Phase synchronization of chaotic oscillators

Transition to phase synchronization in coupled periodically driven chaotic pendulums

We have studied the transition to phase synchronization in the system of two coupled periodically driven pendulums. For the case of identical units, the coupled system has an infinite number of invariant subspaces. The synchronization-desynchronization transition is at the blowout bifurcation which coincides with the hyperchaos-chaos transition. On-off intermittency and intermingled basins of attraction can be observed close to this transition. For the case of nonidentical pendulums, the synchronization-desynchronization transition occurs far beyond the hyperchaos-chaos transition. The basin structure and the statistics of the accompanying intermittency are different from those for identical units.

Propofol anesthesia induces phase synchronization changes in EEG

OBJECTIVE

Phase coupling between EEG channel pairs in various frequency bands was evaluated during propofol anesthetic induction and recovery periods.

METHODS

Twenty-three patients participated in the study. Phase synchronization indices based on the Hilbert transform were investigated on frequency bands 0.05-1 Hz, 1-4 Hz, 4-8 Hz, 8-12 Hz and 12-16 Hz for all pairs of the 9 EEG channels covering midline and frontal areas. A straight line was used to approximate the index values as a function of time and the Sign Test statistics were applied to the slope parameters.

RESULTS

Systematic phase synchronization changes were detected. Generally, phase synchronization in the sub-delta band decreased during the induction and increased during the recovery, while the directions were reversed in the alpha band. The changes were dependent on the channel pair. In the delta, theta and beta bands, the changes were aligned more irregularly than in the sub-delta or in the alpha bands. Highly asymmetric behavior between the induction and the recovery periods was also observed in these bands.

CONCLUSIONS

Induction and recovery from propofol anesthesia changes the phase synchronization between the EEG channels. The passband and location-specific behavior of these changes reveals the effects of the anesthetic to the different neural mechanisms.

Synchronization of homoclinic chaos

Homoclinic chaos is characterized by regular geometric orbits occurring at erratic times. Phase synchronization at the average repetition frequency is achieved by a tiny periodic modulation of a control parameter. An experiment has been carried on a CO(2) laser with feedback, set in a parameter range where homoclinic chaos occurs. Any offset of the modulation frequency from the average induces phase slips over long times. Perfect phase synchronization is recovered by slow changes of the modulation frequency based upon the sign and amplitude of the slip rate. Satellite synchronization regimes are also realized, with variable numbers of homoclinic spikes per period of the modulation.

Experimental phase synchronization of a chaotic convective flow

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.

Phase synchronization with type-II intermittency in chaotic oscillators

We study the phase synchronization (PS) with type-II intermittency showing +/-2pi irregular phase jumping behavior before the PS transition occurs in a system of two coupled hyperchaotic Rossler oscillators. The behavior is understood as a stochastic hopping of an overdamped particle in a potential which has 2pi-periodic minima. We characterize it as type-II intermittency with external noise through the return map analysis. In epsilon(t)<epsilon<epsilon(c) (where epsilon(t) is the bifurcation point of type-II intermittency and epsilon(c) is the PS transition point in coupling strength parameter space), the average length of the time interval between two successive jumps follows <l> approximately exp(|epsilon(t)-epsilon|(2)), which agrees well with the scaling law obtained from the Fokker-Planck equation.

Generalized synchronization versus phase synchronization

The relation between generalized synchronization and phase synchronization is investigated. It was claimed that generalized synchronization always leads to phase synchronization, and phase synchronization is a weaker form than generalized synchronization. We propose examples that generalized synchronization can be weaker than phase synchronization, depending on parameter misfits. Moreover, generalized synchronization does not always lead to phase synchronization.

Reversible transitions between synchronization states of the cardiorespiratory system

Phase synchronization between cardiac and respiratory oscillations is investigated during anesthesia in rats. Synchrograms and time evolution of synchronization indices are used to show that the system passes reversibly through a sequence of different phase-synchronized states as the anesthesia level changes, indicating that it can undergo phase transitionlike phenomena. It appears that the synchronization state may be used to characterize the depth of anesthesia.

Characteristic relations of type-I intermittency in the presence of noise

Near the point of tangent bifurcation, the scaling properties of the laminar length of type-I intermittency are investigated in the presence of noise. Based on analytic and numerical studies, we show that the scaling relation of the laminar length is dramatically deformed from 1/sqrt[epsilon] for epsilon>0 to exp(1/D)|epsilon|(3/2) for epsilon<0 as epsilon passes the bifurcation point (epsilon=0). The results explain why two coupled Rossler oscillators exhibit deformation of the scaling relation of the synchronous length in the nearly synchronous regime.

Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems

The paper presents a qualitative analysis of an array of diffusively coupled identical continuous time dynamical systems. The effects of full, partial, antiphase, and in-phase-antiphase chaotic synchronizations are investigated via the linear invariant manifolds of the corresponding differential equations. The existence of various invariant manifolds, a self-similar behavior, and a hierarchy and embedding of the manifolds of the coupled system are discovered. Sufficient conditions for the stability of the invariant manifolds are obtained via the method of Lyapunov functions. Conditions under which full global synchronization cannot be achieved even for the largest coupling constant are defined. The general rigorous results are illustrated through examples of coupled Lorenz-like and Rossler systems.

Enhancement of phase synchronization through asymmetric couplings

Phase synchronization in lattices of coupled chaotic oscillators is studied. It is found that phase synchronization can be greatly improved by asymmetric biased coupling. The mechanism responsible for this effect is the transition from a localized wave to synchronized flow and nonlocal phase synchronization.

Integral behavior for localized synchronization in nonidentical extended systems

We report the synchronization of two nonidentical spatially extended fields, ruled by one-dimensional complex Ginzburg-Landau equations. The two fields are prepared in different dynamical regimes, and interact via an imperfect coupling consisting of a given number of local controllers N(c). The strength of the coupling is ruled by the parameter varepsilon. We show that, in the limit of three controllers per correlation length, the synchronization behavior is not affected if the product varepsilonN(c)/N is kept constant, providing a sort of integral behavior for localized synchronization.

Asymptotically stable phase synchronization revealed by autoregressive circle maps

A specially designed of nonlinear time series analysis is introduced based on phases, which are defined as polar angles in spaces spanned by a finite number of delayed coordinates. A canonical choice of the polar axis and a related implicit estimation scheme for the potentially underlying autoregressive circle map (next phase map) guarantee the invertibility of reconstructed phase space trajectories to the original coordinates. The resulting Fourier approximated, invertibility enforcing phase space map allows us to detect conditional asymptotic stability of coupled phases. This comparatively general synchronization criterion unites two existing generalizations of the old concept and can successfully be applied, e.g., to phases obtained from electrocardiogram and airflow recordings characterizing cardiorespiratory interaction.

Spatial coupling in cyclic population dynamics: models and data

We use a dynamic random field to model a spatial collection of coupled oscillators with discrete time stochastic dynamics. At each time step the phase of each cyclic local population is subject to random noise, incremented by a common dynamic, and pulled by a coupling force in the direction of some collective mean phase. We define asynchrony and derive expressions for its measurement in this model. We describe robust methods for phase estimation of cyclic population time series, for estimating strength of coupling between local populations, and for measuring variance of locally acting noise from field data. Proposed methods allow intermittently acting phase synchronizing events operating over large spatial scales to be distinguished from more continuous and possibly locally acting coupling, both of which could result in elevated levels of phase synchronization. We demonstrate the utility of this approach by applying it to classical spatial time series data of Canadian lynx. Analysis confirms findings of previous studies and reveals evidence to suggest that interpopulation coupling was weaker over the 20th century than for the 1800s. Analysis supports the notion that synchrony in these populations is maintained by a continuous and locally acting coupling between adjacent regions with large phase adjustments occurring only infrequently. When this coupling is absent, asynchrony develops between populations.

Characterization of intermittent lag synchronization

Intermittent lag synchronization of two nonidentical symmetrically coupled Rossler systems is investigated. This phenomenon can be seen as a process wherein the intermittent bursts away from the lag synchronization regime correspond to jumps of the system toward other lag configurations. During these jumps, the chaotic trajectory visits closely a periodic orbit. The identification of the different lag configurations and the measure of the fraction of time passed by the system in each one of them provide information on the global scenario of transitions undergone by the system before reaching perfect lag synchronization.

Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators

Experimental observations of time-delay-induced amplitude death in two coupled nonlinear electronic circuits that are individually capable of exhibiting limit-cycle oscillations are described. The existence of multiply connected death islands in the parameter space of coupling strength and time delay for coupled identical oscillators is established. The existence of such regions was predicted earlier on theoretical grounds [Phys. Rev. Lett. 80, 5109 (1998); Physica (Amsterdam) 129D, 15 (1999)]. The experiments also reveal the occurrence of multiple frequency states, frequency suppression of oscillations with increased time delay, and the onset of both in-phase and antiphase collective oscillations.

From low-dimensional synchronous chaos to high-dimensional desynchronous spatiotemporal chaos in coupled systems

The dynamic behavior of coupled chaotic oscillators is investigated. For small coupling, chaotic state undergoes a transition from a spatially disordered phase to an ordered phase with an orientation symmetry breaking. For large coupling, a transition from full synchronization to partial synchronization with translation symmetry breaking is observed. Two bifurcation branches, one in-phase branch starting from synchronous chaos and the other antiphase branch bifurcated from spatially random chaos, are identified by varying coupling strength epsilon. Hysteresis, bistability, and first-order transitions between these two branches are observed.

One-side riddled basin below and beyond the blowout bifurcation

In this Rapid Comunication we report a phenomenon of a one-side riddled basin where one side of the basin of attraction of an attractor on an invariant subspace (ISS) is globally riddled, while the other side is only locally riddled. This kind of basin appears due to the symmetry breaking with respect to the ISS. This one-side riddled basin can even persist beyond the blowout bifurcation, contrary to the previously reported riddled basins which exist only below the blowout transition. An experimental situation where this phenomenon can be expected is proposed.

Cortico-muscular synchronization during isometric muscle contraction in humans as revealed by magnetoencephalography

Magnetoencephalographic (MEG) and electromyographic (EMG) signals were recorded from six subjects during isometric contraction of four different muscles. Cortical sources were located from the MEG signal which was averaged time-locked to the onset of motor unit potentials. A spatial filtering algorithm was used to estimate the source activity. Sources were found in the primary motor cortex (M1) contralateral to the contracted muscle. Significant coherence between rectified EMG and M1 activity was seen in the 20 Hz frequency range in all subjects. Interactions between the motor cortex and spinal motoneuron pool were investigated by separately studying the non-stationary phase and amplitude dynamics of M1 and EMG signals. Delays between M1 and EMG signals, computed from their phase difference, were found to be in agreement with conduction times from the primary motor cortex to the respective muscle. The time-dependent cortico-muscular phase synchronization was found to be correlated with the time course of both M1 and EMG signals. The findings demonstrate that the coupling between the primary motor cortex and motoneuron pool is at least partly due to phase synchronization of 20 Hz oscillations which varies over time. Furthermore, the consistent phase lag between M1 and EMG signals, compatible with conduction time between M1 and the respective muscle with the M1 activity preceding EMG activity, supports the conjecture that the motor cortex drives the motoneuron pool.

Periodicity manifestations in the turbulent regime of the globally coupled map lattice

We revisit the globally coupled map lattice. We show that in the so called turbulent regime various periodic cluster attractor states are formed, even though the coupling between the maps are very small relative to the nonlinearity in the element maps. Most outstanding is a maximally symmetric three cluster attractor in period-3 motion, due to the foliation of the period-3 window of the element logistic maps. An analytical approach is proposed which successfully explains the systematics of various periodicity manifestations in the turbulent regime. The linear stability of the period-3 cluster attractors is investigated.

Nonlocal chaotic phase synchronization

A novel synchronization behavior, nonlocal chaotic phase synchronization, is investigated. For two coupled Rossler oscillators with only one forced by an injected periodic signal, the phase of the unforced oscillator can be locked to the phase of the periodic signal while the forced one is well unlocked by the signal; in a chain of coupled chaotic oscillators with nearest coupling, the phase of an oscillator (or a cluster) can be locked to another nonneighbor one. Moreover, the mechanism underlying the transition to nonlocal synchronization is discussed in detail.

Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors

Weakly electric fish generate a periodic electric field as a carrier signal for active location and communication tasks. Highly sensitive P-type receptors on their surface fire in response to carrier amplitude modulations (AM's) in a noisy phase locked fashion. A simple generic model of receptor activity and signal encoding is presented. Its suprathreshold dynamics, memory and receptor noise reproduce observed firing interval distributions and correlations. The model ultimately explains how smooth responses to AM's are compatible with its nonlinear phase locking properties, and reveals how receptor noise can sometimes enhance the encoding of small yet suprathreshold AM's.

Phase synchronization of two-dimensional lattices of coupled chaotic maps

Phase synchronized states can emerge in the collective behavior of an ensemble of two-dimensional chaotic coupled map lattices, due to a nearest-neighbor interaction. A definition of phase is given for iterated systems, which corresponds to the definition of phase in continuous systems. The transition to phase synchronization is characterized in an ensemble of lattices of logistic maps, in terms of the phase synchronization ratio, the average abnormal ratio, and conditional Lyapunov exponents. The largest Lyapunov exponent of the global system lambda(max) depends on both the number of coupled maps and the coupling strength. If the number of coupled maps is over some threshold, lambda(max) depends only on the coupling strength. The approach of nearest-neighbor coupling is robust against a small difference in the map parameters.

Measuring information transfer

An information theoretic measure is derived that quantifies the statistical coherence between systems evolving in time. The standard time delayed mutual information fails to distinguish information that is actually exchanged from shared information due to common history and input signals. In our new approach, these influences are excluded by appropriate conditioning of transition probabilities. The resulting transfer entropy is able to distinguish effectively driving and responding elements and to detect asymmetry in the interaction of subsystems.

Phase synchronization of coupled ginzburg-landau equations

The occurrence of phase synchronization of a pair of unidirectionally coupled nonidentical Ginzburg-Landau equations is demonstrated and characterized using cyclic and extended phases. Furthermore, it is shown that weak coupling first leads to frequency synchronization and later to phase synchronization. For strong coupling there is evidence for generalized synchronization.

Collective phase slips and phase synchronizations in coupled oscillator systems

Phase synchronization dynamics in coupled limit cycles with distributed natural frequencies are explored. A synchronization tree from free oscillations to local clustering and global phase locking is found. A desynchronization-induced transition to chaos is shown. Near the onset of various phase synchronization points, a simultaneous quantized stick-slip feature of the phases of oscillators is observed on the desynchronization side and heuristically interpreted in terms of a heteroclinic path instability.

Intermittency in chaotic rotations

We examine the rotational dynamics associated with bounded chaotic flows, such as those on chaotic attractors, and find that the dynamics typically exhibits on-off intermittency. In particular, a properly defined chaotic rotation tends to follow, approximately, the phase-space rotation of a harmonic oscillator with occasional bursts away from this nearly uniform rotation. The intermittent behavior is identified in several well studied chaotic systems, and an argument is provided for the generality of this behavior.

Phase transition in globally coupled Rössler oscillators

We study a population of identical Rössler oscillators with global coupling. When the coupling constant is increased, an order-disorder-type phase transition occurs. Partial phase synchronization occurs in the ordered phase, although the amplitude of the oscillation is randomly distributed. We analyze the phase transition with a self-consistent method.

Physics of the rhythmic applause

We report on a series of measurements aimed to characterize the development and the dynamics of the rhythmic applause in concert halls. Our results demonstrate that while this process shares many characteristics of other systems that are known to synchronize, it also has features that are unexpected and unaccounted for in many other systems. In particular, we find that the mechanism lying at the heart of the synchronization process is the period doubling of the clapping rhythm. The characteristic interplay between synchronized and unsynchronized regimes during the applause is the result of a frustration in the system. All results are understandable in the framework of the Kuramoto model.

A unifying definition of synchronization for dynamical systems

We propose a unifying definition for synchronization between stationary finite dimensional deterministic dynamical systems. By example, we show that the synchronization phenomena discussed in the dynamical systems literature fits within the framework of this definition, and discuss problems with previous definitions of synchronization. We conclude with a discussion of possible extensions of the definition to infinite dimensional systems described by partial differential equations and/or systems where noise is present. (c) 2000 American Institute of Physics.

Synchronization of mutually coupled self-mixing modulated lasers

Synchronization of mutually coupled chaotic lasers has been demonstrated in a microchip LiNdP4O12 laser array with self-mixing feedback modulation. An abrupt transition to synchronized chaos by way of "phase squeezing" was observed when coupling between the two lasers was increased. This phenomenon is well reproduced by numerical calculations using model equations. It is also shown that low energy variation as well as high disorder are concurrently established in synchronized chaos.

Synchronization and control of coupled ginzburg-landau equations using local coupling

In this paper we discuss the properties of a recently introduced coupling scheme for spatially extended systems based on local spatially averaged coupling signals [see Z. Tasev et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. (to be published); and L. Junge et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, 2265 (1999)]. Using the Ginzburg-Landau model, we performed an extensive numerical examination of this coupling scheme, i.e., a complete scan through the relevant coupling parameters. Furthermore, we demonstrate suppression and control of spatiotemporal chaos, e.g., stabilizing the homogeneous steady state and spatially localized control. As an application all model parameters of the Ginzburg-Landau equation are estimated given only the local information of the system.

Synchronization of chaotic structurally nonequivalent systems

Synchronization features are explored for a pair of chaotic high-dimensional bidirectionally coupled structurally nonequivalent systems. We find two regimes of synchronization in dependence on the coupling strength: creation of a lower dimensional chaotic state, and for larger coupling a transition toward a stable periodic motion. We characterize this new state, showing that it is associated with an abrupt transition in the Lyapunov spectrum. The robustness of this state against noise is discussed, and the use of this dynamical property as a possible approach for the control of chaos is outlined.

Communication through chaotic modeling of languages

We propose a communication technique that uses modeling of language in the encoding-decoding process of message transmission. A temporal partition (time-delay coarse graining of the phase space based on the symbol sequence statistics) is introduced with little if any intervention required for the targeting of the trajectory. Message transmission is performed by means of codeword, i.e., specific targeting instructions are sent to the receiver rather than the explicit message. This approach yields (i) error correction availability for transmission in the presence of noise or dropouts, (ii) transmission in a compressed format, (iii) a high level of security against undesirable detection, and (iv) language recognition.

Phase order in chaotic maps and in coupled map lattices

By defining a direction phase as the direction of two sequential iterations of the logistic map, a transition of a net direction phase M from zero to a finite value as the parameter &mgr; increases is found. Near the transition point &mgr;(0) a scaling M approximately (&mgr;-&mgr;(0))(alpha) with alpha = 0.5 is obtained. The order state of the direction phases in a coupled map lattice is also studied. A phase synchronization of the direction phases is found although the lattices still remain chaotic.

Experiments on arrays of globally coupled chaotic electrochemical oscillators: Synchronization and clustering

Experiments on chaotically oscillating arrays of 64 nickel electrodes in sulfuric acid were carried out. External resistors in parallel and series are added to vary the extent of global coupling among the oscillators without changing the other properties of the system. The array is heterogeneous due to small variations in the properties of the electrodes and there is also a small amount of noise. The addition of global coupling transforms a system of independent elements to a state of complete synchronization. At intermediate coupling strengths stable clusters, or condensates of elements, form. All the elements in a cluster follow the same chaotic trajectory but each cluster has its own dynamics; the system is thus temporally chaotic but spatially ordered. Many cluster configurations occur under the same conditions and transitions among them can be produced. For values of the coupling parameter on either side of the stable cluster region a non-stationary behavior occurs in which clustered and synchronized states alternately form and break up. Some statistical properties of the cluster states are determined. (c) 2000 American Institute of Physics.

Phase synchronization in coupled nonidentical excitable systems and array-enhanced coherence resonance

We study the dynamics of a lattice of coupled nonidentical Fitz Hugh-Nagumo system subject to independent external noise. It is shown that these stochastic oscillators can lead to global synchronization behavior without an external signal. With the increase of the noise intensity, the system exhibits coherence resonance behavior. Coupling can enhance greatly the noise-induced coherence in the system.

Phase synchronization in the forced Lorenz system

We demonstrate that the dynamics of phase synchronization in a chaotic system under weak periodic forcing depends crucially on the distribution of intrinsic characteristic times of this system. Under the external periodic action, the frequency of every unstable periodic orbit is locked to the frequency of the force. In systems which in the autonomous case displays nearly isochronous chaotic rotations, the locking ratio is the same for all periodic orbits; since a typical chaotic orbit wanders between the periodic ones, its phase follows the phase of the force. For the Lorenz attractor with its unbounded times of return onto a Poincaré surface, such state of perfect phase synchronization is inaccessible. Analysis with the help of unstable periodic orbits shows that this state is replaced by another one, which we call "imperfect phase synchronization," and in which we observe alternation of temporal segments, corresponding to different rational values of frequency lockings.

Loss of lag synchronization in coupled chaotic systems

Lag synchronization denotes a particular form of synchronization in which the amplitudes of two interacting, nonidentical chaotic oscillators are correlated but there is a characteristic time delay between them. We study transitions to and between different forms of synchronization for the attractors defined as "in-phase" and "out-of-phase" and investigate the processes by which lag synchronization is lost in two coupled Rössler systems. With a small frequency mismatch between the two systems, these processes are related to the occurrence of a peculiar form of basin structure as more and more periodic orbits embedded into the synchronized chaotic state become unstable in a transverse direction.

Desynchronization of chaos in coupled logistic maps

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average. Considering a system of two coupled logistic maps, we show that the transition to riddling will be soft or hard, depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to locally riddled basins of attraction. We present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction. Each scenario is described by following the type and location of the relevant asynchronous cycles, and determining their stable and unstable invariant manifolds. One scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area. Another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed. Finally, a phase diagram is presented to illustrate the parameter values at which the various transitions occur.

Synchronization of noisy systems by stochastic signals

We study, in terms of synchronization, the nonlinear response of noisy bistable systems to a stochastic external signal, represented by Markovian dichotomic noise. We propose a general kinetic model which allows us to conduct a full analytical study of the nonlinear response, including the calculation of cross-correlation measures, the mean switching frequency, and synchronization regions. Theoretical results are compared with numerical simulations of a noisy overdamped bistable oscillator. We show that dichotomic noise can instantaneously synchronize the switching process of the system. We also show that synchronization is most pronounced at an optimal noise level-this effect connects this phenomenon with aperiodic stochastic resonance. Similar synchronization effects are observed for a stochastic neuron model stimulated by a stochastic spike train.

Synchronization in the human cardiorespiratory system

We investigate synchronization between cardiovascular and respiratory systems in healthy humans under free-running conditions. For this aim we analyze nonstationary irregular bivariate data, namely, electrocardiograms and measurements of respiratory flow. We briefly discuss a statistical approach to synchronization in noisy and chaotic systems and illustrate it with numerical examples; effects of phase and frequency locking are considered. Next, we present and discuss methods suitable for the detection of hidden synchronous epochs from such data. The analysis of the experimental records reveals synchronous regimes of different orders n:m and transitions between them; the physiological significance of this finding is discussed.

Controlling and synchronizing space time chaos

Control and synchronization of continuous space-extended systems is realized by means of a finite number of local tiny perturbations. The perturbations are selected by an adaptive technique, and they are able to restore each of the independent unstable patterns present within a space time chaotic regime, as well as to synchronize two space time chaotic states. The effectiveness of the method and the robustness against external noise is demonstrated for the amplitude and phase turbulent regimes of the one-dimensional complex Ginzburg-Landau equation. The problem of the minimum number of local perturbations necessary to achieve control is discussed as compared with the number of independent spatial correlation lengths.

Complex dynamics and phase synchronization in spatially extended ecological systems

Population cycles that persist in time and are synchronized over space pervade ecological systems, but their underlying causes remain a long-standing enigma. Here we examine the synchronization of complex population oscillations in networks of model communities and in natural systems, where phenomena such as unusual '4- and 10-year cycle' of wildlife are often found. In the proposed spatial model, each local patch sustains a three-level trophic system composed of interacting predators, consumers and vegetation. Populations oscillate regularly and periodically in phase, but with irregular and chaotic peaks together in abundance-twin realistic features that are not found in standard ecological models. In a spatial lattice of patches, only small amounts of local migration are required to induce broad-scale 'phase synchronization, with all populations in the lattice phase-locking to the same collective rhythm. Peak population abundances, however, remain chaotic and largely uncorrelated. Although synchronization is often perceived as being detrimental to spatially structured populations, phase synchronization leads to the emergence of complex chaotic travelling-wave structures which may be crucial for species persistence.

Role of multistability in the transition to chaotic phase synchronization

In this paper we describe the transition to phase synchronization for systems of coupled nonlinear oscillators that individually follow the Feigenbaum route to chaos. A nested structure of phase synchronized regions of different attractor families is observed. With this structure, the transition to nonsynchronous behavior is determined by the loss of stability for the most stable synchronous mode. It is shown that the appearance of hyperchaos and the transition from lag synchronization to phase synchronization are related to the merging of chaotic attractors from different families. Numerical examples using Rossler systems and model maps are given. (c) 1999 American Institute of Physics.