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Kuramoto oscillators are widely used to explain collective phenomena in networks of coupled oscillatory units. We show that simple networks of two populations with a generic coupling scheme, where both coupling strengths and phase lags between and within populations are distinct, can exhibit chaotic dynamics as conjectured by Ott and Antonsen [Chaos 18, 037113 (2008)]. These chaotic mean-field dynamics arise universally across network size, from the continuum limit of infinitely many oscillators down to very small networks with just two oscillators per population. Hence, complicated dynamics are expected even in the simplest description of oscillator networks. [ABSTRACT FROM AUTHOR]

Recent years have witnessed increasing interest in phase-amplitude reduction of limit-cycle dynamics. Adding an amplitude coordinate to the phase coordinate allows us to take into account the dynamics transversal to the limit cycle and thereby overcome the main limitations of classic phase reduction (strong convergence to the limit cycle and weak inputs). While previous studies, mostly focus on local quantities such as infinitesimal responses, a major and limiting challenge of phase-amplitude reduction is to compute amplitude coordinates globally, in the basin of attraction of the limit cycle. In this paper, we propose a method to compute the full set of phase-amplitude coordinates in the large. Our method is based on the so-called Koopman (composition) operator and aims at computing the eigenfunctions of the operator through Laplace averages (in combination with the harmonic balance method). This yields a forward integration method that is not limited to two-dimensional systems. We illustrate the method by computing the so-called isostables of limit cycles in two-, three-, and four-dimensional state spaces, as well as their responses to strong external inputs. [ABSTRACT FROM AUTHOR]

We consider chimera states in a one-dimensional medium of nonlinear nonlocally coupled phase oscillators. In terms of a local coarse-grained complex order parameter, the problem of finding stationary rotating nonhomogeneous solutions reduces to a third-order ordinary differential equation. This allows finding chimera-type and other inhomogeneous states as periodic orbits of this equation. Stability calculations reveal that only some of these states are stable. We demonstrate that an oscillatory instability leads to a breathing chimera, for which the synchronous domain splits into subdomains with different mean frequencies. Further development of instability leads to turbulent chimeras. [ABSTRACT FROM AUTHOR]

We propose a chaos communication scheme based on a chaotic optical phase carrier generated with an optoelectronic oscillator with nonlinear time-delay feedback. The system includes a dedicated non-local nonlinearity, which is a customized three-wave imbalanced interferometer. This particular feature increases the complexity of the chaotic waveform and thus the security of the transmitted information, as these interferometers are characterized by four independent parameters which are part of the secret key for the chaos encryption scheme. We first analyze the route to chaos in the system, and evidence a sequence of period doubling bifurcations from the steady-state to fully developed chaos. Then, in the chaotic regime, we study the synchronization between the emitter and the receiver, and achieve chaotic carrier cancellation with a signal-to-noise ratio up to 20 dB. We finally demonstrate error-free chaos communications at a data rate of 3 Gbit/s. [ABSTRACT FROM AUTHOR]

We present experimental and numerical results about dynamical interactions of topological solitons in a semiconductor laser with coherent injection and feedback. We show different kind of interactions such as repulsion, annihilation, or formation of soliton bound states, depending on laser parameters. Collisions between single structures and bound states conserve momentum and charge. [ABSTRACT FROM AUTHOR]

Thermoacoustic instability, caused by a positive feedback between the unsteady heat release and the acoustic field in a combustor, is a major challenge faced in most practical combustors such as those used in rockets and gas turbines. We employ the synchronization theory for understanding the coupling between the unsteady heat release and the acoustic field of a thermoacoustic system. Interactions between coupled subsystems exhibiting different collective dynamics such as periodic, quasiperiodic, and chaotic oscillations are addressed. Even though synchronization studies have focused on different dynamical states separately, synchronous behaviour of two coupled systems exhibiting a quasiperiodic route to chaos has not been studied. In this study, we report the first experimental observation of different synchronous behaviours between two subsystems of a thermoacoustic system exhibiting such a transition as reported in Kabiraj et al. [Chaos 22, 023129 (2012)]. A rich variety of synchronous behaviours such as phase locking, intermittent phase locking, and phase drifting are observed as the dynamics of such subsystem change. The observed synchronization behaviour is further characterized using phase locking value, correlation coefficient, and relative mean frequency. These measures clearly reveal the boundaries between different states of synchronization. [ABSTRACT FROM AUTHOR]

We present a collective coordinate approach to study the collective behaviour of a finite ensemble of N stochastic Kuramoto oscillators using two degrees of freedom: one describing the shape dynamics of the oscillators and one describing their mean phase. Contrary to the thermodynamic limit N → ∞ in which the mean phase of the cluster of globally synchronized oscillators is constant in time, the mean phase of a finite-size cluster experiences Brownian diffusion with a variance proportional to 1/N. This finite-size effect is quantitatively well captured by our collective coordinate approach. [ABSTRACT FROM AUTHOR]

We study multistability in phase locked states in networks of phase oscillators under both Kuramoto dynamics and swing equation dynamics--a popular model for studying coarse-scale dynamics of an electrical AC power grid. We first establish the existence of geometrically frustrated states in such systems--where although a steady state flow pattern exists, no fixed point exists in the dynamical variables of phases due to geometrical constraints. We then describe the stable fixed points of the system with phase differences along each edge not exceeding p/2 in terms of cycle flows--constant flows along each simple cycle--as opposed to phase angles or flows. The cycle flow formalism allows us to compute tight upper and lower bounds to the number of fixed points in ring networks. We show that long elementary cycles, strong edge weights, and spatially homogeneous distribution of natural frequencies (for the Kuramoto model) or power injections (for the oscillator model for power grids) cause such networks to have more fixed points. We generalize some of these bounds to arbitrary planar topologies and derive scaling relations in the limit of large capacity and large cycle lengths, which we show to be quite accurate by numerical computation. Finally, we present an algorithm to compute all phase locked states--both stable and unstable--for planar networks. [ABSTRACT FROM AUTHOR]

Phase synchronization may emerge from mutually interacting non-linear oscillators, even under weak coupling, when phase differences are bounded, while amplitudes remain uncorrelated. However, the detection of this phenomenon can be a challenging problem to tackle. In this work, we apply the Discrete Complex Wavelet Approach (DCWA) for phase assignment, considering signals from coupled chaotic systems and experimental data. The DCWA is based on the Dual-Tree Complex Wavelet Transform (DT-CWT), which is a discrete transformation. Due to its multi-scale properties in the context of phase characterization, it is possible to obtain very good results from scalar time series, even with non-phase-coherent chaotic systems without state space reconstruction or pre-processing. The method correctly predicts the phase synchronization for a chemical experiment with three locally coupled, non-phase-coherent chaotic processes. The impact of different time-scales is demonstrated on the synchronization process that outlines the advantages of DCWA for analysis of experimental data. [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]

We analyze repulsively coupled Kuramoto oscillators, which are exposed to a distribution of natural frequencies. This source of disorder leads to closed orbits of repetitive temporary patterns of phase-locked motion, providing clocks on macroscopic time scales. The periods can be orders of magnitude longer than the periods of individual oscillators. By construction, the attractor space is quite rich. This may cause long transients until the deterministic trajectories find their stationary orbits. The smaller the width of the distribution about the common natural frequency, the longer are the emerging time scales on average. Among the long-period orbits, we find self-similar sequences of temporary phase-locked motion on different time scales. The ratio of time scales is determined by the ratio of widths of the distributions. The results illustrate a mechanism for how simple systems can provide rather flexible dynamics, with a variety of periods even without external entrainment. [ABSTRACT FROM AUTHOR]

We study the global synchronization of hierarchically-organized Stuart-Landau oscillators, where each subsystem consists of three oscillators with activity-dependent couplings. We considered all possible coupling signs between the three oscillators, and found that they can generate different numbers of phase attractors depending on the network motif. Here, the subsystems are coupled through mean activities of total oscillators. Under weak inter-subsystem couplings, we demonstrate that the synchronization between subsystems is highly correlated with the number of attractors in uncoupled subsystems. Among the network motifs, perfect anti-symmetric ones are unique to generate both single and multiple attractors depending on the activities of oscillators. The flexible local complexity can make global synchronization controllable. [ABSTRACT FROM AUTHOR]

We consider a network of identical pulse-coupled oscillators with delay and all-to-all coupling. We demonstrate that the discontinuous nature of the dynamics induces the appearance of isochronous regions--subsets of the phase space filled with periodic orbits having the same period. For each fixed value of the network parameters, such an isochronous region corresponds to a subset of initial states on an appropriate surface of section with non-zero dimensions such that all periodic orbits in this set have qualitatively similar dynamical behaviour. We analytically and numerically study in detail such an isochronous region, give proof of its existence, and describe its properties. We further describe other isochronous regions that appear in the system. [ABSTRACT FROM AUTHOR]

During the last decade, complex network representations have emerged as a powerful instrument for describing the cross-talk between different brain regions both at rest and as subjects are carrying out cognitive tasks, in healthy brains and neurological pathologies. The transient nature of such cross-talk has nevertheless by and large been neglected, mainly due to the inherent limitations of some metrics, e.g., causality ones, which require a long time series in order to yield statistically significant results. Here, we present a methodology to account for intermittent causal coupling in neural activity, based on the identification of non-overlapping windows within the original time series in which the causality is strongest. The result is a less coarse-grained assessment of the timevarying properties of brain interactions, which can be used to create a high temporal resolution time-varying network. We apply the proposed methodology to the analysis of the brain activity of control subjects and alcoholic patients performing an image recognition task. Our results show that short-lived, intermittent, local-scale causality is better at discriminating both groups than global network metrics. These results highlight the importance of the transient nature of brain activity, at least under some pathological conditions. [ABSTRACT FROM AUTHOR]

It has been established that the count of ordinal patterns, which do not occur in a time series, called forbidden patterns, is an effective measure for the detection of determinism in noisy data. A very recent study has shown that this measure is also partially robust against the effects of irregular sampling. In this paper, we extend said research with an emphasis on exploring the parameter space for the method's sole parameter—the length of the ordinal patterns—and find that the measure is more robust to under-sampling and irregular sampling than previously reported. Using numerically generated data from the Lorenz system and the hyper-chaotic Rössler system, we investigate the reliability of the relative proportion of ordinal patterns in periodic and chaotic time series for various degrees of under-sampling, random depletion of data, and timing jitter. Discussion and interpretation of results focus on determining the limitations of the measure with respect to optimal parameter selection, the quantity of data available, the sampling period, and the Lyapunov and de-correlation times of the system. [ABSTRACT FROM AUTHOR]

We consider the existence of non-synchronized fixed points to the Kuramoto model defined on sparse networks: specifically, networks where each vertex has degree exactly three. We show that "most" such networks support multiple attracting phase-locked solutions that are not synchronized and study the depth and width of the basins of attraction of these phase-locked solutions. We also show that it is common in "large enough" graphs to find phase-locked solutions where one or more of the links have angle difference greater than π/2. [ABSTRACT FROM AUTHOR]

Following the long-lived qualitative-dynamics tradition of explaining behavior in complex systems via the architecture of their attractors and basins, we investigate the patterns of switching between distinct trajectories in a network of synchronized oscillators. Our system, consisting of nonlinear amplitude-phase oscillators arranged in a ring topology with reactive nearest-neighbor coupling, is simple and connects directly to experimental realizations. We seek to understand how the multiple stable synchronized states connect to each other in state space by applying Gaussian white noise to each of the oscillators' phases. To do this, we first analytically identify a set of locally stable limit cycles at any given coupling strength. For each of these attracting states, we analyze the effect of weak noise via the covariance matrix of deviations around those attractors. We then explore the noise-induced attractor switching behavior via numerical investigations. For a ring of three oscillators, we find that an attractor-switching event is always accompanied by the crossing of two adjacent oscillators' phases. For larger numbers of oscillators, we find that the distribution of times required to stochastically leave a given state falls off exponentially, and we build an attractor switching network out of the destination states as a coarse-grained description of the high-dimensional attractor-basin architecture. [ABSTRACT FROM AUTHOR]

The control of network-coupled nonlinear dynamical systems is an active area of research in the nonlinear science community. Coupled oscillator networks represent a particularly important family of nonlinear systems, with applications ranging from the power grid to cardiac excitation. Here, we study the control of network-coupled limit cycle oscillators, extending the previous work that focused on phase oscillators. Based on stabilizing a target fixed point, our method aims to attain complete frequency synchronization, i.e., consensus, by applying control to as few oscillators as possible. We develop two types of controls. The first type directs oscillators towards larger amplitudes, while the second does not. We present numerical examples of both control types and comment on the potential failures of the method. [ABSTRACT FROM AUTHOR]

We investigate the relative merit of phase-based methods for inferring directional couplings in complex networks of weakly interacting dynamical systems from multivariate time-series data. We compare the evolution map approach and its partialized extension to each other with respect to their ability to correctly infer the network topology in the presence of indirect directional couplings for various simulated experimental situations using coupled model systems. In addition, we investigate whether the partialized approach allows for additional or complementary indications of directional interactions in evolving epileptic brain networks using intracranial electroencephalographic recordings from an epilepsy patient. For such networks, both direct and indirect directional couplings can be expected, given the brain's connection structure and effects that may arise from limitations inherent to the recording technique. Our findings indicate that particularly in larger networks (number of nodes ≫ 10), the partialized approach does not provide information about directional couplings extending the information gained with the evolution map approach. [ABSTRACT FROM AUTHOR]

This report concerns Entoptic Rotating Spiral Waves as observed and documented by the author over a period of 46 years (1962-2008). The manifestations of these state-dependent, elusive rotating spiral entities were brief, emerging only during sleep-to-waking arousal epochs (in limbo). The images were seen only with closed lids in favorable ambient lighting—here, termed the umbral view. The clusters of rotating spiral entities emerge briefly to conscious view; their angular subtenses are estimated to be between 1° and 4°, and the rotations at ten-turns per second. Epochs of these activities commonly continued for about 20 s, with longevity of each visible entity up to 4 s. 90% of all observed entities were circular and outwardly levorotary; 5% were elliptical, appearing only as horizontal (prolate) entities. Overlapping units were rare, and were chiefly elliptical. Observations of twin spirals were also rare, seen in counter rotations, each twin inwardly rotating. [ABSTRACT FROM AUTHOR]

Occurrence of bi- and trirhythmicities (coexistence of two or three stable limit cycles, respectively, with distinctly different periods) has been studied experimentally by applying delayed feedback control to the copper-phosphoric acid electrochemical system oscillating close to a Hopf bifurcation point under potentiostatic condition. The oscillating electrode potential is delayed by τ and the difference between the present and delayed values is fed back to the circuit potential with a feedback gain K. The experiments were performed by determining the period of current oscillations T as a function of (both increasing and decreasing) τ at several fixed values of K. With small delay times, the period exhibits a sinusoidal type dependence on s. However, with relatively large delays (typically τ ⪢ T) for each feedback gain K, there exists a critical delay scrit above which birhythmicity emerges. The experiments show that for weak feedback, Kscrit is approximately constant. At very large delays, the dynamics becomes even more complex, and trirhythmicity could be observed. Results of numerical simulations based on a general kinetic model for metal electrodissolution were consistent with the experimental observations. The experimental and numerical results are also interpreted by using a phase model; the model parameters can be obtained from experimental data measured at small delay times. Analytical solutions to the phase model quantitatively predict the parameter regions for the appearance of birhythmicity in the experiments, and explain the almost constant value of Kτcrit for weak feedback. [ABSTRACT FROM AUTHOR]

In this paper, we study dynamical systems in which a large number N of identical Landau-Stuart oscillators are globally coupled via a mean-field. Previously, it has been observed that this type of system can exhibit a variety of different dynamical behaviors. These behaviors include time periodic cluster states in which each oscillator is in one of a small number of groups for which all oscillators in each group have the same state which is different from group to group, as well as a behavior in which all oscillators have different states and the macroscopic dynamics of the mean field is chaotic. We argue that this second type of behavior is "extensive" in the sense that the chaotic attractor in the full phase space of the system has a fractal dimension that scales linearly with N and that the number of positive Lyapunov exponents of the attractor also scales linearly with N. An important focus of this paper is the transition between cluster states and extensive chaos as the system is subjected to slow adiabatic parameter change. We observe discontinuous transitions between the cluster states (which correspond to low dimensional dynamics) and the extensively chaotic states. Furthermore, examining the cluster state, as the system approaches the discontinuous transition to extensive chaos, we find that the oscillator population distribution between the clusters continually evolves so that the cluster state is always marginally stable. This behavior is used to reveal the mechanism of the discontinuous transition. We also apply the Kaplan-Yorke formula to study the fractal structure of the extensively chaotic attractors. [ABSTRACT FROM AUTHOR]

The dynamics of p53 play important roles in the regulation of cell fate decisions in response to various stresses, and programmed cell death 5 (PDCD5) functions as a co-activator of p53 that modulates p53 dynamics. In the present paper, we investigated how p53 dynamics are modulated by PDCD5 during the deoxyribose nucleic acid damage response using methods of bifurcation analysis and potential landscape. Our results revealed that p53 activities display rich dynamics under different PDCD5 levels, including monostability, bistability with two stable steady states, oscillations, and the coexistence of a stable steady state (or two states) and an oscillatory state. The physical properties of the p53 oscillations were further demonstrated by the potential landscape in which the potential force attracts the system state to the limit cycle attractor, and the curl flux force drives coherent oscillation along the cyclic trajectory. We also investigated the efficiency with which PDCD5 induced p53 oscillations. We show that Hopf bifurcation can be induced by increasing the PDCD5 efficiency and that the system dynamics exhibited clear transition features in both barrier height and energy dissipation when the efficiency was close to the bifurcation point. [ABSTRACT FROM AUTHOR]

Occurrence of bi- and trirhythmicities (coexistence of two or three stable limit cycles, respectively, with distinctly different periods) has been studied experimentally by applying delayed feedback control to the copper-phosphoric acid electrochemical system oscillating close to a Hopf bifurcation point under potentiostatic condition. The oscillating electrode potential is delayed by τ and the difference between the present and delayed values is fed back to the circuit potential with a feedback gain K. The experiments were performed by determining the period of current oscillations T as a function of (both increasing and decreasing) τ at several fixed values of K. With small delay times, the period exhibits a sinusoidal type dependence on τ. However, with relatively large delays (typically τ ⪢ T) for each feedback gain K, there exists a critical delay τcrit above which birhythmicity emerges. The experiments show that for weak feedback, Kτcrit is approximately constant. At very large delays, the dynamics becomes even more complex, and trirhythmicity could be observed. Results of numerical simulations based on a general kinetic model for metal electrodissolution were consistent with the experimental observations. The experimental and numerical results are also interpreted by using a phase model; the model parameters can be obtained from experimental data measured at small delay times. Analytical solutions to the phase model quantitatively predict the parameter regions for the appearance of birhythmicity in the experiments, and explain the almost constant value of Kτcrit for weak feedback. [ABSTRACT FROM AUTHOR]

This report concerns Entoptic Rotating Spiral Waves as observed and documented by the author over a period of 46 years (1962-2008). The manifestations of these state-dependent, elusive rotating spiral entities were brief, emerging only during sleep-to-waking arousal epochs (in limbo). The images were seen only with closed lids in favorable ambient lighting—here, termed the umbral view. The clusters of rotating spiral entities emerge briefly to conscious view; their angular subtenses are estimated to be between 1° and 4°, and the rotations at ten-turns per second. Epochs of these activities commonly continued for about 20 s, with longevity of each visible entity up to 4 s. 90% of all observed entities were circular and outwardly levorotary; 5% were elliptical, appearing only as horizontal (prolate) entities. Overlapping units were rare, and were chiefly elliptical. Observations of twin spirals were also rare, seen in counter rotations, each twin inwardly rotating. [ABSTRACT FROM AUTHOR]

Like the inertia of a physical body describes its tendency to resist changes of its state of motion, inertia of an oscillator describes its tendency to resist changes of its frequency. Here, we show that finite inertia of individual oscillators enables nonlinear phase waves in spatially extended coupled systems. Using a discrete model of coupled phase oscillators with inertia, we investigate these wave phenomena numerically, complemented by a continuum approximation that permits the analytical description of the key features of wave propagation in the long-wavelength limit. The ability to exhibit traveling waves is a generic feature of systems with finite inertia and is independent of the details of the coupling function. [ABSTRACT FROM AUTHOR]

Using the model of a generalized Van der Pol oscillator in the regime of subcritical Hopf bifurcation, we investigate the influence of time delay on noise-induced oscillations. It is shown that for appropriate choices of time delay, either suppression or enhancement of coherence resonance can be achieved. Analytical calculations are combined with numerical simulations and experiments on an electronic circuit. [ABSTRACT FROM AUTHOR]

A new approach based on the dual-tree complex wavelet transform is introduced for phase assignment to non-linear oscillators, namely, the Discrete Complex Wavelet Approach--DCWA. This methodology is able to measure phase difference with enough accuracy to track fine variations, even in the presence of Gaussian observational noise and when only a single scalar measure of the oscillator is available. So, it can be an especially interesting tool to deal with experimental data. In order to compare it with other phase detection techniques, a testbed is introduced. This testbed provides time series from dynamics similar to non-linear oscillators, such that a theoretical phase choice is known in advance. Moreover, it allows to tune different types of phase synchronization to test phase detection methods under a variety of scenarios. Through numerical benchmarks, we report that the proposed approach is a reliable alternative and that it is particularly effective compared with other methodologies in the presence of moderate to large noises. [ABSTRACT FROM AUTHOR]

For an ensemble of globally coupled oscillators with time-delayed interactions, an explicit relation for the frequency of synchronized dynamics corresponding to different phase behaviors is obtained. One class of solutions corresponds to globally synchronized in-phase oscillations. The other class of solutions have mixed phases, and these can be either randomly distributed or can be a splay state, namely with phases distributed uniformly on a circle. In the strong coupling limit and for larger networks, the in-phase synchronized configuration alone remains. Upon variation of the coupling strength or the size of the system, the frequency can change discontinuously, when there is a transition from one class of solutions to another. This can be from the in-phase state to a mixed-phase state, but can also occur between two in-phase configurations of different frequency. Analytical and numerical results are presented for coupled Landau-Stuart oscillators, while numerical results are shown for Rossler and FitzHugh-Nagumo systems. [ABSTRACT FROM AUTHOR]

The non-stationary dynamics of a bouncing ball, comprising both periodic as well as chaotic behavior, is studied through wavelet transform. The multi-scale characterization of the time series displays clear signatures of self-similarity, complex scaling behavior, and periodicity. Self-similar behavior is quantified by the generalized Hurst exponent, obtained through both wavelet based multi-fractal detrended fluctuation analysis and Fourier methods. The scale dependent variable window size of the wavelets aptly captures both the transients and non-stationary periodic behavior, including the phase synchronization of different modes. The optimal time-frequency localization of the continuous Morlet wavelet is found to delineate the scales corresponding to neutral turbulence, viscous dissipation regions, and different time varying periodic modulations. [ABSTRACT FROM AUTHOR]

We investigate the relative merit of phase-based methods--mean phase coherence, unweighted and weighted phase lag index--for estimating the strength of interactions between dynamical systems from empirical time series which are affected by common sources and noise. By numerically analyzing the interaction dynamics of coupled model systems, we compare these methods to each other with respect to their ability to distinguish between different levels of coupling for various simulated experimental situations. We complement our numerical studies by investigating consistency and temporal variations of the strength of interactions within and between brain regions using intracranial electroencephalographic recordings from an epilepsy patient. Our findings indicate that the unweighted and weighted phase lag index are less prone to the influence of common sources but that this advantage may lead to constrictions limiting the applicability of these methods. [ABSTRACT FROM AUTHOR]

Clustering phase synchronization (CPS) is a common scenario to the global phase synchronization of coupled dynamical systems. In this work, a novel scenario, the non-clustering phase synchronization (NPS), is reported. It is found that coupled systems do not transit to the global synchronization until a certain sufficiently large coupling is attained, and there is no clustering prior to the global synchronization. To reveal the relationship between CPS and NPS, we further analyze the noise effect on coupled phase oscillators and find that the coupled oscillator system can change from CPS to NPS with the increase of noise intensity or system disorder. These findings are expected to shed light on the mechanism of various intriguing self-organized behaviors in coupled systems. [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]

Using a model system of FitzHugh-Nagumo type in the excitable regime, the similarity between synchronization of self-sustained and noise-induced oscillations is studied for the case of more than one main frequency in the spectrum. It is shown that this excitable system undergoes the same frequency lockings as a self-sustained quasiperiodic oscillator. The presence of noise-induced both stable and unstable limit cycles and tori, as well as their tangential bifurcations, are discussed. As the FitzHugh-Nagumo oscillator represents one of the basic neural models, the obtained results are of high importance for neuroscience. [ABSTRACT FROM AUTHOR]

Coherence and information theoretic analyses were applied to quantitate the response properties and the encoding of time-varying stimuli in paddlefish electroreceptors (ERs), studied in vivo. External electrical stimuli were Gaussian noise waveforms of varied frequency band and strength, including naturalistic waveforms derived from zooplankton prey. Our coherence analyses elucidated the role of internal oscillations and transduction processes in shaping the 0.5-20 Hz best frequency tuning of these electroreceptors, to match the electrical signals emitted by zooplankton prey. Stimulus-response coherence fell off above approximately 20 Hz, apparently due to intrinsic limits of transduction, but was detectable up to 40-50 Hz. Aligned with this upper fall off was a narrow band of intense internal noise at ∼25 Hz, due to prominent membrane potential oscillations in cells of sensory epithelia, which caused a narrow deadband of external insensitivity. Using coherence analysis, we showed that more than 76% of naturalistic stimuli of weak strength, ∼1 μV/cm, was linearly encoded into an afferent spike train, which transmitted information at a rate of ∼30 bits/s. Stimulus transfer to afferent spike timing became essentially nonlinear as the stimulus strength was increased to induce bursting firing. Strong stimuli, as from nearby zooplankton prey, acted to synchronize the bursting responses of afferents, including across populations of electroreceptors, providing a plausible mechanism for reliable information transfer to higher-order neurons through noisy synapses. [ABSTRACT FROM AUTHOR]

Signal transmission time delays in a network of nonlinear oscillators are known to be responsible for a variety of interesting dynamic behaviors including phase-flip transitions leading to synchrony or out of synchrony. Here, we uncover that phase-flip transitions are general phenomena and can occur in a network of coupled bursting neurons with a variety of coupling types. The transitions are marked by nonlinear changes in both temporal and phase-space characteristics of the coupled system. We demonstrate these phase-transitions with Hindmarsh-Rose and Leech-Heart interneuron models and discuss the implications of these results in understanding collective dynamics of bursting neurons in the brain. [ABSTRACT FROM AUTHOR]

The ring-loop oscillator consisting of two coupled klystrons which is capable of generating hyperbolic chaotic signal in the microwave band is considered. The system of delayed-differential equations describing the dynamics of the oscillator is derived. This system is further reduced to the two-dimensional return map under the assumption of the instantaneous build-up of oscillations in the cavities. The results of detailed numerical simulation for both models are presented showing that there exists large enough range of control parameters where the sustained regime corresponds to the structurally stable hyperbolic chaos. [ABSTRACT FROM AUTHOR]

We theoretically study the synchronization between collective oscillations exhibited by two weakly interacting groups of nonidentical phase oscillators with internal and external global sinusoidal couplings of the groups. Coupled amplitude equations describing the collective oscillations of the oscillator groups are obtained by using the Ott-Antonsen ansatz, and then coupled phase equations for the collective oscillations are derived by phase reduction of the amplitude equations. The collective phase coupling function, which determines the dynamics of macroscopic phase differences between the groups, is calculated analytically. We demonstrate that the groups can exhibit effective antiphase collective synchronization even if the microscopic external coupling between individual oscillator pairs belonging to different groups is in-phase, and similarly effective in-phase collective synchronization in spite of microscopic antiphase external coupling between the groups. [ABSTRACT FROM AUTHOR]

Many neuronal systems and models display a certain class of mixed mode oscillations (MMOs) consisting of periods of small amplitude oscillations interspersed with spikes. Various models with different underlying mechanisms have been proposed to generate this type of behavior. Stochastic versions of these models can produce similarly looking time series, often with noise-driven mechanisms different from those of the deterministic models. We present a suite of measures which, when applied to the time series, serves to distinguish models and classify routes to producing MMOs, such as noise-induced oscillations or delay bifurcation. By focusing on the subthreshold oscillations, we analyze the interspike interval density, trends in the amplitude, and a coherence measure. We develop these measures on a biophysical model for stellate cells and a phenomenological FitzHugh-Nagumo-type model and apply them on related models. The analysis highlights the influence of model parameters and resets and return mechanisms in the context of a novel approach using noise level to distinguish model types and MMO mechanisms. Ultimately, we indicate how the suite of measures can be applied to experimental time series to reveal the underlying dynamical structure, while exploiting either the intrinsic noise of the system or tunable extrinsic noise. [ABSTRACT FROM AUTHOR]

We study the chaotic behavior of order parameters in two coupled ensembles of self-sustained oscillators. Coupling within each of these ensembles is switched on and off alternately, while the mutual interaction between these two subsystems is arranged through quadratic nonlinear coupling. We show numerically that in the course of alternating Kuramoto transitions to synchrony and back to asynchrony, the exchange of excitations between two subpopulations proceeds in such a way that their collective phases are governed by an expanding circle map similar to the Bernoulli map. We perform the Lyapunov analysis of the dynamics and discuss finite-size effects. [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]

We have proposed a model of deformable self-propelled particles in which the time-evolution equations are given in terms of the center-of-mass velocity and a nematic order parameter representing the motion-induced deformation [T. Ohta and T. Ohkuma, Phys. Rev. Lett. 102, 154101 (2009)]. We investigate its many-body problem applying a global orientational coupling. Depending on the strength of the interaction, the self-propelled particles exhibit various kinds of collective dynamics and chaotic behavior: a ballistic procession state, a scattered state, a coherently phase synchronized state, two types of in-phase synchronized state, and an anomalously diffusive state. The phase reduction method for the weak coupling regime reveals the bifurcations between the secular collective motions. The phase boundary among the chaos regime and the synchronized regimes is determined by the linear stability analysis of the synchronized states. [ABSTRACT FROM AUTHOR]

This paper examines the interplay of the effect of cross immunity and antibody-dependent enhancement (ADE) in multistrain diseases. Motivated by dengue fever, we study a model for the spreading of epidemics in a population with multistrain interactions mediated by both partial temporary cross immunity and ADE. Although ADE models have previously been observed to cause chaotic outbreaks, we show analytically that weak cross immunity has a stabilizing effect on the system. That is, the onset of disease fluctuations requires a larger value of ADE with small cross immunity than without. However, strong cross immunity is shown numerically to cause oscillations and chaotic outbreaks even for low values of ADE. [ABSTRACT FROM AUTHOR]

Results on the dynamics of certain predominantly feedforward networks of phase oscillators are presented. We start with a purely feedforward chain and show numerically that in certain parameter ranges, independent of the length of the chain, there is a global attractor that is effectively a two-dimensional torus. Specifically, we find that after the system has reached its steady state, the phases of all the oscillators in the chain at any one moment in time are determined entirely by the phases of the first two oscillators. In the cases tested this phenomenon is found to persist with the addition of feedback couplings whose strengths may be up to a significant fraction of the feedforward strengths. [ABSTRACT FROM AUTHOR]

Nonlinear oscillators that are mutually coupled via dissimilar (or conjugate) variables display distinct regimes of synchronous behavior. In identical chaotic oscillators diffusively coupled in this manner, complete synchronization occurs only by chaos suppression when the coupled subsystems drive each other into a regime of periodic dynamics. Furthermore, the coupling does not vanish but acts as an “internal” drive. When the oscillators are mismatched, phase synchronization occurs, while in a master slave configuration, generalized synchrony results. These effects are demonstrated in a system of coupled chaotic Rössler oscillators. [ABSTRACT FROM AUTHOR]

Phase synchronization is an important phenomenon of nonlinear dynamics and has recently received much scientific attention. In this work a method for identifying phase synchronization epochs is described which focuses on estimating the gradient of segments of the generalized phase differences between phase slips in an experimental time series. In phase synchronized systems, there should be a zero gradient of the generalized phase differences even if the systems are contaminated by noise. A method which tests if the gradient of the generalized phase difference is statistically different from zero is reported. The method has been validated by numerical studies on model systems and by comparing the results to those published previously. The method is applied to cardiorespiratory time series from a human volunteer measured in clinical settings and compared to synchrogram analysis for the same data. Potential problems with synchrogram analysis of experimental data are discussed. [ABSTRACT FROM AUTHOR]

A framework for detecting nonlinear oscillatory activity in broadband time series is presented. First, a narrow-band oscillatory mode is extracted from a broadband background. Second, it is tested whether the extracted mode is significantly different from linearly filtered noise, modeled as a linear stochastic process possibly passed through a static nonlinear transformation. If a nonlinear oscillatory mode is positively detected, it can be further analyzed using nonlinear approaches such as phase synchronization analysis. For linear processes standard approaches, such as the coherence analysis, are more appropriate. The method is illustrated in a numerical example and applied to analyze experimentally obtained human electroencephalogram time series from a sleeping subject. [ABSTRACT FROM AUTHOR]

The phenomenon of phase synchronization for a particle in a periodic ratchet potential is studied. We consider the deterministic dynamics in the underdamped case where the inertia plays an important role since the dynamics can become chaotic. The ratchet potential is tilted due to a constant external force and is rocking by an external periodic forcing. This potential has to be tilted in order to obtain a rotator or self-sustained nonlinear oscillator in the absence of the external periodic forcing; this oscillator then acquires an intrinsic frequency that can be locked with the frequency of the external driving. We introduced an instantaneous linear phase, using a set of discrete time markers, and the associated average frequency, and show that this frequency can be synchronized with the frequency of the driving. We calculate Arnold tongues in a two-dimensional parameter space and discuss their implications for the chaotic transport in ratchets. We show that the local maxima in the current correspond to the borders of these Arnold tongues; in this way we established a link between optimal transport in ratchets and phase synchronization. [ABSTRACT FROM AUTHOR]

We consider a Hodgkin–Huxley-type model of oscillatory activity in neurons of the snail Helix pomatia. This model has a distinctive feature: It demonstrates multistability in oscillatory and silent modes that is typical for the thalamocortical neurons. A single neuron cell can demonstrate a variety of oscillatory activity: Regular and chaotic spiking and bursting behavior. We study collective phenomena in small and large arrays of nonidentical cells coupled by models of electrical and chemical synapses. Two single elements coupled by electrical coupling show different types of synchronous behavior, in particular in-phase and antiphase synchronous regimes. In an ensemble of three inhibitory synaptically coupled elements, the phenomenon of sequential synchronous dynamics is observed. We study the synchronization phenomena in the chain of nonidentical neurons at different oscillatory behavior coupled with electrical and chemical synapses. Various regimes of phase synchronization are observed: (i) Synchronous regular and chaotic spiking; (ii) synchronous regular and chaotic bursting; and (iii) synchronous regular and chaotic bursting with different numbers of spikes inside the bursts. We detect and study the effect of collective synchronous burst generation due to the cluster formation and the oscillatory death. [ABSTRACT FROM AUTHOR]

We study waves in a chain of dispersively coupled phase oscillators. Two approaches—a quasicontinuous approximation and an iterative numerical solution of the lattice equation—allow us to characterize different types of traveling waves: compactons, kovatons, solitary waves with exponential tails as well as a novel type of semicompact waves that are compact from one side. Stability of these waves is studied using numerical simulations of the initial value problem. [ABSTRACT FROM AUTHOR]