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A method for the estimation of the Lyapunov exponent corresponding to enslaved phase dynamics from time series has been proposed. It is valid for both nonautonomous systems demonstrating periodic dynamics in the presence of noise and coupled chaotic oscillators and allows us to estimate precisely enough the value of this Lyapunov exponent in the supercritical region of the control parameters. The main results are illustrated with the help of the examples of the noised circle map, the nonautonomous Van der Pole oscillator in the presence of noise, and coupled chaotic Rössler systems.

The dynamical responses of a complex system to external perturbations are of both fundamental interest and practical significance. Here, by the model of networked chaotic oscillators, we investigate how the synchronization behavior of a complex network is influenced by an externally added periodic driving. Interestingly, it is found that by a slight change of the properties of the external driving, e.g., the frequency or phase lag between its intrinsic oscillation and external driving, the network synchronizability could be significantly modified. We demonstrate this phenomenon by different network models and, based on the method of master stability function, give an analysis on the underlying mechanisms. Our studies highlight the importance of external perturbations on the collective behaviors of complex networks, and also provide an alternate approach for controlling network synchronization.

Existing works on coherence resonance, i.e., the phenomenon of noise-enhanced temporal regularity, focus on excitable dynamical systems such as those described by the FitzHugh-Nagumo equations. We extend the scope of coherence resonance to an important class of dynamical systems: coupled chaotic oscillators. In particular, we show that, when a system of coupled chaotic oscillators is under the influence of noise, the degree of temporal regularity of dynamical variables characterizing the difference among the oscillators can increase and reach a maximum value at some optimal noise level. We present numerical results illustrating the phenomenon and give a physical theory to explain it.

We report experimental evidence of phase synchronization of high dimensional chaotic oscillators in a laboratory experiment. The experiment consists of a thermocapillary driven convective cell in a time dependent chaotic regime. The synchronized states emerge as a consequence of a localized temperature perturbation to the heater. The transition to phase synchronization is studied as a function of the external perturbations. The existence and stability conditions for this phenomenon are discussed.

We study the effect of impurities introduced into a lattice and their ability to control the dynamical behavior of arrays of coupled nonlinear chaotic oscillators. In particular we show that under certain conditions a single impurity can produce simple spatiotemporal patterns in place of complex chaotic behavior for very long chains of oscillators. Under the same conditions we also examine the effect of disorder in the lengths of the pendula and explain the observed patterns and periodic behavior to be the result of inadvertent impurities.

When two identical chaotic oscillators interact, one or more intervals of coupling parameters generally exist in which the synchronized state is weakly stable, and its basin of attraction is riddled with holes that are repelled from it. The paper discusses the role of the absorbing area for the emergence of local vs global riddling and for controlling the dynamics, once synchronization breaks down.

We study phase synchronization in a chain of weakly coupled chaotic oscillators. In the synchronous state, the phases of oscillators are locked, while the amplitudes remain chaotic. We demonstrate that the coexistence of several clusters of mutually synchronized elements and global synchronization of all oscillators is possible. Two mechanisms of the transition to global synchronization are shown. The dynamics of spatiotemporal defects is discussed for the cases of phase-coherent and funnel R\"ossler oscillators.

We investigate the coevolution dynamics of node activities and coupling strengths in coupled chaotic oscillators via a simple threshold adaptive scheme. The coupling strength is synchronous activity regulated, which in turn is able to boost the synchronization remarkably. In the case of weak coupling, the globally coupled oscillators present a highly clustered functional connectivity with a power-law distribution in the tail with $\ensuremath{\gamma}\ensuremath{\simeq}3.1$, while for strong coupling, they self-organize into a network with a heterogeneously rich connectivity at the onset of synchronization but exhibit rather sparse structure to maintain the synchronization in noisy environment. The relevance of the results is briefly discussed.

Synchronizing nonidentical chaotic oscillators is very often achieved by using various types of couplings. In the practice of synchronization the ``right choice'' of the coupling variable\char22{}$y$ for the R\"ossler system, $x$ for the Lorenz equations, and so on\char22{}is usually stated rather than explained or justified. Using the R\"ossler and Rucklidge system, in this paper, it is shown that such ``optimal'' choices are strongly related to observability properties which, in turn, can be quantified. In this paper it will also be shown that synchronizability does not only depend on the observability of the system but it is also a consequence of the dynamical regimes under study. The paper aims at providing important insight into the critical problem of making the ``right choice'' when it comes to choosing the coupling variable in synchronization schemes.

We apply the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators, including H\'enon maps, logistic maps, and Lorentz oscillators. As the coupling strength $\ensuremath{\epsilon}$ 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 $\ensuremath{\epsilon}$. Our result is useful for understanding the synchronization process in complex networks.

In this article we report that there are two different types of destruction of the phase synchronization 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., Comment: 10 pages, 12 figures

Anti-phase synchronization (AS) in coupled chaotic oscillators is investigated. The necessary condition for AS is given and the stability of AS is studied. Results are demonstrated with numerical simulations and electronic circuits.

The phase difference in coupled chaotic oscillators exhibits a small noiselike fast fluctuation in long-term phase dynamics. The fast fluctuation can cause an error in the measurement of the period of the temporary phase locking state. We discuss why we should filter out the fast fluctuation on determination of periodic phase synchronization.

We report the occurrence of spontaneous synchronizing events between two semiconductor lasers, when the emission of a frequency- and intensity-chaotic driving laser is unidirectionally coupled into a second stable response laser. The driving laser is driven chaotic by delayed optical feedback, the response laser is a device-identical solitary laser. We demonstrate the onset of an episodic synchronization regime when the two lasers are spectrally detuned with respect to each other. By a joint experimental and modeling analysis we can attribute the onset and the duration of the episodes to properties of spectral overlap of both lasers. This effect can even give rise to seemingly anticorrelated intensity behavior. We expect episodic synchronization to be a generic scenario for the loss of synchronization of chaotic oscillators exhibiting frequency cycles.

We study the route to synchronization in two noisy, nonisochronous oscillators. Anomalous phase synchronization arises if both oscillators differ in their respective value of nonisochronicity and it is characterized by a strong detuning of the oscillator frequencies with the onset of coupling. Here we show that anomalous synchronization, both in limit-cycle or chaotic oscillators, can considerably be enlarged under the influence of asymmetrical coupling and noise. In these systems we describe a number of noise induced effects, such as an inversion of the natural frequency difference and coupling induced desynchronization of two identical oscillators. Our results can be explained in terms of a noisy particle in a tilted washboard potential.

We have investigated the dynamics of the coupled Lorenz oscillators numerically and theoretically. We find the partial amplitude death when the interaction is strong enough. The linear stability analysis of the partial amplitude death is proposed.

We study the dynamic stabilities of unidirectionally coupled linear arrays of chaotic oscillators with time-delay feedbacks in star configuration, and find that if all oscillators in the network are identical, then the oscillators in the linear arrays can anticipate the driving oscillators, and simultaneously the oscillators in the linear arrays with the same position with respect to the central one are in synchronous chaotic state. Compared with the anticipated synchronization, the layered synchronization is first generated and last destroyed as the coupling constant is increased. This coexistence of anticipated and layered chaotic synchronization is destroyed by long time feedback. If the driving and driven oscillators are different, then only layered chaotic synchronization is possible.

We study phase synchronization effects of chaotic oscillators with a type-I intermittency behavior. The external and mutual locking of the average length of the laminar stage for coupled discrete and continuous in time systems is shown and the mechanism of this synchronization is explained. We demonstrate that this phenomenon can be described by using results of the parametric resonance theory and that this correspondence enables one to predict and derive all zones of synchronization.

In this paper we present a method based on a generalized Hamiltonian formalism to associate to a chaotic system of known dynamics a function of the phase space variables with the characteristics of an energy. Using this formalism we have found energy functions for the Lorenz, Rössler, and Chua families of chaotic oscillators. We have theoretically analyzed the flow of energy in the process of synchronizing two chaotic systems via feedback coupling and used the previously found energy functions for computing the required energy to maintain a synchronized regime between systems of these families. We have calculated the flows of energy at different coupling strengths covering cases of both identical as well as nonidentical synchronization. The energy dissipated by the guided system seems to be sensitive to the transitions in the stability of its equilibrium points induced by the coupling.

Synchronization of three-coupled chaotic oscillators was studied with the use of a coupled map system derived for interacting kicked relaxators. Partial synchronization (PS), in which two of the three were synchronized, was observed in addition to complete synchronization. An intermittency associated with the breakdown of the PS, seemingly different from the conventional on-off intermittency, was found. We elucidated the statistics, observing the burst-size distribution, the laminar duration distribution, etc. It was found that the breakdown of the PS generated an anomalous diffusion different from that associated with on-off intermittency.

Whether common noise can induce complete synchronization in chaotic systems has been a topic of great relevance and long-standing controversy. We first clarify the mechanism of this phenomenon and show that the existence of a significant contraction region, where nearby trajectories converge, plays a decisive role. Second, we demonstrate that, more generally, common noise can induce phase synchronization in nonidentical chaotic systems. Such a noise-induced synchronization and synchronization transitions are of special significance for understanding neuron encoding in neurobiology.

Generalized synchronization in an array of mutually (bidirectionally) coupled nonidentical chaotic oscillators is studied. Coupled Lorenz oscillators and coupled Lorenz-Rossler oscillators are adopted as our working models. With increasing the coupling strengths, the system experiences a cascade of transitions from the partial to the global generalized synchronizations, i.e., different oscillators are gradually entrained through a clustering process. This scenario of transitions reveals an intrinsic self-organized order in groups of interacting units, which generalizes the idea of generalized synchronizations in drive-response systems.

Dynamic behaviors of coupled ring and linear arrays of unidirectionally coupled Lorenz oscillators are studied numerically. It is found that the chaotic rotating waves generated from the ring propagate with spatial periodic synchronization along the linear array, that is to say, two chaotic oscillators in the linear array are synchronized if the number of oscillators (spatial distance) between them is a multiple of oscillator number in the ring. Numerically it is shown that the stabilities of the synchronized states are enhanced by chaos, and degraded when the oscillators are far from the ring.

The effect of impurities in a two-dimensional lattice of coupled nonlinear chaotic oscillators and their ability to control the dynamical behavior of the system are studied. We show that a single impurity can produce synchronized spatio-temporal patterns, even though all oscillators and the impurity are chaotic when uncoupled. When a small number of impurities is arranged in a way, that the lattice is divided into two disjoint parts, synchronization is enforced even for small coupling. The synchronization is not affected as the size of the lattice increases, although the impurity concentration tends to zero., 4 pages, 4 figures