Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system

Impact of Cavity Geometry on Microlaser Dynamics

We experimentally investigate spatiotemporal lasing dynamics in semiconductor microcavities with various geometries, featuring integrable or chaotic ray dynamics. The classical ray dynamics directly impacts the lasing dynamics, which is primarily determined by the local directionality of long-lived ray trajectories. The directionality of optical propagation dictates the characteristic length scales of intensity variations, which play a pivotal role in nonlinear light-matter interactions. While wavelength-scale intensity variations tend to stabilize lasing dynamics, modulation on much longer scales causes spatial filamentation and irregular pulsation. Our results will pave the way to control the lasing dynamics by engineering the cavity geometry and ray dynamical properties.

Chaotic mode-competition dynamics in a multimode semiconductor laser with optical feedback and injection

Photonic computing has attracted increasing interest for the acceleration of information processing in machine learning applications. The mode-competition dynamics of multimode semiconductor lasers are useful for solving the multi-armed bandit problem in reinforcement learning for computing applications. In this study, we numerically evaluate the chaotic mode-competition dynamics in a multimode semiconductor laser with optical feedback and injection. We observe the chaotic mode-competition dynamics among the longitudinal modes and control them by injecting an external optical signal into one of the longitudinal modes. We define the dominant mode as the mode with the maximum intensity; the dominant mode ratio for the injected mode increases as the optical injection strength increases. We deduce that the characteristics of the dominant mode ratio in terms of the optical injection strength are different among the modes owing to the different optical feedback phases. We propose a control technique for the characteristics of the dominant mode ratio by precisely tuning the initial optical frequency detuning between the optical injection signal and injected mode. We also evaluate the relationship between the region of the large dominant mode ratios and the injection locking range. The region with the large dominant mode ratios does not correspond to the injection-locking range. The control technique of chaotic mode-competition dynamics in multimode lasers is promising for applications in reinforcement learning and reservoir computing in photonic artificial intelligence.

Coherent Structures in Plane Channel Flow of Dilute Polymer Solutions with Vanishing Inertia

When subjected to sufficiently strong velocity gradients, solutions of long, flexible polymers exhibit flow instabilities and chaotic motion, often referred to as elastic turbulence. Its mechanism differs from the familiar, inertia-driven turbulence in Newtonian fluids and is poorly understood. Here, we demonstrate that the dynamics of purely elastic pressure-driven channel flows of dilute polymer solutions are organized by exact coherent structures that take the form of two-dimensional traveling waves. Our results demonstrate that no linear instability is required to sustain such traveling wave solutions and that their origin is purely elastic in nature. We show that the associated stress profiles are characterized by thin, filamentlike arrangements of polymer stretch, which is sustained by a solitary pair of vortices. We discuss the implications of the traveling wave solutions for the transition to elastic turbulence in straight channels and propose ways for their detection in experiments.

Synchronization of complex human networks

The synchronization of human networks is essential for our civilization and understanding its dynamics is important to many aspects of our lives. Human ensembles were investigated, but in noisy environments and with limited control over the network parameters which govern the network dynamics. Specifically, research has focused predominantly on all-to-all coupling, whereas current social networks and human interactions are often based on complex coupling configurations. Here, we study the synchronization between violin players in complex networks with full and accurate control over the network connectivity, coupling strength, and delay. We show that the players can tune their playing period and delete connections by ignoring frustrating signals, to find a stable solution. These additional degrees of freedom enable new strategies and yield better solutions than are possible within current models such as the Kuramoto model. Our results may influence numerous fields, including traffic management, epidemic control, and stock market dynamics.

Extraction of slow and fast dynamics of multiple time scale systems using wavelet techniques

A methodology is presented based on wavelet techniques to approximate fast and slow dynamics present in time-series whose behavior is characterized by different local scales in time. These approximations are useful to understand the global dynamics of the original full systems, especially in experimental situations where all information is contained in a one-dimensional time-series. Wavelet analysis is a natural approach to handle these approximations because each dynamical behavior manifests its specific subset in frequency domain, for example, with two time scales, the slow and fast dynamics, present in low and high frequencies, respectively. The proposed procedure is illustrated by the analysis of a complex experimental time-series of iron electrodissolution where the slow chaotic dynamics is interrupted by fast irregular spiking. The method can be used to first filter the time-series data and then separate the fast and slow dynamics even when clear maxima and/or minima in the corresponding global wavelet spectrum are missing. The results could find applications in the analysis of synchronization of complex systems through multi-scale analysis.

Identification of minimal parameters for optimal suppression of chaos in dissipative driven systems

Taming chaos arising from dissipative non-autonomous nonlinear systems by applying additional harmonic excitations is a reliable and widely used procedure nowadays. But the suppressory effectiveness of generic non-harmonic periodic excitations continues to be a significant challenge both to our theoretical understanding and in practical applications. Here we show how the effectiveness of generic suppressory excitations is optimally enhanced when the impulse transmitted by them (time integral over two consecutive zeros) is judiciously controlled in a not obvious way. Specifically, the effective amplitude of the suppressory excitation is minimal when the impulse transmitted is maximum. Also, by lowering the impulse transmitted one obtains larger regularization areas in the initial phase difference-amplitude control plane, the price to be paid being the requirement of larger amplitudes. These two remarkable features, which constitute our definition of optimum control, are demonstrated experimentally by means of an analog version of a paradigmatic model, and confirmed numerically by simulations of such a damped driven system including the presence of noise. Our theoretical analysis shows that the controlling effect of varying the impulse is due to a subsequent variation of the energy transmitted by the suppressory excitation.

Transient dynamics and their control in time-delay autonomous Boolean ring networks

Biochemical systems with switch-like interactions, such as gene regulatory networks, are well modeled by autonomous Boolean networks. Specifically, the topology and logic of gene interactions can be described by systems of continuous piecewise-linear differential equations, enabling analytical predictions of the dynamics of specific networks. However, most models do not account for time delays along links associated with spatial transport, mRNA transcription, and translation. To address this issue, we have developed an experimental test bed to realize a time-delay autonomous Boolean network with three inhibitory nodes, known as a repressilator, and use it to study the dynamics that arise as time delays along the links vary. We observe various nearly periodic oscillatory transient patterns with extremely long lifetime, which emerge in small network motifs due to the delay, and which are distinct from the eventual asymptotically stable periodic attractors. For repeated experiments with a given network, we find that stochastic processes give rise to a broad distribution of transient times with an exponential tail. In some cases, the transients are so long that it is doubtful the attractors will ever be approached in a biological system that has a finite lifetime. To counteract the long transients, we show experimentally that small, occasional perturbations applied to the time delays can force the trajectories to rapidly approach the attractors.

From chemical systems to systems chemistry: Patterns in space and time

We present a brief, idiosyncratic overview of the past quarter century of progress in nonlinear chemical dynamics and discuss what we view as the most exciting recent developments and some challenges and likely areas of progress in the next 25 years.

Pinning synchronization of fractional-order complex networks with Lipschitz-type nonlinear dynamics

This paper deals with pinning synchronization problem of fractional-order complex networks with Lipschitz-type nonlinear nodes and directed communication topology. We first reformulate the problem as a global asymptotic stability problem by describing network evolution in terms of error dynamics. Then, a novel frequency domain approach is developed by using Laplace transform, algebraic graph theory and generalized Gronwall inequality. We show that pinning synchronization can be ensured if the extended network topology contains a spanning tree and the coupling strength is large enough. Furthermore, we provide an easily testable criterion for global pinning synchronization depending on fractional-order, network topology, oscillator dynamics and state feedback. Numerical simulations are provided to illustrate the effectiveness of the theoretical analysis.

Deconvolution of reacting-flow dynamics using proper orthogonal and dynamic mode decompositions

Analytical and computational studies of reacting flows are extremely challenging due in part to nonlinearities of the underlying system of equations and long-range coupling mediated by heat and pressure fluctuations. However, many dynamical features of the flow can be inferred through low-order models if the flow constituents (e.g., eddies or vortices) and their symmetries, as well as the interactions among constituents, are established. Modal decompositions of high-frequency, high-resolution imaging, such as measurements of species-concentration fields through planar laser-induced florescence and of velocity fields through particle-image velocimetry, are the first step in the process. A methodology is introduced for deducing the flow constituents and their dynamics following modal decomposition. Proper orthogonal (POD) and dynamic mode (DMD) decompositions of two classes of problems are performed and their strengths compared. The first problem involves a cellular state generated in a flat circular flame front through symmetry breaking. The state contains two rings of cells that rotate clockwise at different rates. Both POD and DMD can be used to deconvolve the state into the two rings. In POD the contribution of each mode to the flow is quantified using the energy. Each DMD mode can be associated with an energy as well as a unique complex growth rate. Dynamic modes with the same spatial symmetry but different growth rates are found to be combined into a single POD mode. Thus, a flow can be approximated by a smaller number of POD modes. On the other hand, DMD provides a more detailed resolution of the dynamics. Two classes of reacting flows behind symmetric bluff bodies are also analyzed. In the first, symmetric pairs of vortices are released periodically from the two ends of the bluff body. The second flow contains von Karman vortices also, with a vortex being shed from one end of the bluff body followed by a second shedding from the opposite end. The way in which DMD can be used to deconvolve the second flow into symmetric and von Karman vortices is demonstrated. The analyses performed illustrate two distinct advantages of DMD: (1) Unlike proper orthogonal modes, each dynamic mode is associated with a unique complex growth rate. By comparing DMD spectra from multiple nominally identical experiments, it is possible to identify "reproducible" modes in a flow. We also find that although most high-energy modes are reproducible, some are not common between experimental realizations; in the examples considered, energy fails to differentiate between reproducible and nonreproducible modes. Consequently, it may not be possible to differentiate reproducible and nonreproducible modes in POD. (2) Time-dependent coefficients of dynamic modes are complex. Even in noisy experimental data, the dynamics of the phase of these coefficients (but not their magnitude) are highly regular. The phase represents the angular position of a rotating ring of cells and quantifies the downstream displacement of vortices in reacting flows. Thus, it is suggested that the dynamical characterizations of complex flows are best made through the phase dynamics of reproducible DMD modes.

Dynamics in hybrid complex systems of switches and oscillators

While considerable progress has been made in the analysis of large systems containing a single type of coupled dynamical component (e.g., coupled oscillators or coupled switches), systems containing diverse components (e.g., both oscillators and switches) have received much less attention. We analyze large, hybrid systems of interconnected Kuramoto oscillators and Hopfield switches with positive feedback. In this system, oscillator synchronization promotes switches to turn on. In turn, when switches turn on, they enhance the synchrony of the oscillators to which they are coupled. Depending on the choice of parameters, we find theoretically coexisting stable solutions with either (i) incoherent oscillators and all switches permanently off, (ii) synchronized oscillators and all switches permanently on, or (iii) synchronized oscillators and switches that periodically alternate between the on and off states. Numerical experiments confirm these predictions. We discuss how transitions between these steady state solutions can be onset deterministically through dynamic bifurcations or spontaneously due to finite-size fluctuations.

Dynamical-systems analysis and unstable periodic orbits in reacting flows behind symmetric bluff bodies

Dynamical systems analysis is performed for reacting flows stabilized behind four symmetric bluff bodies to determine the effects of shape on the nature of flame stability, acoustic coupling, and vortex shedding. The task requires separation of regular, repeatable aspects of the flow from experimental noise and highly irregular, nonrepeatable small-scale structures caused primarily by viscous-mediated energy cascading. The experimental systems are invariant under a reflection, and symmetric vortex shedding is observed throughout the parameter range. As the equivalence ratio-and, hence, acoustic coupling-is reduced, a symmetry-breaking transition to von Karman vortices is initiated. Combining principal-components analysis with a symmetry-based filtering, we construct bifurcation diagrams for the onset and growth of von Karman vortices. We also compute Lyapunov exponents for each flame holder to help quantify the transitions. Furthermore, we outline changes in the phase-space orbits that accompany the onset of von Karman vortex shedding and compute unstable periodic orbits (UPOs) embedded in the complex flows prior to and following the bifurcation. For each flame holder, we find a single UPO in flows without von Karman vortices and a pair of UPOs in flows with von Karman vortices. These periodic orbits organize the dynamics of the flow and can be used to reduce or control flow irregularities. By subtracting them from the overall flow, we are able to deduce the nature of irregular facets of the flows.

A unified model for the dynamics of driven ribbon with strain and magnetic order parameters

We develop a unified model to explain the dynamics of driven one dimensional ribbon for materials with strain and magnetic order parameters. We show that the model equations in their most general form explain several results on driven magnetostrictive metallic glass ribbons such as the period doubling route to chaos as a function of a dc magnetic field in the presence of a sinusoidal field, the quasiperiodic route to chaos as a function of the sinusoidal field for a fixed dc field, and induced and suppressed chaos in the presence of an additional low amplitude near resonant sinusoidal field. We also investigate the influence of a low amplitude near resonant field on the period doubling route. The model equations also exhibit symmetry restoring crisis with an exponent close to unity. The model can be adopted to explain certain results on magnetoelastic beam and martensitic ribbon under sinusoidal driving conditions. In the latter case, we find interesting dynamics of a periodic one orbit switching between two equivalent wells as a function of an ac magnetic field that eventually makes a direct transition to chaos under resonant driving condition. The model is also applicable to magnetomartensites and materials with two order parameters.

Dynamics of strain bifurcations in a magnetostrictive ribbon

We develop a coupled nonlinear oscillator model involving magnetization and strain to explain several experimentally observed dynamical features exhibited by forced magnetostrictive ribbon. Here we show that the model recovers the observed period-doubling route to chaos as function of the dc field for a fixed ac field and quasiperiodic route to chaos as a function of the ac field, keeping the dc field constant. The model also predicts induced and suppressed chaos under the influence of an additional small-amplitude near-resonant ac field. Our analysis suggests rich dynamics in coupled order-parameter systems such as magnetomartensitic and magnetoelectric materials.

Chaos suppression in a Nd:YVO4 laser by biharmonical pump modulation

This work demonstrates the suppression of chaos in a Nd:YVO(4) laser by biharmonical pump modulation (the first for chaos-inducing and the second for chaos-suppressing). The laser exhibits chaotic behavior when only the first signal is applied for pump modulation and its frequency is adjusted close to the relaxation-oscillation frequency. Adding the second signal with subhamonic and a specific phase difference to the first modulation signal will reshape the modulated waveform of the pump beam to suppress the aforementioned chaotic behavior. The initial phase of the second harmonic perturbation plays an important role in the suppression of chaos. This result is confirmed by numerical simulation.

Controlling spatiotemporal chaos in chains of dissipative Kapitza pendula

The control of chaos (suppression and enhancement) of a damped pendulum subjected to two perpendicular periodic excitations of its pivot (one chaos inducing and the other chaos controlling) is investigated. Analytical (Melnikov analysis) and numerical (Lyapunov exponents) results show that the initial phase difference between the two excitations plays a fundamental role in the control scenario. We demonstrate the effectiveness of the method in suppressing spatiotemporal chaos of chains of identical chaotic coupled pendula where homogeneous regularization is obtained under localized control on a minimal number of pendula. Additionally, we demonstrate the robustness of the control scenario against changes in the coupling function. In particular, synchronization-induced homogeneous regularization of chaotic chains can be highly enhanced by considering time-varying couplings instead of stationary couplings.

Design strategies for the creation of aperiodic nonchaotic attractors

Parametric modulation in nonlinear dynamical systems can give rise to attractors on which the dynamics is aperiodic and nonchaotic, namely, with largest Lyapunov exponent being nonpositive. We describe a procedure for creating such attractors by using random modulation or pseudorandom binary sequences with arbitrarily long recurrence times. As a consequence the attractors are geometrically fractal and the motion is aperiodic on experimentally accessible time scales. A practical realization of such attractors is demonstrated in an experiment using electronic circuits.

On control of chaos and synchronization in the vibronic laser

It is shown theoretically that the method of time delayed incoherent optical feedback ensures control of chaotic dynamics in the vibronic alexandrite laser. The numerical solutions of the laser equations including the optical delayed feedback term are presented and the conditions for stabilization of the laser output are discussed. The possibility of synchronization of two chaotic vibronic lasers is reported when one of them is driven by the output of the other, thus giving the basis for secure communication.

An investigation of EEG dynamics in an animal model of temporal lobe epilepsy using the maximum Lyapunov exponent

Analysis of intracranial electroencephalographic (iEEG) recordings in patients with temporal lobe epilepsy (TLE) has revealed characteristic dynamical features that distinguish the interictal, ictal, and postictal states and inter-state transitions. Experimental investigations into the mechanisms underlying these observations require the use of an animal model. A rat TLE model was used to test for differences in iEEG dynamics between well-defined states and to test specific hypotheses: 1) the short-term maximum Lyapunov exponent (STL(max)), a measure of signal order, is lowest and closest in value among cortical sites during the ictal state, and highest and most divergent during the postictal state; 2) STL(max) values estimated from the stimulated hippocampus are the lowest among all cortical sites; and 3) the transition from the interictal to ictal state is associated with a convergence in STL(max) values among cortical sites. iEEGs were recorded from bilateral frontal cortices and hippocampi. STL(max) and T-index (a measure of convergence/divergence of STL(max) between recorded brain areas) were compared among the four different periods. Statistical tests (ANOVA and multiple comparisons) revealed that ictal STL(max) was lower (p<0.05) than other periods, STL(max) values corresponding to the stimulated hippocampus were lower than those estimated from other cortical regions, and T-index values were highest during the postictal period and lowest during the ictal period. Also, the T-index values corresponding to the preictal period were lower than those during the interictal period (p<0.05). These results indicate that a rat TLE model demonstrates several important dynamical signal characteristics similar to those found in human TLE and support future use of the model to study epileptic state transitions.

Refined fiber laser model

With modification, a recently proposed laser array model is found to agree quantitatively with fiber laser experiments. Comparisons of transient behavior, stable dynamical states, and transitions are made using both previously published and new experiments. While the original model agrees well for fibers with relatively low losses, achieving quantitative agreement over a wide range of operating conditions requires more physically appropriate descriptions of gain dynamics. The refined model is derived, and its predictions are found to be in excellent agreement with experiments.

Control of rare intense events in spatiotemporally chaotic systems

We address the problem of using feedback control for the purpose of suppressing rare intense events in spatially extended systems. As an example, we investigate the use of control to suppress turbulent spikes in the complex Ginzburg-Landau equation in the limit of small dissipation. We explore how information obtained by forecasting can be used to implement spatially and temporally localized control parameter changes and how control strength and cost are related to effectiveness in this framework. The effects of model error and imperfect state measurement are also considered.

Adaptive dynamical networks via neighborhood information: synchronization and pinning control

In this paper, we introduce a model of an adaptive dynamical network by integrating the complex network model and adaptive technique. In this model, the adaptive updating laws for each vertex in the network depend only on the state information of its neighborhood, besides itself and external controllers. This suggests that an adaptive technique be added to a complex network without breaking its intrinsic existing network topology. The core of adaptive dynamical networks is to design suitable adaptive updating laws to attain certain aims. Here, we propose two series of adaptive laws to synchronize and pin a complex network, respectively. Based on the Lyapunov function method, we can prove that under several mild conditions, with the adaptive technique, a connected network topology is sufficient to synchronize or stabilize any chaotic dynamics of the uncoupled system. This implies that these adaptive updating laws actually enhance synchronizability and stabilizability, respectively. We find out that even though these adaptive methods can succeed for all networks with connectivity, the underlying network topology can affect the convergent rate and the terminal average coupling and pinning strength. In addition, this influence can be measured by the smallest nonzero eigenvalue of the corresponding Laplacian. Moreover, we provide a detailed study of the influence of the prior parameters in this adaptive laws and present several numerical examples to verify our theoretical results and further discussion.

Local perturbations do not affect stability of laboratory fruitfly metapopulations

Background

A large number of theoretical studies predict that the dynamics of spatially structured populations (metapopulations) can be altered by constant perturbations to local population size. However, these studies presume large metapopulations inhabiting noise-free, zero-extinction environments, and their predictions have never been empirically verified.

Methodology/Principal Findings

Here we report an empirical study on the effects of localized perturbations on global dynamics and stability, using fruitfly metapopulations in the laboratory. We find that constant addition of individuals to a particular subpopulation in every generation stabilizes that subpopulation locally, but does not have any detectable effect on the dynamics and stability of the metapopulation. Simulations of our experimental system using a simple but widely applicable model of population dynamics were able to recover the empirical findings, indicating the generality of our results. We then simulated the possible consequences of perturbing more subpopulations, increasing the strength of perturbations, and varying the rate of migration, but found that none of these conditions were expected to alter the outcomes of our experiments. Finally, we show that our main results are robust to the presence of local extinctions in the metapopulation.

Conclusions/Significance

Our study shows that localized perturbations are unlikely to affect the dynamics of real metapopulations, a finding that has cautionary implications for ecologists and conservation biologists faced with the problem of stabilizing unstable metapopulations in nature.

Controlling spatio-temporal chaos in the scenario of the one-dimensional complex Ginzburg-Landau equation

We discuss some issues related with the process of controlling space-time chaotic states in the one-dimensional complex Ginzburg-Landau equation. We address the problem of gathering control over turbulent regimes with the use of only a limited number of controllers, each one of them implementing, in parallel, a local control technique for restoring an unstable plane-wave solution. We show that the system extension does not influence the density of controllers needed in order to achieve control.

Melnikov method approach to control of homoclinic/heteroclinic chaos by weak harmonic excitations

A review on the application of Melnikov's method to control homoclinic and heteroclinic chaos in low-dimensional, non-autonomous and dissipative oscillator systems by weak harmonic excitations is presented, including diverse applications, such as chaotic escape from a potential well, chaotic solitons in Frenkel-Kontorova chains and chaotic-charged particles in the field of an electrostatic wave packet.

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.

Laser stabilization with multiple-delay feedback control

Stabilization of chaotic intensity fluctuations of intracavity frequency-doubled solid-state (Nd: YAG) lasers using multiple-delay feedback control (MDFC) is demonstrated by numerical simulations. It is shown that MDFC not only provides stable (cw) output for constant pump rates but also works with slowly varying pump currents, resulting in corresponding (nonchaotic) intensity modulations.

Chaos control using notch filter feedback

A method for stabilizing periodic orbits and steady states of chaotic systems is presented using specifically filtered feedback signals. The efficiency of this control technique is illustrated with simulations (Rössler system, laser model) and a successful experimental application for stabilizing intensity fluctuations of an intracavity frequency-doubled Nd:YAG laser.

Generalized synchronization in coupled Ginzburg-Landau equations and mechanisms of its arising

Generalized chaotic synchronization regime is observed in the unidirectionally coupled one-dimensional Ginzburg-Landau equations. The mechanism resulting in the generalized synchronization regime arising in the coupled spatially extended chaotic systems demonstrating spatiotemporal chaotic oscillations has been described. The cause of the generalized synchronization occurrence is studied with the help of the modified Ginzburg-Landau equation with additional dissipation.

Chaos in an electromagnetically induced transparent medium inside an optical cavity

We theoretically predict and experimentally demonstrate chaotic behaviors in a system comprising of three-level atoms inside an optical ring cavity. This electromagnetically induced transparency (EIT) system is driven to chaos through period-doubling route by reducing the frequency detuning of the coupling laser beam. The chaos occurs in a different parametric regime as previously predicted and is believed to be caused by the enhanced dispersion and nonlinearity due to induced atomic coherence in such EIT system.

The phase-modulated logistic map

We study the logistic mapping with the nonlinearity parameter varied through a delayed feedback mechanism. This history dependent modulation through a phaselike variable offers an enhanced possibility for stabilization of periodic dynamics. Study of the system as a function of nonlinearity and modulation parameters reveals new phenomena: In addition to period-doubling and tangent bifurcations, there can be bifurcations where the period increases by unity. These are extensions of crises that arise in nonlinear dynamical systems. Periodic orbits in this system can be systematized via the kneading theory, which in the present case extends the analysis of Metropolis, Stein, and Stein for unimodal maps.

Enhancing chaoticity of spatiotemporal chaos

In some practical situations strong chaos is needed. This introduces the task of chaos control with enhancing chaoticity rather than suppressing chaoticity. In this paper a simple method of linear amplifications incorporating modulo operations is suggested to make spatiotemporal systems, which may be originally chaotic or nonchaotic, strongly chaotic. Specifically, this control can eliminate periodic windows, increase the values and the number of positive Lyapunov exponents, make the probability distributions of the output chaotic sequences more homogeneous, and reduce the correlations of chaotic outputs for different times and different space units. The applicability of the method to practical tasks, in particular to random number generators and secure communications, is briefly discussed.

Stabilizing unstable steady states using multiple delay feedback control

Feedback control with different and independent delay times is introduced and shown to be an efficient method for stabilizing fixed points (equilibria) of dynamical systems. In comparison to other delay based chaos control methods multiple delay feedback control is superior for controlling steady states and works also for relatively large delay times (sometimes unavoidable in experiments due to system dead times). To demonstrate this approach for stabilizing unstable fixed points we present numerical simulations of Chua's circuit and a successful experimental application for stabilizing a chaotic frequency doubled Nd-doped yttrium aluminum garnet laser.

BDNF boosts spike fidelity in chaotic neural oscillations

Oscillatory activity and its nonlinear dynamics are of fundamental importance for information processing in the central nervous system. Here we show that in aperiodic oscillations, brain-derived neurotrophic factor (BDNF), a member of the neurotrophin family, enhances the accuracy of action potentials in terms of spike reliability and temporal precision. Cultured hippocampal neurons displayed irregular oscillations of membrane potential in response to sinusoidal 20-Hz somatic current injection, yielding wobbly orbits in the phase space, i.e., a strange attractor. Brief application of BDNF suppressed this unpredictable dynamics and stabilized membrane potential fluctuations, leading to rhythmical firing. Even in complex oscillations induced by external stimuli of 40 Hz (gamma) on a 5-Hz (theta) carrier, BDNF-treated neurons generated more precisely timed spikes, i.e., phase-locked firing, coupled with theta-phase precession. These phenomena were sensitive to K252a, an inhibitor of tyrosine receptor kinases and appeared attributable to BDNF-evoked Na(+) current. The data are the first indication of pharmacological control of endogenous chaos. BDNF diminishes the ambiguity of spike time jitter and thereby might assure neural encoding, such as spike timing-dependent synaptic plasticity.

Detecting and controlling unstable periodic orbits that are not part of a chaotic attractor

A method for controlling unstable periodic orbits (UPOs) that have not been controllable before is presented. The method is based on detecting UPOs that are situated outside the skeleton of a chaotic attractor. The main idea is to exploit flexible parts of the attractor, which under weak external perturbations allow variable excursions of the trajectory away from its originally determined path. After the perturbation, the trajectory of the autonomous system seeks its path back to the chaotic attractor and reveals additional UPOs that are otherwise not used by the system. It is shown that these UPOs can be controlled as easily as the UPOs that form the basic chaotic attractor. The effectiveness of the proposed method is demonstrated on two different chaotic systems with very distinct response abilities to external perturbations. Additionally, some applications of the method in the fields of laser technology, information encoding, and biomedical engineering are discussed.

Suppression of the pulsed regimes appearing in free-electron lasers using feedback control of an unstable stationary state

We show that the pulsed regimes observed in free-electron lasers (FELs) can be suppressed using feedback control. By applying tiny parameter perturbations, the feedback allows to keep the systems onto a stationary state that is naturally existing in phase space, but is usually inaccessible because of its unstable nature. We test this method numerically on a master equation derived from the classical iterative model. Then we present the experimental results obtained on the super-ACO FEL. This method is in principle directly applicable to the other free-electron lasers, whose instabilities have a dynamical (deterministic) origin.

Continuous control of chaos based on the stability criterion

A method of chaos control based on stability criterion is proposed in the present paper. This method can stabilize chaotic systems onto a desired periodic orbit by a small time-continuous perturbation nonlinear feedback. This method does not require linearization of the system around the stabilized orbit and only an approximate location of the desired periodic orbit is required which can be automatically detected in the control process. The control can be started at any moment by choosing appropriate perturbation restriction condition. It seems that more flexibility and convenience are the main advantages of this method. The discussions on control of attitude motion of a spacecraft, Rössler system, and two coupled Duffing oscillators are given as numerical examples.

Controlling fast chaos in delay dynamical systems

We introduce a novel approach for controlling fast chaos in time-delay dynamical systems and use it to control a chaotic photonic device with a characteristic time scale of approximately 12 ns. Our approach is a prescription for how to implement existing chaos-control algorithms in a way that exploits the system's inherent time delay and allows control even in the presence of substantial control-loop latency (the finite time it takes signals to propagate through the components in the controller). This research paves the way for applications exploiting fast control of chaos, such as chaos-based communication schemes and stabilizing the behavior of ultrafast lasers.

Spatiotemporal chaos control with a target wave in the complex Ginzburg-Landau equation system

An effective method for controlling spiral turbulence in spatially extended systems is realized by introducing a spatially localized inhomogeneity into a two-dimensional system described by the complex Ginzburg-Landau equation. Our numerical simulations show that with the introduction of the inhomogeneity, a target wave can be produced, which will sweep all spiral defects out of the boundary of the system. The effects exist in certain parameter regions where the spiral waves are absolutely unstable. A theoretical explanation is given to reveal the underlying mechanism.

Propagation of desynchronous disturbances in synchronized chaotic one-way coupled map lattices

Propagations of desynchronous perturbations in synchronization processes of spatiotemporal chaos are investigated by considering chaotic one-way coupled map lattices. Under large coupling approximation the desynchronous area in time space is analytically calculated, based on the concepts of comoving Lyapunov exponent and absolute largest Lyapunov exponent. The ideas used in this paper are expected to be applicable to synchronizations of spatiotemporally chaotic systems of coupled maps and coupled oscillators with convective instability.

Controlling chaos in a Lorenz-like system using feedback

We demonstrate that the dynamics of an autonomous chaotic laser can be controlled to a periodic or steady state under self-synchronization. In general, past the chaos threshold the dependence of the laser output on feedback applied to the pump is submerged in the Lorenz-like chaotic pulsation. However there exist specific feedback delays that stabilize the chaos to periodic behavior or even steady state. The range of control depends critically on the feedback delay time and amplitude. Our experimental results are compared with the complex Lorenz equations which show good agreement.

Experimental demonstration of attractor annihilation in a multistable fiber laser

We report on the experimental open-loop control of generalized multistability in a system with coexisting attractors. The experimental system is an erbium-doped fiber laser with pump modulation of the diode laser. We demonstrate that additional weak harmonic modulation of the diode current annihilates one or two stable limit cycles in the laser. The ability of the method to select a desired state is illustrated through a codimension-two bifurcation diagram in the parameter space of the frequency and amplitude of the control modulation. We identify main resonances on the bifurcation lines (annihilation curves) and evaluate conditions for attractor annihilation.

Control of chaos by capture and release

We propose and demonstrate a different method for the control of flip saddles in dissipative chaotic systems. Due to the dynamics of a flip saddle, the stable manifolds of a target orbit and its perturbation can be modeled as a pair of concentric Möbius bands. Over the period of the target orbit, these bands rotate relative to one another. The method of capture and release (CR) takes advantage of this rotation, and captures a nearby system state in the perturbed stable manifold, releasing it when the rotation of the Möbius bands brings them into alignment. Unlike the method of Ott, Grebogi, and Yorke and most of its variants, CR does not rely on the unstable component of the flow to push the system state onto the stable manifold; the system state is evolving in a stable subspace for the duration of the control perturbation.

Turbulence control by developing a spiral wave with a periodic signal injection in the complex Ginzburg-Landau equation

Turbulence control in the two-dimensional complex Ginzburg-Landau equation is investigated. An approach is proposed for the purpose of control. In the presence of a small spiral wave seed initiation, a fully developed turbulence can be completely annihilated by injecting a single periodic signal into a small fixed space area around the spiral wave tip. The control is achieved in a parameter region where the spiral wave of the uncontrolled system is absolutely unstable. The robustness, convenience, and high control efficiency of this method are emphasized, and the mechanism underlying these practical advantages is intuitively understood.

Transforming complex multistability to controlled monostability

Multistability, a commonly observed feature among nonlinear systems, could be inconvenient under various circumstances. We demonstrate that a control in the form of slow and weak periodic parameter modulation can be effectively applied to transform a complex multistable system to a controlled monostable one. For the representative of a nonlinear system, we choose the Hénon map as the standard model. The number of coexisting stable states is known to increase as the dissipativity reduces. We show that even in the low dissipative limit, when the number of coexisting states could be arbitrarily large, the periodic parameter modulation can destroy the states coexisting with stable period 1. Thus, the system can be brought from any other branch to period-1 branch, leading to controlled monostability. This method works in the presence of noise as well.

Chaotic function generator: complex dynamics and its control in a loss-modulated Nd:YAG laser

The complex dynamics resulting from electronic feedback of a laser's intensity are explored and characterized. Distinct stable and chaotic regimes can be elicited from the laser by tuning the bias of the feedback loop. An additional branch of the feedback loop, containing a derivative filter, provides access to new kinds of dynamics, including a more gradual transition to chaos. The whole feedback network together allows the laser dynamics to be selected from among a wide range of chaotic wave forms distinguished by statistical or spectral information. In other words, this laser system can be used as a tunable generator of chaotic functions.

Synchronization of chaos in microchip lasers by using incoherent feedback

We propose a chaos-synchronization scheme using incoherent feedback to the pumping power in two microchip lasers. The population inversion of the slave laser is controlled for synchronization by using the detected signals of the peak heights of chaotic pulse intensities in the two lasers. Matching of the optical frequencies between the two lasers (i.e., injection locking) is not required for synchronization using this method. We numerically demonstrate the incoherent feedback method and investigate synchronization regions against parameter mismatching between the two lasers. Synchronization is maintained within a mismatching of 1% for all laser parameters, which implies that the difficulty in reproducing the synchronized laser pulses is very useful for applications of secure optical communications.

Controlling dynamics in spatially extended systems

Spatially extended systems exhibit a variety of spatiotemporal dynamics--from stable to chaotic. These dynamics can change under pathological conditions and impair normal functions. Thus, having the ability to control the altered dynamics for improved functioning has the potential for wide ranging applications in real and artificial systems. Here we propose a simple and general method that can be used to target the spatiotemporal dynamics, both globally and in spatially localized regions, in either direction--i.e., towards the stable or unstable manifold-by simply changing the strength and the sign of an externally applied perturbation or pinning. The method is applicable to both chaotic and nonchaotic systems, with discrete and continuous local dynamics, and for different topologies of interactions. We also apply it to simulate an experiment on epileptogenic neuronal activity in rat hippocampal tissue [B. J. Gluckman et al., J. Neurophys. 76, 6202 (1996)]. This unified approach for differential targeting of global and local dynamics promises to be useful for systems spanning large spatial scales and having structural and functional heterogeneity.

Controlling wake turbulence

This Letter introduces a control strategy for taming the wake turbulence behind a cylinder. An angular momentum injection scheme is proposed to synchronize the vertical velocity field. We show that the base suction, wake formation length, absolute instability, and the Kármán vortex street are effectively controlled by the angular momentum injection. A control equation is designed to implement the injection. The Navier-Stokes equations, along with the control equation, are solved. The occurrence of a new recirculation free zone is identified.

Transforming chaos to periodic oscillations

We demonstrate that the dynamics of an autonomous chaotic class C laser can be controlled to a periodic state via external modulation of the pump. In the absence of modulation, above the chaos threshold, the laser exhibits Lorenz-like chaotic pulsations. The average amplitude and frequency of these pulsations depend on the pump power. We find that there exist parameter windows where modulation of the pump power extinguishes the chaos in favor of simpler periodic behavior. Moreover we find a number of locking ratios between the pump and laser output follow the Farey sequence.