Phase synchronization of chaotic oscillations in terms of periodic orbits

Collective dynamics of coupled Lorenz oscillators near the Hopf boundary: Intermittency and chimera states

We study collective dynamics of networks of mutually coupled identical Lorenz oscillators near a subcritical Hopf bifurcation. Such systems exhibit induced multistable behavior with interesting spatiotemporal dynamics including synchronization, desynchronization, and chimera states. For analysis, we first consider a ring topology with nearest-neighbor coupling and find that the system may exhibit intermittent behavior due to the complex basin structures and dynamical frustration, where temporal dynamics of the oscillators in the ensemble switches between different attractors. Consequently, different oscillators may show a dynamics that is intermittently synchronized (or desynchronized), giving rise to intermittent chimera states. The behavior of the intermittent laminar phases is characterized by the characteristic time spent in the synchronization manifold, which decays as a power law. Such intermittent dynamics is quite general and is also observed in an ensemble of a large number of oscillators arranged in variety of network topologies including nonlocal, scale-free, random, and small-world networks.

Cardiorespiratory coupling in mechanically ventilated patients studied via synchrogram analysis

Respiration and cardiac activity are strictly interconnected with reciprocal influences. They act as weakly coupled oscillators showing varying degrees of phase synchronization and their interactions are affected by mechanical ventilation. The study aims at differentiating the impact of three ventilatory modes on the cardiorespiratory phase coupling in critically ill patients. The coupling between respiration and heartbeat was studied through cardiorespiratory phase synchronization analysis carried out via synchrogram during pressure control ventilation (PCV), pressure support ventilation (PSV), and neurally adjusted ventilatory assist (NAVA) in critically ill patients. Twenty patients were studied under all the three ventilatory modes. Cardiorespiratory phase synchronization changed significantly across ventilatory modes. The highest synchronization degree was found during PCV session, while the lowest one with NAVA. The percentage of all epochs featuring synchronization regardless of the phase locking ratio was higher with PCV (median: 33.9%, first-third quartile: 21.3-39.3) than PSV (median: 15.7%; first-third quartile: 10.9-27.8) and NAVA (median: 3.7%; first-third quartile: 3.3-19.2). PCV induces a significant amount of cardiorespiratory phase synchronization in critically ill mechanically ventilated patients. Synchronization induced by patient-driven ventilatory modes was weaker, reaching the minimum with NAVA. Findings can be explained as a result of the more regular and powerful solicitation of the cardiorespiratory system induced by PCV. The degree of phase synchronization between cardiac and respiratory activities in mechanically ventilated humans depends on the ventilatory mode.

Phase and frequency linear response theory for hyperbolic chaotic oscillators

We formulate a linear phase and frequency response theory for hyperbolic flows, which generalizes phase response theory for autonomous limit cycle oscillators to hyperbolic chaotic dynamics. The theory is based on a shadowing conjecture, stating the existence of a perturbed trajectory shadowing every unperturbed trajectory on the system attractor for any small enough perturbation of arbitrary duration and a corresponding unique time isomorphism, which we identify as phase such that phase shifts between the unperturbed trajectory and its perturbed shadow are well defined. The phase sensitivity function is the solution of an adjoint linear equation and can be used to estimate the average change of phase velocity to small time dependent or independent perturbations. These changes in frequency are experimentally accessible, giving a convenient way to define and measure phase response curves for chaotic oscillators. The shadowing trajectory and the phase can be constructed explicitly in the tangent space of an unperturbed trajectory using co-variant Lyapunov vectors. It can also be used to identify the limits of the regime of linear response.

Topological synchronization of chaotic systems

A chaotic dynamics is typically characterized by the emergence of strange attractors with their fractal or multifractal structure. On the other hand, chaotic synchronization is a unique emergent self-organization phenomenon in nature. Classically, synchronization was characterized in terms of macroscopic parameters, such as the spectrum of Lyapunov exponents. Recently, however, we attempted a microscopic description of synchronization, called topological synchronization, and showed that chaotic synchronization is, in fact, a continuous process that starts in low-density areas of the attractor. Here we analyze the relation between the two emergent phenomena by shifting the descriptive level of topological synchronization to account for the multifractal nature of the visited attractors. Namely, we measure the generalized dimension of the system and monitor how it changes while increasing the coupling strength. We show that during the gradual process of topological adjustment in phase space, the multifractal structures of each strange attractor of the two coupled oscillators continuously converge, taking a similar form, until complete topological synchronization ensues. According to our results, chaotic synchronization has a specific trait in various systems, from continuous systems and discrete maps to high dimensional systems: synchronization initiates from the sparse areas of the attractor, and it creates what we termed as the 'zipper effect', a distinctive pattern in the multifractal structure of the system that reveals the microscopic buildup of the synchronization process. Topological synchronization offers, therefore, a more detailed microscopic description of chaotic synchronization and reveals new information about the process even in cases of high mismatch parameters.

Synchronization in populations of electrochemical bursting oscillators with chaotic slow dynamics

We investigate the synchronization of coupled electrochemical bursting oscillators using the electrodissolution of iron in sulfuric acid. The dynamics of a single oscillator consisted of slow chaotic oscillations interrupted by a burst of fast spiking, generating a multiple time-scale dynamical system. A wavelet analysis first decomposed the time series data from each oscillator into a fast and a slow component, and the corresponding phases were also obtained. The phase synchronization of the fast and slow dynamics was analyzed as a function of electrical coupling imposed by an external coupling resistance. For two oscillators, a progressive transition was observed: With increasing coupling strength, first, the fast bursting intervals overlapped, which was followed by synchronization of the fast spiking, and finally, the slow chaotic oscillations synchronized. With a population of globally coupled 25 oscillators, the coupling eliminated the fast dynamics, and only the synchronization of the slow dynamics can be observed. The results demonstrated the complexities of synchronization with bursting oscillations that could be useful in other systems with multiple time-scale dynamics, in particular, in neuronal networks.

Intermittent route to generalized synchronization in bidirectionally coupled chaotic oscillators

The type of transition from asynchronous behavior to the generalized synchronization regime in mutually coupled chaotic oscillators has been studied. To separate the epochs of the synchronous and asynchronous motion in time series of mutually coupled chaotic oscillators, a method based on the local Lyapunov exponent calculation has been proposed. The efficiency of the method has been testified using the examples of unidirectionally coupled dynamical systems for which the type of transition is well known. The transition to generalized synchronization regime in mutually coupled systems has been shown to be an on-off intermittency as well as in the case of the unidirectional coupling.

Jump intermittency as a second type of transition to and from generalized synchronization

The transition from asynchronous dynamics to generalized chaotic synchronization and then to completely synchronous dynamics is known to be accompanied by on-off intermittency. We show that there is another (second) type of the transition called jump intermittency which occurs near the boundary of generalized synchronization in chaotic systems with complex two-sheeted attractors. Although this transient behavior also exhibits intermittent dynamics, it differs sufficiently from on-off intermittency supposed hitherto to be the only type of motion corresponding to the transition to generalized synchronization. This type of transition has been revealed and the underling mechanism has been explained in both unidirectionally and mutually coupled chaotic Lorenz and Chen oscillators. To detect the epochs of synchronous and asynchronous motion in mutually coupled oscillators with complex topology of an attractor a technique based on finding time intervals when the phase trajectories are located on equal or different sheets of chaotic attractors of coupled oscillators has been developed. We have also shown that in the unidirectionally coupled systems the proposed technique gives the same results that may obtained with the help of the traditional method using the auxiliary system approach.

Cross frequency coupling in next generation inhibitory neural mass models

Coupling among neural rhythms is one of the most important mechanisms at the basis of cognitive processes in the brain. In this study, we consider a neural mass model, rigorously obtained from the microscopic dynamics of an inhibitory spiking network with exponential synapses, able to autonomously generate collective oscillations (COs). These oscillations emerge via a super-critical Hopf bifurcation, and their frequencies are controlled by the synaptic time scale, the synaptic coupling, and the excitability of the neural population. Furthermore, we show that two inhibitory populations in a master-slave configuration with different synaptic time scales can display various collective dynamical regimes: damped oscillations toward a stable focus, periodic and quasi-periodic oscillations, and chaos. Finally, when bidirectionally coupled, the two inhibitory populations can exhibit different types of θ-γ cross-frequency couplings (CFCs): phase-phase and phase-amplitude CFC. The coupling between θ and γ COs is enhanced in the presence of an external θ forcing, reminiscent of the type of modulation induced in hippocampal and cortex circuits via optogenetic drive.

Causal stability and synchronization

Synchronization of chaos arises between coupled dynamical systems and is very well understood as a temporal phenomenon, which leads the coupled systems to converge or develop a dependence with time. In this work, we provide a complementary spatial perspective to this phenomenon by introducing the novel idea of causal stability. We then propose and prove a causal stability synchronization theorem as a necessary and sufficient condition for complete synchronization. We also provide an empirical criterion to identify synchronizing variables in coupled identical chaotic dynamical systems based on intrasystem causal influences estimated using time series data of the driving system alone. For this, a recently proposed measure, Compression-Complexity Causality (CCC), is used. The sign and magnitude of the estimated CCC value capture the nature of dynamical influences from each variable to rest of the subsystem and are thus able to determine whether or not the variable, when used to couple another system, will drive that system to synchronization.

Entrainment of aperiodic and periodic oscillations in the Mercury Beating Heart system using external periodic forcing

We report experimental results indicating entrainment of aperiodic and periodic oscillatory dynamics in the Mercury Beating Heart (MBH) system under the influence of superimposed periodic forcing. Aperiodic oscillations in MBH were controlled to generate stable topological modes, namely, circle, ellipse, and triangle, evolving in a periodic fashion at different parameters of the forcing signal. These periodic dynamics show 1:1 entrainment for circular and elliptical modes, and additionally the controlled system exhibits 1:2 entrainment for elliptical and triangular modes at a different set of parameters. The external periodic forcing of the periodic MBH system reveals the existence of domains of entrainment (1:1, 1:2, 1:3, and 1:4) represented in the Arnold tongue structures. Moreover, Devil's staircase is obtained when the amplitude-frequency space of parameters of the applied signal is scanned.

Phase definition to assess synchronization quality of nonlinear oscillators

This paper proposes a phase definition, named the vector field phase, which can be defined for systems with arbitrary finite dimension and is a monotonically increasing function of time. The proposed definition can properly quantify the dynamics in the flow direction, often associated with the null Lyapunov exponent. Numerical examples that use benchmark periodic and chaotic oscillators are discussed to illustrate some of the main features of the definition, which are that (i) phase information can be obtained either from the vector field or from a time series, (ii) it permits not only detection of phase synchronization but also quantification of it, and (iii) it can be used in the phase synchronization of very different oscillators.

Sensory Stream Adaptation in Chaotic Networks

Implicit expectations induced by predictable stimuli sequences affect neuronal response to upcoming stimuli at both single cell and neural population levels. Temporally regular sensory streams also phase entrain ongoing low frequency brain oscillations but how and why this happens is unknown. Here we investigate how random recurrent neural networks without plasticity respond to stimuli streams containing oddballs. We found the neuronal correlates of sensory stream adaptation emerge if networks generate chaotic oscillations which can be phase entrained by stimulus streams. The resultant activity patterns are close to critical and support history dependent response on long timescales. Because critical network entrainment is a slow process stimulus response adapts gradually over multiple repetitions. Repeated stimuli generate suppressed responses but oddball responses are large and distinct. Oscillatory mismatch responses persist in population activity for long periods after stimulus offset while individual cell mismatch responses are strongly phasic. These effects are weakened in temporally irregular sensory streams. Thus we show that network phase entrainment provides a biologically plausible mechanism for neural oddball detection. Our results do not depend on specific network characteristics, are consistent with experimental studies and may be relevant for multiple pathologies demonstrating altered mismatch processing such as schizophrenia and depression.

Separation of coexisting dynamical regimes in multistate intermittency based on wavelet spectrum energies in an erbium-doped fiber laser

We propose a method for the detection and localization of different types of coexisting oscillatory regimes that alternate with each other leading to multistate intermittency. Our approach is based on consideration of wavelet spectrum energies. The proposed technique is tested in an erbium-doped fiber laser with four coexisting periodic orbits, where external noise induces intermittent switches between the coexisting states. Statistical characteristics of multistate intermittency, such as the mean duration of the phases for every oscillation type, are examined with the help of the developed method. We demonstrate strong advantages of the proposed technique over previously used amplitude methods.

Coexistence of intermittencies in the neuronal network of the epileptic brain

Intermittent behavior occurs widely in nature. At present, several types of intermittencies are known and well-studied. However, consideration of intermittency has usually been limited to the analysis of cases when only one certain type of intermittency takes place. In this paper, we report on the temporal behavior of the complex neuronal network in the epileptic brain, when two types of intermittent behavior coexist and alternate with each other. We prove the presence of this phenomenon in physiological experiments with WAG/Rij rats being the model living system of absence epilepsy. In our paper, the deduced theoretical law for distributions of the lengths of laminar phases prescribing the power law with a degree of -2 agrees well with the experimental neurophysiological data.

Intermittency of intermittencies

A phenomenon of intermittency of intermittencies is discovered in the temporal behavior of two coupled complex systems. We observe for the first time the coexistence of two types of intermittent behavior taking place simultaneously near the boundary of the synchronization regime of coupled chaotic oscillators. This phenomenon is found both in the numerical and physiological experiments. The laws for both the distribution and mean length of laminar phases versus the control parameter values are analytically deduced. A very good agreement between the theoretical results and simulation is shown.

A "saddle-node" bifurcation scenario for birth or destruction of a Smale-Williams solenoid

Formation or destruction of hyperbolic chaotic attractor under parameter variation is considered with an example represented by Smale-Williams solenoid in stroboscopic Poincaré map of two alternately excited non-autonomous van der Pol oscillators. The transition occupies a narrow but finite parameter interval and progresses in such way that periodic orbits constituting a "skeleton" of the attractor undergo saddle-node bifurcation events involving partner orbits from the attractor and from a non-attracting invariant set, which forms together with its stable manifold a basin boundary of the attractor.

Chaotic phase synchronization in bursting-neuron models driven by a weak periodic force

We investigate the entrainment of a neuron model exhibiting a chaotic spiking-bursting behavior in response to a weak periodic force. This model exhibits two types of oscillations with different characteristic time scales, namely, long and short time scales. Several types of phase synchronization are observed, such as 1:1 phase locking between a single spike and one period of the force and 1:l phase locking between the period of slow oscillation underlying bursts and l periods of the force. Moreover, spiking-bursting oscillations with chaotic firing patterns can be synchronized with the periodic force. Such a type of phase synchronization is detected from the position of a set of points on a unit circle, which is determined by the phase of the periodic force at each spiking time. We show that this detection method is effective for a system with multiple time scales. Owing to the existence of both the short and the long time scales, two characteristic phenomena are found around the transition point to chaotic phase synchronization. One phenomenon shows that the average time interval between successive phase slips exhibits a power-law scaling against the driving force strength and that the scaling exponent has an unsmooth dependence on the changes in the driving force strength. The other phenomenon shows that Kuramoto's order parameter before the transition exhibits stepwise behavior as a function of the driving force strength, contrary to the smooth transition in a model with a single time scale.

Clustering in cell cycle dynamics with general response/signaling feedback

Motivated by experimental and theoretical work on autonomous oscillations in yeast, we analyze ordinary differential equations models of large populations of cells with cell-cycle dependent feedback. We assume a particular type of feedback that we call responsive/signaling (RS), but do not specify a functional form of the feedback. We study the dynamics and emergent behavior of solutions, particularly temporal clustering and stability of clustered solutions. We establish the existence of certain periodic clustered solutions as well as "uniform" solutions and add to the evidence that cell-cycle dependent feedback robustly leads to cell-cycle clustering. We highlight the fundamental differences in dynamics between systems with negative and positive feedback. For positive feedback systems the most important mechanism seems to be the stability of individual isolated clusters. On the other hand we find that in negative feedback systems, clusters must interact with each other to reinforce coherence. We conclude from various details of the mathematical analysis that negative feedback is most consistent with observations in yeast experiments.

The resonance frequency shift, pattern formation, and dynamical network reorganization via sub-threshold input

We describe a novel mechanism that mediates the rapid and selective pattern formation of neuronal network activity in response to changing correlations of sub-threshold level input. The mechanism is based on the classical resonance and experimentally observed phenomena that the resonance frequency of a neuron shifts as a function of membrane depolarization. As the neurons receive varying sub-threshold input, their natural frequency is shifted in and out of its resonance range. In response, the neuron fires a sequence of action potentials, corresponding to the specific values of signal currents, in a highly organized manner. We show that this mechanism provides for the selective activation and phase locking of the cells in the network, underlying input-correlated spatio-temporal pattern formation, and could be the basis for reliable spike-timing dependent plasticity. We compare the selectivity and efficiency of this pattern formation to a supra-threshold network activation and a non-resonating network/neuron model to demonstrate that the resonance mechanism is the most effective. Finally we show that this process might be the basis of the phase precession phenomenon observed during firing of hippocampal place cells, and that it may underlie the active switching of neuronal networks to locking at various frequencies.

Characterizing the phase synchronization transition of chaotic oscillators

The chaotic phase synchronization transition is studied in connection with the zero Lyapunov exponent. We propose a hypothesis that it is associated with a switching of the maximal finite-time zero Lyapunov exponent, which is introduced in the framework of a large deviation analysis. A noisy sine circle map is investigated to introduce this hypothesis and it is tested in an unidirectionally coupled Rössler system by using the covariant Lyapunov vector associated with the zero Lyapunov exponent.

Ring intermittency near the boundary of the synchronous time scales of chaotic oscillators

In this Brief Report we study both experimentally and numerically the intermittent behavior taking place near the boundary of the synchronous time scales of chaotic oscillators being in the regime of time scale synchronization. We have shown that the observed type of the intermittent behavior should be classified as the ring intermittency.

Synchronized states in chaotic systems coupled indirectly through a dynamic environment

We consider synchronization of chaotic systems coupled indirectly through common environment where the environment has an intrinsic dynamics of its own modulated via feedback from the systems. We find that a rich variety of synchronization behavior, such as in-phase, antiphase, complete, and antisynchronization, is possible. We present an approximate stability analysis for the different synchronization behaviors. The transitions to different states of synchronous behavior are analyzed in the parameter plane of coupling strengths by numerical studies for specific cases such as Rössler and Lorenz systems and are characterized using various indices such as correlation, average phase difference, and Lyapunov exponents. The threshold condition obtained from numerical analysis is found to agree with that from the stability analysis.

Zero Lyapunov exponent in the vicinity of the saddle-node bifurcation point in the presence of noise

We consider a behavior of the zero Lyapunov exponent in the vicinity of the bifurcation point that occurs as the result of the interplay between dynamical mechanisms and random dynamics. We analytically deduce the laws for the dependence of this Lyapunov exponent on the control parameter both above and below the bifurcation point. The developed theory is applicable both to the systems with the random force and to the deterministic chaotic oscillators. We find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations. We also discuss how the revealed regularities are expected to take place in other relevant physical circumstances.

Paths to globally generalized synchronization in scale-free networks

We apply the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators, including Hénon maps, logistic maps, and Lorentz oscillators. As the coupling strength epsilon between nodes of the network is increased, transitions from partially to globally generalized synchronization and intermittent behaviors near the synchronization thresholds, are found. The generalized synchronization starts from the hubs of the network and then spreads throughout the whole network with the increase of epsilon . Our result is useful for understanding the synchronization process in complex networks.

Intermittency transition to generalized synchronization in coupled time-delay systems

We report the nature of the transition to generalized synchronization (GS) in a system of two coupled scalar piecewise linear time-delay systems using the auxiliary system approach. We demonstrate that the transition to GS occurs via an on-off intermittency route and that it also exhibits characteristically distinct behaviors for different coupling configurations. In particular, the intermittency transition occurs in a rather broad range of coupling strength for the error feedback coupling configuration and in a narrow range of coupling strength for the direct feedback coupling configuration. It is also shown that the intermittent dynamics displays periodic bursts of periods equal to the delay time of the response system in the former case, while they occur in random time intervals of finite duration in the latter case. The robustness of these transitions with system parameters and delay times has also been studied for both linear and nonlinear coupling configurations. The results are corroborated analytically by suitable stability conditions for asymptotically stable synchronized states and numerically by the probability of synchronization and by the transition of sub-Lyapunov exponents of the coupled time-delay systems. We have also indicated the reason behind these distinct transitions by referring to the unstable periodic orbit theory of intermittency synchronization in low-dimensional systems.

Two types of phase synchronization destruction

In this paper we report that there are two different types of destruction of the phase synchronization (PS) regime of chaotic oscillators depending on the parameter mismatch as well as in the case of the classical synchronization of periodic oscillators. When the parameter mismatch of the interacting chaotic oscillators is small enough, the PS breaking takes place without the destruction of the phase coherence of chaotic attractors of oscillators. Alternatively, due to the large frequency detuning, the PS breaking is accomplished by loss of the phase coherence of the chaotic attractors.

Analysis of phase synchronization of coupled chaotic oscillators with empirical mode decomposition

Empirical mode decomposition is investigated as a tool to determine the phase and frequency and to study phase synchronization between complex chaotic oscillators. Within this approach, the oscillator is characterized by a spectrum of frequencies corresponding to the empirical modes. First, the phase and frequency of the oscillators resulting from two well-known methods, based on modified variables and the Poincaré surface of section, are compared with those obtained using empirical mode decomposition. Next, for both parametrically and essentially different chaotic oscillators coupled as a drive-response system, transition to phase synchronization between corresponding empirical modes is investigated, defined as an adjustment of the mode frequencies of the response oscillator to those of the drive oscillator as the coupling is increased. In particular, anomalous and imperfect phase synchronization between modes is observed.

Ring intermittency in coupled chaotic oscillators at the boundary of phase synchronization

A new type of intermittent behavior is described to occur near the boundary of the phase synchronization regime of coupled chaotic oscillators. This mechanism, called ring intermittency, arises for sufficiently high initial mismatches in the frequencies of the two coupled systems. The laws for both the distribution and the mean length of the laminar phases versus the coupling strength are analytically deduced. Very good agreement between the theoretical results and the numerically calculated data is shown. We discuss how this mechanism is expected to take place in other relevant physical circumstances.

Comment on "Periodic phase synchronization in coupled chaotic oscillators"

Kye [Phys. Rev. E 68, 025201 (2003)] have recently claimed that, before the onset of chaotic phase synchronization in coupled phase coherent oscillators, there exists a temporally coherent state called periodic phase synchronization (PPS). Here we give evidence that some of their numerical calculations are flawed, while we provide theoretical arguments that indicate that PPS is not to be expected generically in this type of systems.

Controlling chaos in spatially extended beam-plasma system by the continuous delayed feedback

In this paper we discuss the control of complex spatio-temporal dynamics in a spatially extended nonlinear system (fluid model of Pierce diode) based on the concepts of controlling chaos in the systems with few degrees of freedom. A presented method is connected with stabilization of unstable homogeneous equilibrium state and the unstable spatio-temporal periodical states analogous to unstable periodic orbits of chaotic dynamics of the systems with few degrees of freedom. We show that this method is effective and allows to achieve desired regular dynamics chosen from a number of possible in the considered system.

Synchronization of spectral components and its regularities in chaotic dynamical systems

The chaotic synchronization regime in coupled dynamical systems is considered. It has been shown that the onset of a synchronous regime is based on the appearance of a phase relation between the interacting chaotic oscillator frequency components of Fourier spectra. The criterion of synchronization of spectral components as well as the measure of synchronization has been discussed. The universal power law has been described. The main results are illustrated by coupled Rössler systems, Van der Pol and Van der Pol-Duffing oscillators.

Generalized synchronization in time-delayed systems

We investigate the generalized synchronization between two unidirectionally linearly and nonlinearly coupled chaotic nonidentical Ikeda models and find existence conditions of the generalized synchronization. Also we study the chaos synchronization between nonidentical Ikeda models with variable feedback-delay times and find the existence and sufficient stability conditions for the retarded synchronization manifold with the coupling-delay lag time. Generalization of the approach to the wide class of nonlinear chaotic systems is also presented.

Deterministic stochastic resonance in a Rössler oscillator

We discuss the characteristics of stochastic resonancelike behavior observed in a deterministic system. If a periodically forced Rössler oscillator strays from the phase locking state, it exhibits the intermittent behavior known as phase slips. When the periodic force is modulated by a weak signal, the phase slips synchronize with the weak signal statistically. We numerically demonstrate, in terms of interslip intervals and signal to noise ratio, that the maximum synchronization can be achieved with the optimum intensity of chaotic fluctuations. It is shown that the stochastic resonancelike behavior can be observed regardless of the choice of parameters. The frequency dependence of the signal indicates that there is an optimum frequency for the maximum resonance. The phase slip rate is derived based on the fact that the phase slips are caused by a boundary crisis caused by an unstable-unstable pair bifurcation. The interslip distributions obtained from the derived slip rate and the approximation theory of the time-dependent Poisson process agree with those obtained by numerical simulations. In addition, the maximum enhancement of a weak signal is shown to be achieved by adjusting the chaotic fluctuations even if a signal becomes mixed with noise.

Three types of transitions to phase synchronization in coupled chaotic oscillators

We study the effect of noncoherence on the onset of phase synchronization of two coupled chaotic oscillators. Depending on the coherence properties of oscillations characterized by the phase diffusion, three types of transitions to phase synchronization are found. For phase-coherent attractors this transition occurs shortly after one of the zero Lyapunov exponents becomes negative. At rather strong phase diffusion, phase locking manifests a strong degree of generalized synchronization, and occurs only after one positive Lyapunov exponent becomes negative. For intermediate phase diffusion, phase synchronization sets in via an interior crises of the hyperchaotic set.

Transition from phase synchronization to complete synchronization in mutually coupled nonidentical Nd:YAG lasers

Using mutually coupled nonidentical continuous-wave Nd:YAG lasers, we experimentally confirmed the recently proposed transition route from phase synchronization to complete synchronization. As evidence of this transition we obtained the probability distribution of the intermittent synchronization time near the threshold of the complete synchronization transition.

Detectability of nondifferentiable generalized synchrony

Generalized synchronization of chaos is a type of cooperative behavior in directionally coupled oscillators that is characterized by existence of stable and persistent functional dependence of response trajectories from the chaotic trajectory of driving oscillator. In many practical cases this function is nondifferentiable and has a very complex shape. The generalized synchrony in such cases seems to be undetectable, and only the cases in which a differentiable synchronization function exists are considered to make sense in practice. We show that this viewpoint is not always correct and the nondifferentiable generalized synchrony can be revealed in many practical cases. Conditions for detection of generalized synchrony are derived analytically, and illustrated numerically with a simple example of nondifferentiable generalized synchronization.

A geometric theory of chaotic phase synchronization

A rigorous mathematical treatment of chaotic phase synchronization is still lacking, although it has been observed in many numerical and experimental studies. In this article we address the extension of results on phase synchronization in periodic oscillators to systems with phase coherent chaotic attractors with small phase diffusion. As models of such systems we consider special flows over diffeomorphisms in which the neutral direction is periodically perturbed. A generalization of the Averaging Theorem for periodic systems is used to extend Kuramoto's geometric theory of phase locking in periodically forced limit cycle oscillators to this class of systems. This approach results in reduced equations describing the dynamics of the phase difference between drive and response systems over long time intervals. The reduced equations are used to illustrate how the structure of a chaotic attractor is important in its response to a periodic perturbation, and to conclude that chaotic phase coherent systems may not always be treated as noisy periodic oscillators in this context. Although this approach is strictly justified for periodic perturbations affecting only the phase variable of a chaotic oscillator, we argue that these ideas are applicable much more generally.

Role of unstable periodic orbits in phase and lag synchronization between coupled chaotic oscillators

An increase of the coupling strength in the system of two coupled Rössler oscillators leads from a nonsynchronized state through phase synchronization to the regime of lag synchronization. The role of unstable periodic orbits in these transitions is investigated. Changes in the structure of attracting sets are discussed. We demonstrate that the onset of phase synchronization is related to phase-lockings on the surfaces of unstable tori, whereas transition from phase to lag synchronization is preceded by a decrease in the number of unstable periodic orbits.

Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling

The effect of noise on phase synchronization in small sets and larger populations of weakly coupled chaotic oscillators is explored. Both independent and correlated noise are found to enhance phase synchronization of two coupled chaotic oscillators below the synchronization threshold; this is in contrast to the behavior of two coupled periodic oscillators. This constructive effect of noise results from the interplay between noise and the locking features of unstable periodic orbits. We show that in a population of nonidentical chaotic oscillators, correlated noise enhances synchronization in the weak coupling region. The interplay between noise and weak coupling induces a collective motion in which the coherence is maximal at an optimal noise intensity. Both the noise-enhanced phase synchronization and the coherence resonance numerically observed in coupled chaotic Rössler oscillators are verified experimentally with an array of chaotic electrochemical oscillators.

Chaotic synchronization and evolution of optical phase in a bidirectional solid-state ring laser

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.

An analysis of the reliability phenomenon in the FitzHugh-Nagumo model

The reliability of single neurons on realistic stimuli has been experimentally confirmed in a wide variety of animal preparations. We present a theoretical study of the reliability phenomenon in the FitzHugh-Nagumo model on white Gaussian stimulation. The analysis of the model's dynamics is performed in three regimes-the excitable, bistable, and oscillatory ones. We use tools from the random dynamical systems theory, such as the pullbacks and the estimation of the Lyapunov exponents and rotation number. The results show that for most stimulus intensities, trajectories converge to a single stochastic equilibrium point, and the leading Lyapunov exponent is negative. Consequently, in these regimes the discharge times are reliable in the sense that repeated presentation of the same aperiodic input segment evokes similar firing times after some transient time. Surprisingly, for a certain range of stimulus intensities, unreliable firing is observed due to the onset of stochastic chaos, as indicated by the estimated positive leading Lyapunov exponents. For this range of stimulus intensities, stochastic chaos occurs in the bistable regime and also expands in adjacent parts of the excitable and oscillating regimes. The obtained results are valuable in the explanation of experimental observations concerning the reliability of neurons stimulated with broad-band Gaussian inputs. They reveal two distinct neuronal response types. In the regime where the first Lyapunov has negative values, such inputs eventually lead neurons to reliable firing, and this suggests that any observed variance of firing times in reliability experiments is mainly due to internal noise. In the regime with positive Lyapunov exponents, the source of unreliable firing is stochastic chaos, a novel phenomenon in the reliability literature, whose origin and function need further investigation.

Experimental characterization of the transition to phase synchronization of chaotic CO2 laser systems

We investigate the transition route to phase synchronization in a chaotic laser with external modulation. Such a transition is characterized by the presence of a regime of periodic phase synchronization, in which phase slips occur with maximal coherence in the phase difference between output signal and external modulation. We provide the first experimental evidence of such a regime and demonstrate that it occurs at the crossover point between two different scaling laws of the intermittent-type behavior of phase slips.

Noise-enhanced phase synchronization of chaotic oscillators

The effects of noise on phase synchronization (PS) of coupled chaotic oscillators are explored. In contrast to coupled periodic oscillators, noise is found to enhance phase synchronization significantly below the threshold of PS. This constructive role of noise has been verified experimentally with chaotic electrochemical oscillators of the electrodissolution of Ni in sulfuric acid solution.