Zero lag synchronization of chaotic systems with time delayed couplings

Synchronization of polarization chaos in mutually coupled free-running VCSELs

We numerically demonstrate and analyze polarization chaos synchronization between two free-running vertical cavity surface emitting semiconductor lasers (VCSELs) in the mutual coupling configuration under two scenarios: parallel injection and orthogonal injection. Specifically, we investigate the effect of external parameters (the bias current, frequency detuning and coupling coefficient) and internal parameters (the linewidth enhancement factor, spin-flip relaxation rate, field decay rate, carrier decay rate, birefringence and dichroism) on the synchronization quality. Finally simulation results confirm that in the parallel injection, chaotic synchronization can reach a cross-correlation coefficient of 0.99 within a range of parameter mismatch ±12%. On the other hand, the chaos synchronization for orthogonal injection only reaches a cross-correlation coefficient of 0.95 within a range of parameter mismatch ±3%.

Numerical investigations of synchronization and communication based on an electro-optic phase chaos system with concealment of time delay

A modified electro-optic phase chaos system that can conceal time delay (TD) and allows for unidirectional message transmission, is numerically investigated. The configuration includes two cascaded delay loops, and the parallel-coupled microresonators (PCMRs) in one of two loops result in a frequency-dependent group delay. The largest Lyapunov exponent (LLE), Lempel-Ziv complexity (LZC) and permutation entropy (PE) are used to distinguish the chaotic behavior and the degree of complexity in a time series, and the autocorrelation function (ACF) and the delayed mutual information (DMI) are plotted to extract the TD. The corresponding diagrams show that in the electro-optic system phase chaos with high complexity can occur within a certain range of feedback strength. The diagrams also show that, at a fixed feedback strength, the effect of the TD concealment becomes quite good with an increase in the number of PCMRs. The numerical simulation also reveals that the delayed chaotic dynamics can be identically synchronized, and the synchronization solution is robust. Moreover, based on the coherence of Mach-Zehnder interferometers, we convert the phase variations of the transmitter outputs and the receiver into the corresponding intensity variations, so the synchronization error of the two-phase chaotic series can be monitored. At last, we can successfully decipher the message introduced on the transmitting end of a link. In this scheme, the feedback TD has been concealed, which prevents eavesdroppers from listening and makes the proposed chaotic communication system secure.

Synchronization in networks with heterogeneous coupling delays

Synchronization in networks of identical oscillators with heterogeneous coupling delays is studied. A decomposition of the network dynamics is obtained by block diagonalizing a newly introduced adjacency lag operator which contains the topology of the network as well as the corresponding coupling delays. This generalizes the master stability function approach, which was developed for homogenous delays. As a result the network dynamics can be analyzed by delay differential equations with distributed delay, where different delay distributions emerge for different network modes. Frequency domain methods are used for the stability analysis of synchronized equilibria and synchronized periodic orbits. As an example, the synchronization behavior in a system of delay-coupled Hodgkin-Huxley neurons is investigated. It is shown that the parameter regions where synchronized periodic spiking is unstable expand when increasing the delay heterogeneity.

Quantifying cardio-respiratory phase synchronization-a comparison of five methods using ECGs of post-infarction patients

OBJECTIVE

Phase synchronization between two weakly coupled oscillators occurs in many natural systems. Since it is difficult to unambiguously detect such synchronization in experimental data, several methods have been proposed for this purpose. Five popular approaches are systematically optimized and compared here.

APPROACH

We study and apply the automated synchrogram method, the reduced synchrogram method, two variants of a gradient method, and the Fourier mode method, analyzing 24h data records from 1455 post-infarction patients, the same data with artificial inaccuracies, and corresponding surrogate data generated by Fourier phase randomization.

MAIN RESULTS

We find that the automated synchrogram method is the most robust of all studied approaches when applied to records with missing data or artifacts, whereas the gradient methods should be preferred for noisy data and low-accuracy R-peak positions. We also show that a strong circadian rhythm occurs with much more frequent phase synchronization episodes observed during night time than during day time by all five methods.

SIGNIFICANCE

In specific applications, the identified characteristic differences as well as strengths and weaknesses of each method in detecting episodes of cardio-respiratory phase synchronization will be useful for selecting an appropriate method with respect to the type of systematic and dynamical noise in the data.

Interaction Control to Synchronize Non-synchronizable Networks

Synchronization constitutes one of the most fundamental collective dynamics across networked systems and often underlies their function. Whether a system may synchronize depends on the internal unit dynamics as well as the topology and strength of their interactions. For chaotic units with certain interaction topologies synchronization might be impossible across all interaction strengths, meaning that these networks are non-synchronizable. Here we propose the concept of interaction control, generalizing transient uncoupling, to induce desired collective dynamics in complex networks and apply it to synchronize even such non-synchronizable systems. After highlighting that non-synchronizability prevails for a wide range of networks of arbitrary size, we explain how a simple binary control may localize interactions in state space and thereby synchronize networks. Intriguingly, localizing interactions by a fixed control scheme enables stable synchronization across all connected networks regardless of topological constraints. Interaction control may thus ease the design of desired collective dynamics even without knowledge of the networks’ exact interaction topology and consequently have implications for biological and self-organizing technical systems.

Chaos synchronization by resonance of multiple delay times

Chaos synchronization may arise in networks of nonlinear units with delayed couplings. We study complete and sublattice synchronization generated by resonance of two large time delays with a specific ratio. As it is known for single-delay networks, the number of synchronized sublattices is determined by the greatest common divisor (GCD) of the network loop lengths. We demonstrate analytically the GCD condition in networks of iterated Bernoulli maps with multiple delay times and complement our analytic results by numerical phase diagrams, providing parameter regions showing complete and sublattice synchronization by resonance for Tent and Bernoulli maps. We compare networks with the same GCD with single and multiple delays, and we investigate the sensitivity of the correlation to a detuning between the delays in a network of coupled Stuart-Landau oscillators. Moreover, the GCD condition also allows detection of time-delay resonances, leading to high correlations in nonsynchronizable networks. Specifically, GCD-induced resonances are observed both in a chaotic asymmetric network and in doubly connected rings of delay-coupled noisy linear oscillators.

Zero-lag synchronization despite inhomogeneities in a relay system

A novel proposal for the zero-lag synchronization of the delayed coupled neurons, is to connect them indirectly via a third relay neuron. In this study, we develop a Poincaré map to investigate the robustness of the synchrony in such a relay system against inhomogeneity in the neurons and synaptic parameters. We show that when the inhomogeneity does not violate the symmetry of the system, synchrony is maintained and in some cases inhomogeneity enhances synchrony. On the other hand if the inhomogeneity breaks the symmetry of the system, zero lag synchrony can not be preserved. In this case we give analytical results for the phase lag of the spiking of the neurons in the stable state.

Synchronization of networks of oscillators with distributed delay coupling

This paper studies the stability of synchronized states in networks, where couplings between nodes are characterized by some distributed time delay, and develops a generalized master stability function approach. Using a generic example of Stuart-Landau oscillators, it is shown how the stability of synchronized solutions in networks with distributed delay coupling can be determined through a semi-analytic computation of Floquet exponents. The analysis of stability of fully synchronized and of cluster or splay states is illustrated for several practically important choices of delay distributions and network topologies.

Generalized synchronization in relay systems with instantaneous coupling

We demonstrate the existence of generalized synchronization in systems that act as mediators between two dynamical units that, in turn, show complete synchronization with each other. These are the so-called relay systems. Specifically, we analyze the Lyapunov spectrum of the full system to elucidate when complete and generalized synchronization appear. We show that once a critical coupling strength is achieved, complete synchronization emerges between the systems to be synchronized, and at the same point, generalized synchronization with the relay system also arises. Next, we use two nonlinear measures based on the distance between phase-space neighbors to quantify the generalized synchronization in discretized time series. Finally, we experimentally show the robustness of the phenomenon and of the theoretical tools here proposed to characterize it.

Chaos in networks with time-delayed couplings

Networks of nonlinear units coupled by time-delayed signals can show chaos. In the limit of long delay times, chaos appears in two ways: strong and weak, depending on how the maximal Lyapunov exponent scales with the delay time. Only for weak chaos, a network can synchronize completely, without time shift. The conditions for strong and weak chaos and synchronization in networks with multiple delay times are investigated.

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.

Effect of multiple time-delay on vibrational resonance

We report our investigation on the effect of multiple time-delay on vibrational resonance in a single Duffing oscillator and in a system of n Duffing oscillators coupled unidirectionally and driven by both a low- and a high-frequency periodic force. For the single oscillator, we obtain analytical expressions for the response amplitude Q and the amplitude g of the high-frequency force at which resonance occurs. The regions in parameter space of enhanced Q at resonance, as compared to the case in absence of time-delay, show a bands-like structure. For the two-coupled oscillators, we explain all the features of variation of Q with the control parameter g. For the system of n-coupled oscillators with a single time-delay coupling, the response amplitudes of the oscillators are shown to be independent of the time-delay. In the case of a multi time-delayed coupling, undamped signal propagation takes place for coupling strength (δ) above a certain critical value (denoted as δu). Moreover, the response amplitude approaches a limiting value QL with the oscillator number i. We obtain analytical expressions for both δu and QL.

Analytical limitation for time-delayed feedback control in autonomous systems

We prove an analytical limitation on the use of time-delayed feedback control for the stabilization of periodic orbits in autonomous systems. This limitation depends on the number of real Floquet multipliers larger than unity, and is therefore similar to the well-known odd number limitation of time-delayed feedback control. Recently, a two-dimensional example has been found, which explicitly demonstrates that the unmodified odd number limitation does not apply in the case of autonomous systems. We show that our limitation correctly predicts the stability boundaries in this case.

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.

Spectral and correlation properties of rings of delay-coupled elements: comparing linear and nonlinear systems

The dynamical properties of delay-coupled systems are currently of great interest. So far the analysis has concentrated primarily on identical synchronization properties. Here we study the dynamics of rings of delay-coupled nodes, a topology that cannot show identical synchronization, and compare its properties to those of linear stochastic maps. We find that, in the long delay limit, the correlation functions and spectra of delay-coupled rings of nonlinear systems obey the same scaling laws as linear systems, indicating that important properties of the emerging solution result from network topology.

Bidirectional chaos communication between two outer semiconductor lasers coupled mutually with a central semiconductor laser

Based on a linear chain composed of a central semiconductor laser and two outer semiconductor lasers, chaos synchronization and bidirectional communication between two outer lasers have been investigated under the case that the central laser and the two outer lasers are coupled mutually, whereas there exists no coupling between the two outer lasers. The simulation results show that high-quality and stable isochronal synchronization between the two outer lasers can be achieved, while the cross-correlation coefficients between the two outer lasers and the central laser are very low under proper operation condition. Based on the high performance chaos synchronization between the two outer lasers, message bidirectional transmissions of bit rates up to 20 Gbit/s can be realized through adopting a novel decoding scheme which is different from that based on chaos pass filtering effect. Furthermore, the security of bidirectional communication is also analyzed.

Complete chaotic synchronization and exclusion of mutual Pyragas control in two delay-coupled Rössler-type oscillators

Two identical chaotic oscillators that are mutually coupled via time delayed signals show very complex patterns of completely synchronized dynamics including stationary states and periodic as well as chaotic oscillations. We have experimentally observed these synchronized states in delay-coupled electronic circuits and have analyzed their stability by numerical simulations and analytical calculations. We found that the conditions for longitudinal and transversal stability largely exclude each other and prevent, e.g., the synchronization of Pyragas-controlled orbits. Most striking is the observation of complete chaotic synchronization for large delay times, which should not be allowed in the given coupling scheme on the background of the actual paradigm.

Numerical characterization of transient polarization square-wave switching in two orthogonally coupled VCSELs

We study the dynamics of two vertical-cavity surface-emitting lasers (VCSELs) mutually coupled such that the natural lasing polarization of each laser is rotated by 90 degrees and then is injected into the other laser. Simulations based on the spin-flip model show transient square-wave polarization switchings before a stationary state is reached. The influence of various model parameters on the duration of the stochastic transient time and on the lasers' dynamics in the stationary state is investigated.

Synchronization in a chaotic neural network with time delay depending on the spatial distance between neurons

The synchronization is investigated in a two-dimensional Hindmarsh-Rose neuronal network by introducing a global coupling scheme with time delay, where the length of time delay is proportional to the spatial distance between neurons. We find that the time delay always disturbs synchronization of the neuronal network. When both the coupling strength and length of time delay per unit distance (i.e., enlargement factor) are large enough, the time delay induces the abnormal membrane potential oscillations in neurons. Specifically, the abnormal membrane potential oscillations for the symmetrically placed neurons form an antiphase, so that the large coupling strength and enlargement factor lead to the desynchronization of the neuronal network. The complete and intermittently complete synchronization of the neuronal network are observed for the right choice of parameters. The physical mechanism underlying these phenomena is analyzed.

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.

Synchronization transitions in coupled time-delay electronic circuits with a threshold nonlinearity

Experimental observations of typical kinds of synchronization transitions are reported in unidirectionally coupled time-delay electronic circuits with a threshold nonlinearity and two time delays, namely feedback delay τ(1) and coupling delay τ(2). We have observed transitions from anticipatory to lag via complete synchronization and their inverse counterparts with excitatory and inhibitory couplings, respectively, as a function of the coupling delay τ(2). The anticipating and lag times depend on the difference between the feedback and the coupling delays. A single stability condition for all the different types of synchronization is found to be valid as the stability condition is independent of both the delays. Further, the existence of different kinds of synchronizations observed experimentally is corroborated by numerical simulations and from the changes in the Lyapunov exponents of the coupled time-delay systems.

Chaos synchronization in mutually coupled semiconductor lasers with asymmetrical bias currents

We experimentally and numerically investigated the chaos synchronization characteristics of mutually coupled semiconductor lasers (MCSLs) with asymmetrical bias currents. Experimental results show that, asymmetrical bias current level of two MCSLs has obvious influence on chaos synchronization between them, and stable leader-laggard chaos synchronization can be realized under relatively large asymmetrical bias current levels. Moreover, the influences of frequency detuning and mutually coupling strength between the two lasers on chaos synchronization performance have also been discussed. Theoretical simulations basically conform to our experimental observations.

Synchronization of chaotic networks with time-delayed couplings: an analytic study

Networks of nonlinear units with time-delayed couplings can synchronize to a common chaotic trajectory. Although the delay time may be very large, the units can synchronize completely without time shift. For networks of coupled Bernoulli maps, analytic results are derived for the stability of the chaotic synchronization manifold. For a single delay time, chaos synchronization is related to the spectral gap of the coupling matrix. For networks with multiple delay times, analytic results are obtained from the theory of polynomials. Finally, the analytic results are compared with networks of iterated tent maps and Lang-Kobayashi equations, which imitate the behavior of networks of semiconductor lasers.

Role of delay for the symmetry in the dynamics of networks

The symmetry in a network of oscillators determines the spatiotemporal patterns of activity that can emerge. We study how a delay in the coupling affects symmetry-breaking and -restoring bifurcations. We are able to draw general conclusions in the limit of long delays. For one class of networks we derive a criterion that predicts that delays have a symmetrizing effect. Moreover, we demonstrate that for any network admitting a steady-state solution, a long delay can solely advance the first bifurcation point as compared to the instantaneous-coupling regime.