Bubbling in delay-coupled lasers

Leader-laggard synchronization of polarization chaos in mutually coupled free-running VCSELs

We systematically study the leader-laggard synchronization of polarization chaos in mutually coupled free-running vertical cavity surface emitting semiconductor lasers in two cases of parallel and orthogonal injection. Specifically, we quantitatively investigate the effect of critical external parameter mismatch such as the coupling intensity and frequency detuning on the leader-laggard relationship utilizing the cross-correlation function. When the difference between two main cross-correlation peak values exceeds 0.1, the leader-laggard relationship can be viewed to be stable. Our results demonstrate that compared with the coupling strength, the frequency detuning is the dominant factor in determining the stability of the leader-laggard relationship. The exchange of the leader-laggard role occurs within a frequency detuning region from -5 GHz to 5 GHz for both parallel and orthogonal injection. Once the leader-laggard relationship is stable, the difference between the two cross-correlation values can reach 0.242 for negative frequency detuning, but the corresponding value is only 0.146 under positive frequency detuning.

Random heterogeneity outperforms design in network synchronization

A widely held assumption on network dynamics is that similar components are more likely to exhibit similar behavior than dissimilar ones and that generic differences among them are necessarily detrimental to synchronization. Here, we show that this assumption does not generally hold in oscillator networks when communication delays are present. We demonstrate, in particular, that random parameter heterogeneity among oscillators can consistently rescue the system from losing synchrony. This finding is supported by electrochemical-oscillator experiments performed on a multielectrode array network. Remarkably, at intermediate levels of heterogeneity, random mismatches are more effective in promoting synchronization than parameter assignments specifically designed to facilitate identical synchronization. Our results suggest that, rather than being eliminated or ignored, intrinsic disorder in technological and biological systems can be harnessed to help maintain coherence required for function.

Stochastic resonance in deformable potential with time-delayed feedback

This paper reports the stochastic resonance (SR) phenomenon with memory effects for a Brownian particle in a potential whose shape is subjected to deformation. We model the deformation in the system by the Remoissenet-Peyrard potential and the memory effects by the time-delayed feedback. The question of the possible influence of time-delayed feedback on the occurrence of SR is then of our interest. We examine numerically the effect of feedback strength as well as time delay on SR phenomenon in terms of hysteresis loop area. It is found that time-delayed feedback has a significant effect on SR and can induce double resonances in the system. We show that the properties of SR are varying, depending on interdependence between feedback strength, time delay and shape parameter. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 1)'.

Mutual coupling and synchronization of optically coupled quantum-dot micropillar lasers at ultra-low light levels

Synchronization of coupled oscillators at the transition between classical physics and quantum physics has become an emerging research topic at the crossroads of nonlinear dynamics and nanophotonics. We study this unexplored field by using quantum dot microlasers as optical oscillators. Operating in the regime of cavity quantum electrodynamics (cQED) with an intracavity photon number on the order of 10 and output powers in the 100 nW range, these devices have high β-factors associated with enhanced spontaneous emission noise. We identify synchronization of mutually coupled microlasers via frequency locking associated with a sub-gigahertz locking range. A theoretical analysis of the coupling behavior reveals striking differences from optical synchronization in the classical domain with negligible spontaneous emission noise. Beyond that, additional self-feedback leads to zero-lag synchronization of coupled microlasers at ultra-low light levels. Our work has high potential to pave the way for future experiments in the quantum regime of synchronization.

Spontaneous exchange of leader-laggard relationship in mutually coupled synchronized semiconductor lasers

We investigate the instantaneous behavior of synchronized temporal wave forms in two mutually coupled semiconductor lasers numerically and experimentally. The temporal wave forms of two lasers are synchronized with a propagation delay time, with one laser oscillating in advance of the other, known as the leader-laggard relationship. The leader-laggard relationship can be determined by measuring the cross-correlation between the two temporal wave forms with the propagation delay time. The leader can be identified when the optical carrier frequency of the leader laser is higher than that of the other laser. However, spontaneous exchange between the leader and laggard lasers can be observed in low-frequency fluctuations by short-term cross-correlation measurements, even for a fixed initial optical frequency detuning. The spontaneous exchange of the leader-laggard relationship originates from alternation of partial optical frequency locking between the two lasers. This observation is analyzed using a phase space trajectory on steady-state solutions for mutually coupled lasers with optical frequency detuning.

Extreme events in FitzHugh-Nagumo oscillators coupled with two time delays

We study two identical FitzHugh-Nagumo oscillators which are coupled with one or two different time delays. If only a single-delay coupling is used, the length of the delay determines whether the synchronization manifold is transversally stable or unstable, exhibiting mixed-mode or chaotic oscillations in which the small amplitude oscillations are always in phase but the large amplitude oscillations are in phase or out of phase, respectively. For two delays we find an intricate dynamics which comprises an irregular alteration of small amplitude oscillations, in-phase and out-of-phase large amplitude oscillations, also called extreme events. This transient chaotic dynamics is sandwiched between a bubbling transition and a blowout bifurcation.

Encryption key distribution via chaos synchronization

We present a novel encryption scheme, wherein an encryption key is generated by two distant complex nonlinear units, forced into synchronization by a chaotic driver. The concept is sufficiently generic to be implemented on either photonic, optoelectronic or electronic platforms. The method for generating the key bitstream from the chaotic signals is reconfigurable. Although derived from a deterministic process, the obtained bit series fulfill the randomness conditions as defined by the National Institute of Standards test suite. We demonstrate the feasibility of our concept on an electronic delay oscillator circuit and test the robustness against attacks using a state-of-the-art system identification method.

Bidirectional private key exchange using delay-coupled semiconductor lasers

We experimentally demonstrate a key exchange cryptosystem based on the phenomenon of identical chaos synchronization. In our protocol, the private key is symmetrically generated by the two communicating partners. It is built up from the synchronized bits occurring between two current-modulated bidirectionally coupled semiconductor lasers with additional self-feedback. We analyze the security of the exchanged key and discuss the amplification of its privacy. We demonstrate private key generation rates up to 11  Mbit/s over a public channel.

Local instability driving extreme events in a pair of coupled chaotic electronic circuits

For a long time, extreme events happening in complex systems, such as financial markets, earthquakes, and neurological networks, were thought to follow power-law size distributions. More recently, evidence suggests that in many systems the largest and rarest events differ from the other ones. They are dragon kings, outliers that make the distribution deviate from a power law in the tail. Understanding the processes of formation of extreme events and what circumstances lead to dragon kings or to a power-law distribution is an open question and it is a very important one to assess whether extreme events will occur too often in a specific system. In the particular system studied in this paper, we show that the rate of occurrence of dragon kings is controlled by the value of a parameter. The system under study here is composed of two nearly identical chaotic oscillators which fail to remain in a permanently synchronized state when coupled. We analyze the statistics of the desynchronization events in this specific example of two coupled chaotic electronic circuits and find that modifying a parameter associated to the local instability responsible for the loss of synchronization reduces the occurrence of dragon kings, while preserving the power-law distribution of small- to intermediate-size events with the same scaling exponent. Our results support the hypothesis that the dragon kings are caused by local instabilities in the phase space.

Experimental synchrony of semiconductor lasers in coupled networks

The emission and synchronization of mutually-coupled semiconductor lasers with short cavities has been already recorded, with transversely unstable solutions existing within the chaotic synchronization manifold. Noise and laser-mismatch induced instabilities cause short de-synchronization events within the overall generalized synchronization, influencing the pragmatism of using these signals in secure data exchange applications. However, such operation can be functional for user authentication and sensing applications by assessing a time-averaged performance of synchrony. Until now, this has not been examined either in large-scale laser network configurations or in large transmission coupling paths, as real network implementations oblige. Here we present the first implementation of a fully-coupled fiber network with up to 16 semiconductor lasers, independently controlled and coupled through long interacting cavities. High level of consistent global or cluster synchrony via chaotic signals is demonstrated among all devices of the same origin and under appropriate operation. Devices that are not identical fail to synchronize at any condition, when coupled to the network. Under multiplexed operation, groups of lasers that emit at spectral distances as low as 50pm are shown to preserve intra-cluster synchronization when transmitted in the same fiber-optic channel, despite their large bandwidth of emitted signals.

Trajectory-probed instability and statistics of desynchronization events in coupled chaotic systems

Complex systems, such as financial markets, earthquakes, and neurological networks, exhibit extreme events whose mechanisms of formation are not still completely understood. These mechanisms may be identified and better studied in simpler systems with dynamical features similar to the ones encountered in the complex system of interest. For instance, sudden and brief departures from the synchronized state observed in coupled chaotic systems were shown to display non-normal statistical distributions similar to events observed in the complex systems cited above. The current hypothesis accepted is that these desynchronization events are influenced by the presence of unstable object(s) in the phase space of the system. Here, we present further evidence that the occurrence of large events is triggered by the visitation of the system's phase-space trajectory to the vicinity of these unstable objects. In the system studied here, this visitation is controlled by a single parameter, and we exploit this feature to observe the effect of the visitation rate in the overall instability of the synchronized state. We find that the probability of escapes from the synchronized state and the size of those desynchronization events are enhanced in attractors whose shapes permit the chaotic trajectories to approach the region of strong instability. This result shows that the occurrence of large events requires not only a large local instability to amplify noise, or to amplify the effect of parameter mismatch between the coupled subsystems, but also that the trajectories of the system wander close to this local instability.

Manipulating coherence resonance in a quantum dot semiconductor laser via electrical pumping

Excitability and coherence resonance are studied in a semiconductor quantum dot laser under short optical self-feedback. For low pump levels, these are observed close to a homoclinic bifurcation, which is in correspondence with earlier observations in quantum well lasers. However, for high pump levels, we find excitability close to a boundary crisis of a chaotic attractor. We demonstrate that in contrast to the homoclinic bifurcation the crisis and thus the excitable regime is highly sensitive to the pump current. The excitability threshold increases with the pump current, which permits to adjust the sensitivity of the excitable unit to noise as well as to shift the optimal noise strength, at which maximum coherence is observed. The shift adds up to more than one order of magnitude, which strongly facilitates experimental realizations.

Multistabilities and symmetry-broken one-color and two-color states in closely coupled single-mode lasers

We theoretically investigate the dynamics of two mutually coupled, identical single-mode semi-conductor lasers. For small separation and large coupling between the lasers, symmetry-broken one-color states are shown to be stable. In this case the light outputs of the lasers have significantly different intensities while at the same time the lasers are locked to a single common frequency. For intermediate coupling we observe stable symmetry-broken two-color states, where both lasers lase simultaneously at two optical frequencies which are separated by up to 150 GHz. Using a five-dimensional model, we identify the bifurcation structure which is responsible for the appearance of symmetric and symmetry-broken one-color and two-color states. Several of these states give rise to multistabilities and therefore allow for the design of all-optical memory elements on the basis of two coupled single-mode lasers. The switching performance of selected designs of optical memory elements is studied numerically.

Tunable power law in the desynchronization events of coupled chaotic electronic circuits

We study the statistics of the amplitude of the synchronization error in chaotic electronic circuits coupled through linear feedback. Depending on the coupling strength, our system exhibits three qualitatively different regimes of synchronization: weak coupling yields independent oscillations; moderate to strong coupling produces a regime of intermittent synchronization known as attractor bubbling; and stronger coupling produces complete synchronization. In the regime of moderate coupling, the probability distribution for the sizes of desynchronization events follows a power law, with an exponent that can be adjusted by changing the coupling strength. Such power-law distributions are interesting, as they appear in many complex systems. However, most of the systems with such a behavior have a fixed value for the exponent of the power law, while here we present an example of a system where the exponent of the power law is easily tuned in real time.

Suppression of deterministic and stochastic extreme desynchronization events using anticipated synchronization

We numerically show that extreme events induced by parameter mismatches or noise in coupled oscillatory systems can be anticipated and suppressed before they actually occur. We show this in a main system unidirectionally coupled to an auxiliary system subject to a negative delayed feedback. Each system consists of two electronic oscillators coupled in a master-slave configuration. Extreme events are observed in this coupled system as large and sporadic desynchronization events. Under certain conditions, the auxiliary system can predict the dynamics of the main system. We use this to efficiently suppress the extreme events by applying a direct corrective reset to the main system.

Characterizing the deterministic nature of individual power dropouts in semiconductor lasers subject to delayed feedback

We implement a method to identify the deterministic nature of specific events in the dynamics of a semiconductor laser subject to time-delayed optical feedback. Specifically, we study the power dropouts in the low-frequency fluctuations regime on an individual event basis and identify whether the underlying dominant mechanism is deterministic. Our approach is based on sychronization with a twin system in a symmetric relay configuration. We investigate the dependence of the fraction of deterministically driven (i.e., synchronized) dropouts on the laser's pump current as a key parameter. Our experimental results are corroborated by numerical modeling based on rate equations. Our numerical findings also provide insights into the influence of spontaneous emission noise.

Amplitude and phase dynamics in oscillators with distributed-delay coupling

This paper studies the effects of distributed-delay coupling on the dynamics in a system of non-identical coupled Stuart-Landau oscillators. For uniform and gamma delay distribution kernels, the conditions for amplitude death are obtained in terms of average frequency, frequency detuning and the parameters of the coupling, including coupling strength and phase, as well as the mean time delay and the width of the delay distribution. To gain further insights into the dynamics inside amplitude death regions, the eigenvalues of the corresponding characteristic equations are computed numerically. Oscillatory dynamics of the system is also investigated, using amplitude and phase representation. Various branches of phase-locked solutions are identified, and their stability is analysed for different types of delay distributions.

Bifurcation analysis of delay-induced patterns in a ring of Hodgkin-Huxley neurons

Rings of delay-coupled neurons possess a striking capability to produce various stable spiking patterns. In order to reveal the mechanisms of their appearance, we present a bifurcation analysis of the Hodgkin-Huxley (HH) system with delayed feedback as well as a closed loop of HH neurons. We consider mainly the effects of external currents and communication delays. It is shown that typically periodic patterns of different spatial form (wavenumber) appear via Hopf bifurcations as the external current or time delay changes. The Hopf bifurcations are shown to occur in relatively narrow regions of the external current values, which are independent of the delays. Additional patterns, which have the same wavenumbers as the existing ones, appear via saddle-node bifurcations of limit cycles. The obtained bifurcation diagrams are evidence for the important role of communication delays for the emergence of multiple coexistent spiking patterns. The effects of a short-cut, which destroys the rotational symmetry of the ring, are also briefly discussed.

Chaos pass filter: linear response of synchronized chaotic systems

The linear response of synchronized time-delayed chaotic systems to small external perturbations, i.e., the phenomenon of chaos pass filter, is investigated for iterated maps. The distribution of distances, i.e., the deviations between two synchronized chaotic units due to external perturbations on the transferred signal, is used as a measure of the linear response. It is calculated numerically and, for some special cases, analytically. Depending on the model parameters this distribution has power law tails in the region of synchronization leading to diverging moments of distances. This is a consequence of multiplicative and additive noise in the corresponding linear equations due to chaos and external perturbations. The linear response can also be quantified by the bit error rate of a transmitted binary message which perturbs the synchronized system. The bit error rate is given by an integral over the distribution of distances and is calculated analytically and numerically. It displays a complex nonmonotonic behavior in the region of synchronization. For special cases the distribution of distances has a fractal structure leading to a devil's staircase for the bit error rate as a function of coupling strength. The response to small harmonic perturbations shows resonances related to coupling and feedback delay times. A bidirectionally coupled chain of three units can completely filter out the perturbation. Thus the second moment and the bit error rate become zero.

Cluster and group synchronization in delay-coupled networks

We investigate the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics. Using a master stability approach, we find that the master stability function shows a discrete rotational symmetry depending on the number of groups. The coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum, which helps to simplify the evaluation of the master stability function. Our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times, multiple coupling functions, but also with multiple kinds of local dynamics in the networks' nodes. We illustrate our results by calculating stability in the example of delay-coupled semiconductor lasers and in a model for neuronal spiking dynamics.

Transition to complete synchronization and global intermittent synchronization in an array of time-delay systems

We report the nature of transitions from the nonsynchronous to a complete synchronization (CS) state in arrays of time-delay systems, where the systems are coupled with instantaneous diffusive coupling. We demonstrate that the transition to CS occurs distinctly for different coupling configurations. In particular, for unidirectional coupling, locally (microscopically) synchronization transition occurs in a very narrow range of coupling strength but for a global one (macroscopically) it occurs sequentially in a broad range of coupling strength preceded by an intermittent synchronization. On the other hand, in the case of mutual coupling, a very large value of coupling strength is required for local synchronization and, consequently, all the local subsystems synchronize immediately for the same value of the coupling strength and, hence, globally, synchronization also occurs in a narrow range of the coupling strength. In the transition regime, we observe a type of synchronization transition where long intervals of high-quality synchronization which are interrupted at irregular times by intermittent chaotic bursts simultaneously in all the systems and which we designate as global intermittent synchronization. We also relate our synchronization transition results to the above specific types using unstable periodic orbit theory. The above studies are carried out in a well-known piecewise linear time-delay system.

Zero-lag synchronization and bubbling in delay-coupled lasers

We show experimentally that two semiconductor lasers mutually coupled via a passive relay fiber loop exhibit chaos synchronization at zero lag, and study how this synchronized regime is lost as the lasers' pump currents are increased. We characterize the synchronization properties of the system with high temporal resolution in two different chaotic regimes, namely, low-frequency fluctuations and coherence collapse, identifying significant differences between them. In particular, a marked decrease in synchronization quality develops as the lasers enter the coherence collapse regime. Our high-resolution measurements allow us to establish that synchronization loss is associated with bubbling events, the frequency of which increases with increasing pump current.

Isochronal chaos synchronization of delay-coupled optoelectronic oscillators

We study experimentally chaos synchronization of nonlinear optoelectronic oscillators with time-delayed mutual coupling and self-feedback. Coupling three oscillators in a chain, we find that the outer two oscillators always synchronize. In contrast, isochronal synchronization of the mediating middle oscillator is found only when self-feedback is added to the middle oscillator. We show how the stability of the isochronal solution of any network, including the case of three coupled oscillators, can be determined by measuring the synchronization threshold of two unidirectionally coupled systems. In addition, we provide a sufficient condition that guarantees global asymptotic stability of the synchronized solution.

Theory and numerics of vibrational resonance in Duffing oscillators with time-delayed feedback

The influence of linear time-delayed feedback on vibrational resonance is investigated in underdamped and overdamped Duffing oscillators with double-well and single-well potentials driven by both low frequency and high frequency periodic forces. This task is performed through both theoretical approach and numerical simulation. Theoretically determined values of the amplitude of the high frequency force and the delay time at which resonance occurs are in very good agreement with the numerical simulation. A major consequence of time-delayed feedback is that it gives rise to a periodic or quasiperiodic pattern of vibrational resonance profile with respect to the time-delayed parameter. An appropriate time delay is shown to induce a resonance in an overdamped single-well system which is otherwise not possible. For a range of values of the time-delayed parameters, the response amplitude is found to be larger than in delay-time feedback-free systems.

Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers

We study the synchronization properties of the delay dynamics of two identical semiconductor lasers coupled through a semitransparent mirror. Via an analytical and numerical approach, we investigate the influence of asymmetries, in particular mismatches of self- and cross-coupling strength and differences in self- and cross-coupling delay. We show that the former mismatch affects the stability of the zero-lag state but not the dynamics within the synchronization manifold, while the latter mismatch does not affect the quality of synchronization but alters the dynamics significantly. Our results are extended to different unidirectional coupling schemes. This is highly relevant for communication schemes utilizing chaotic dynamics. Finally, the influence of nonlinear gain saturation on the dynamics and stability of synchronization is discussed.

Synchronization of random bit generators based on coupled chaotic lasers and application to cryptography

Random bit generators (RBGs) constitute an important tool in cryptography, stochastic simulations and secure communications. The later in particular has some difficult requirements: high generation rate of unpredictable bit strings and secure key-exchange protocols over public channels. Deterministic algorithms generate pseudo-random number sequences at high rates, however, their unpredictability is limited by the very nature of their deterministic origin. Recently, physical RBGs based on chaotic semiconductor lasers were shown to exceed Gbit/s rates. Whether secure synchronization of two high rate physical RBGs is possible remains an open question. Here we propose a method, whereby two fast RBGs based on mutually coupled chaotic lasers, are synchronized. Using information theoretic analysis we demonstrate security against a powerful computational eavesdropper, capable of noiseless amplification, where all parameters are publicly known. The method is also extended to secure synchronization of a small network of three RBGs.

Controlling synchrony by delay coupling in networks: from in-phase to splay and cluster states

We study synchronization in delay-coupled oscillator networks using a master stability function approach. Within a generic model of Stuart-Landau oscillators (normal form of supercritical or subcritical Hopf bifurcation), we derive analytical stability conditions and demonstrate that by tuning the coupling phase one can easily control the stability of synchronous periodic states. We propose the coupling phase as a crucial control parameter to switch between in-phase synchronization or desynchronization for general network topologies or between in-phase, cluster, or splay states in unidirectional rings. Our results are robust even for slightly nonidentical elements of the network.

Delay stabilization of periodic orbits in coupled oscillator systems

We study diffusively coupled oscillators in Hopf normal form. By introducing a non-invasive delay coupling, we are able to stabilize the inherently unstable anti-phase orbits. For the super- and subcritical cases, we state a condition on the oscillator's nonlinearity that is necessary and sufficient to find coupling parameters for successful stabilization. We prove these conditions and review previous results on the stabilization of odd-number orbits by time-delayed feedback. Finally, we illustrate the results with numerical simulations.