand, in ensembles of general limit cycle oscillators, such as Van der Pol oscillators, due to an anharmonic phase response function as well as additional amplitude dynamics, multiclusters can occur naturally. [ABSTRACT FROM AUTHOR]

This article studies the rotational dynamics of three identical coupled pendulums. There exist two parameter areas where the in-phase rotational motion is unstable and out-of-phase rotations are realized. Asymptotic theory is developed that allows us to analytically identify borders of instability areas of in-phase rotation motion. It is shown that out-of-phase rotations are the result of the parametric instability of in-phase motion. Complex out-of-phase rotations are numerically found and their stability and bifurcations are defined. It is demonstrated that the emergence of chaotic dynamics happens due to the period doubling bifurcation cascade. The detailed scenario of symmetry breaking is presented. The development of chaotic dynamics leads to the origin of two chaotic attractors of different types. The first one is characterized by the different phases of all pendulums. In the second case, the phases of the two pendulums are equal, and the phase of the third one is different. This regime can be interpreted as a drum-head mode in star-networks. It may also indicate the occurrence of chimera states in chains with a greater number of nearest-neighbour interacting elements and in analogical systems with global coupling. [ABSTRACT FROM AUTHOR]

We propose a framework for the analysis of the integro-differential delay Ikeda equations ruling the dynamics of bandpass optoelectronic oscillators (OEOs). Our framework is based on the normal form reduction of OEOs and helps in the determination of the amplitude and the frequency of the primary Hopf limit-cycles as a function of the time delay and other parameters. The study is carried for both the negative and the positive slopes of the sinusoidal transfer function, and our analytical results are confirmed by the numerical and experimental data. [ABSTRACT FROM AUTHOR]

Many systems in biology, physics, and chemistry can be modeled through ordinary differential equations (ODEs), which are piecewise smooth, but switch between different states according to a Markov jump process. In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we suppose that this limit ODE supports a stable limit cycle. We demonstrate that a set of such oscillators can synchronize when they are uncoupled, but they share the same switching Markov jump process. The latter is taken to represent the effect of a common randomly switching environment. We determine the leading order of the Lyapunov coefficient governing the rate of decay of the phase difference in the fast switching limit. The analysis bears some similarities to the classical analysis of synchronization of stochastic oscillators subject to common white noise. However, the discrete nature of the Markov jump process raises some difficulties: in fact, we find that the Lyapunov coefficient from the quasi-steady-state approximation differs from the Lyapunov coefficient one obtains from a second order perturbation expansion in the waiting time between jumps. Finally, we demonstrate synchronization numerically in the radial isochron clock model and show that the latter Lyapunov exponent is more accurate. [ABSTRACT FROM AUTHOR]

Nonparametric detection of coupling delay in unidirectionally and bidirectionally coupled nonlinear dynamical systems is examined. Both continuous and discrete-time systems are considered. Two methods of detection are assessed--the method based on conditional mutual information--the CMI method (also known as the transfer entropy method) and the method of convergent cross mapping--the CCM method. Computer simulations show that neither method is generally reliable in the detection of coupling delays. For continuous-time chaotic systems, the CMI method appears to be more sensitive and applicable in a broader range of coupling parameters than the CCM method. In the case of tested discrete-time dynamical systems, the CCM method has been found to be more sensitive, while the CMI method required much stronger coupling strength in order to bring correct results. However, when studied systems contain a strong oscillatory component in their dynamics, results of both methods become ambiguous. The presented study suggests that results of the tested algorithms should be interpreted with utmost care and the nonparametric detection of coupling delay, in general, is a problem not yet solved. [ABSTRACT FROM AUTHOR]

Statistically distinguishing between phase-coherent and noncoherent chaotic dynamics from time series is a contemporary problem in nonlinear sciences. In this work, we propose different measures based on recurrence properties of recorded trajectories, which characterize the underlying systems from both geometric and dynamic viewpoints. The potentials of the individual measures for discriminating phase-coherent and noncoherent chaotic oscillations are discussed. A detailed numerical analysis is performed for the chaotic Rössler system, which displays both types of chaos as one control parameter is varied, and the Mackey-Glass system as an example of a time-delay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral- and screw-type chaos, a common route from phase-coherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given. [ABSTRACT FROM AUTHOR]

We numerically study a directed small-world network consisting of attractively coupled, identical phase oscillators. While complete synchronization is always stable, it is not always reachable from random initial conditions. Depending on the shortcut density and on the asymmetry of the phase coupling function, there exists a regime of persistent chaotic dynamics. By increasing the density of shortcuts or decreasing the asymmetry of the phase coupling function, we observe a discontinuous transition in the ability of the system to synchronize. Using a control technique, we identify the bifurcation scenario of the order parameter. We also discuss the relation between dynamics and topology and remark on the similarity of the synchronization transition to directed percolation. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the special forms of multimode dynamics that one can observe in systems with resource-mediated coupling, i.e., systems of self-sustained oscillators in which the coupling takes place via the distribution of primary resources that controls the oscillatory state of the individual unit. With this coupling, a spatially inhomogenous state with mixed high and low-amplitude oscillations in the individual units can arise. To examine generic phenomena associated with this type of interaction we consider a chain of resistively coupled electronic oscillators connected to a common power supply. The two-oscillator system displays antiphase synchronization, and it is interesting to note that two-mode oscillations continue to exist outside of the parameter range in which oscillations occur for the individual unit. At low coupling strengths, the multi-oscillator system shows high dimensional quasiperiodicity with little tendency for synchronization. At higher coupling strengths, one typically observes spatial clustering involving a few oscillating units. We describe three different scenarios according to which the cluster can slide along the chain as the bias voltage changes. [ABSTRACT FROM AUTHOR]

Mixed mode oscillations (MMOs) occur when a dynamical system switches between fast and slow motion and small and large amplitude. MMOs appear in a variety of systems in nature, and may be simple or complex. This focus issue presents a series of articles on theoretical, numerical, and experimental aspects of MMOs. The applications cover physical, chemical, and biological systems. [ABSTRACT FROM AUTHOR]

Spiking sequences emerging from dynamical interaction in a pair of oscillatory neurons are investigated theoretically and experimentally. The model comprises two unidirectionally coupled FitzHugh–Nagumo units with modified excitability (MFHN). The first (master) unit exhibits a periodic spike sequence with a certain frequency. The second (slave) unit is in its excitable mode and responds on the input signal with a complex (chaotic) spike trains. We analyze the dynamic mechanisms underlying different response behavior depending on interaction strength. Spiking phase maps describing the response dynamics are obtained. Complex phase locking and chaotic sequences are investigated. We show how the response spike trains can be effectively controlled by the interaction parameter and discuss the problem of neuronal information encoding. [ABSTRACT FROM AUTHOR]

The paper investigates the special clustering phenomena that one can observe in systems of nonlinear oscillators that are coupled via a shared flow of primary resources (or a common power supply). This type of coupling, which appears to be quite frequent in nature, implies that one can no longer separate the inherent dynamics of the individual oscillator from the properties of the coupling network. Illustrated by examples from microbiological population dynamics, renal physiology, and electronic oscillator theory, we show how competition for primary resources in a resource distribution chain leads to a number of new generic phenomena, including partial synchronization, sliding of the synchronization region with the resource supply, and coupling-induced inhomogeneity. [ABSTRACT FROM AUTHOR]

The dynamics of a Bose-Einstein condensate trapped in a periodic potential is governed by a discrete nonlinear equation. The interplay/competition between discreteness (introduced by the lattice) and nonlinearity (due to the interatomic interaction) manifests itself on nontrivial dynamical regimes which disappear in the continuum (translationally invariant) limit, and have been recently observed experimentally. We review some recent efforts on this highly interdisciplinary field, with the goal of stimulating interexchanges among the communities of condensed matter, quantum optics, and nonlinear physics. [ABSTRACT FROM AUTHOR]

We study the dynamics of nonlinear oscillators indirectly coupled through a dynamical environment or a common medium. We observed that this form of indirect coupling leads to synchronization and phase-flip transition in periodic as well as chaotic regime of oscillators. The phase-flip transition from in- to anti-phase synchronization or vise-versa is analyzed in the parameter plane with examples of Landau-Stuart and Rössler oscillators. The dynamical transitions are characterized using various indices such as average phase difference, frequency, and Lyapunov exponents. Experimental evidence of the phase-flip transition is shown using an electronic version of the van der Pol oscillators. [ABSTRACT FROM AUTHOR]