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.

The transition between strong and weak chaos in delay systems: Stochastic modeling approach.

We investigate the scaling behavior of the maximal Lyapunov exponent in chaotic systems with time delay. In the large-delay limit, it is known that one can distinguish between strong and weak chaos depending on the delay scaling, analogously to strong and weak instabilities for steady states and periodic orbits. Here we show that the Lyapunov exponent of chaotic systems shows significant differences in its scaling behavior compared to constant or periodic dynamics due to fluctuations in the linearized equations of motion.

Stochastic switching in delay-coupled oscillators.

A delay is known to induce multistability in periodic systems. Under influence of noise, coupled oscillators can switch between coexistent orbits with different frequencies and different oscillation patterns. For coupled phase oscillators we reduce the delay system to a nondelayed Langevin equation, which allows us to analytically compute the distribution of frequencies and their corresponding residence times.

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.

Strong and weak chaos in networks of semiconductor lasers with time-delayed couplings.

Nonlinear networks with time-delayed couplings may show strong and weak chaos, depending on the scaling of their Lyapunov exponent with the delay time. We study strong and weak chaos for semiconductor lasers, either with time-delayed self-feedback or for small networks. We examine the dependence on the pump current and consider the question of whether strong and weak chaos can be identified from the shape of the intensity trace, the autocorrelations, and the external cavity modes.

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.

Discontinuous attractor dimension at the synchronization transition of time-delayed chaotic systems.

The attractor dimension at the transition to complete synchronization in a network of chaotic units with time-delayed couplings is investigated. In particular, we determine the Kaplan-Yorke dimension from the spectrum of Lyapunov exponents for iterated maps and for two coupled semiconductor lasers. We argue that the Kaplan-Yorke dimension must be discontinuous at the transition and compare it to the correlation dimension.

Pulsed chaos synchronization in networks with adaptive couplings.

Networks of chaotic units with static couplings can synchronize to a common chaotic trajectory. The effect of dynamic adaptive couplings on the cooperative behavior of chaotic networks is investigated. The couplings adjust to the activities of its two units by two competing mechanisms: An exponential decrease of the coupling strength is compensated for by an increase due to desynchronized activity.

Strong and weak chaos in nonlinear networks with time-delayed couplings.

We study chaotic synchronization in networks with time-delayed coupling. We introduce the notion of strong and weak chaos, distinguished by the scaling properties of the maximum Lyapunov exponent within the synchronization manifold for large delay times, and relate this to the condition for stable or unstable chaotic synchronization, respectively. In simulations of laser models and experiments with electronic circuits, we identify transitions from weak to strong and back to weak chaos upon monotonically increasing the coupling strength.

Observing chaos for quantum-dot microlasers with external feedback.

Chaos presents a striking and fascinating phenomenon of nonlinear systems. A common aspect of such systems is the presence of feedback that couples the output signal partially back to the input. Feedback coupling can be well controlled in optoelectronic devices such as conventional semiconductor lasers that provide bench-top platforms for the study of chaotic behaviour and high bit rate random number generation.

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.

Zero-lag synchronization and multiple time delays in two coupled chaotic systems.

Zero-lag synchronization (ZLS) between two chaotic systems coupled by a portion of their signal is achieved for restricted ratios between the delays of the self-feedback and the mutual coupling. We extend this scenario to the case of a set of multiple self-feedbacks {Ndi} and a set of multiple mutual couplings {Ncj}. We demonstrate both analytically and numerically that ZLS can be achieved when SigmaliNdi+igmamjNcj=0, where li,mj(epsilon)Z.

Pulses of chaos synchronization in coupled map chains with delayed transmission.

Pulses of synchronization in chaotic coupled map lattices are discussed in the context of transmission of information. Synchronization and desynchronization propagate along the chain with different velocities which are calculated analytically from the spectrum of convective Lyapunov exponents. Since the front of synchronization travels slower than the front of desynchronization, the maximal possible chain length for which information can be transmitted by modulating the first unit of the chain is bounded.

Dynamics of recurrent neural networks with delayed unreliable synapses: metastable clustering.

The influence of unreliable synapses on the dynamic properties of a neural network is investigated for a homogeneous integrate-and-fire network with delayed inhibitory synapses. Numerical and analytical calculations show that the network relaxes to a state with dynamic clusters of identical size which permanently exchange neurons. We present analytical results for the number of clusters and their distribution of firing times which are determined by the synaptic properties.

Public channel cryptography: chaos synchronization and Hilbert's tenth problem.

The synchronization process of two mutually delayed coupled deterministic chaotic maps is demonstrated both analytically and numerically. The synchronization is preserved when the mutually transmitted signals are concealed by two commutative private filters, a convolution of the truncated time-delayed output signals or some powers of the delayed output signals. The task of a passive attacker is mapped onto Hilbert's tenth problem, solving a set of nonlinear Diophantine equations, which was proven to be in the class of NP-complete problems [problems that are both NP (verifiable in nondeterministic polynomial time) and NP-hard (any NP problem can be translated into this problem)].

Patterns of chaos synchronization.

Small networks of chaotic units which are coupled by their time-delayed variables are investigated. In spite of the time delay, the units can synchronize isochronally, i.e.

Space representation of stochastic processes with delay.

We show that a time series x(t) evolving by a nonlocal update rule x(t) =f (x(t-n),x(t-k)) with two different delays k < n can be mapped onto a local process in two dimensions with special time-delayed boundary conditions, provided that n and k are coprime. For certain stochastic update rules exhibiting a nonequilibrium phase transition, this mapping implies that the critical behavior does not depend on the short delay k . In these cases, the autocorrelation function of the time series is related to the critical properties of the corresponding two-dimensional model.

Spiking optical patterns and synchronization.

We analyze the time resolved spike statistics of a solitary and two mutually interacting chaotic semiconductor lasers whose chaos is characterized by apparently random, short intensity spikes. Repulsion between two successive spikes is observed, resulting in a refractory period, which is largest at laser threshold. For time intervals between spikes greater than the refractory period, the distribution of the intervals follows a Poisson distribution.

Sublattice synchronization of chaotic networks with delayed couplings.

Synchronization of chaotic units coupled by their time-delayed variables is investigated analytically. A type of cooperative behavior is found: Sublattice synchronization. Although the units of one sublattice are not directly coupled to each other, they completely synchronize without time delay.

Dynamics of neural cryptography.

Synchronization of neural networks has been used for public channel protocols in cryptography. In the case of tree parity machines the dynamics of both bidirectional synchronization and unidirectional learning is driven by attractive and repulsive stochastic forces. Thus it can be described well by a random walk model for the overlap between participating neural networks.

On the stationary state of a network of inhibitory spiking neurons.

The background activity of a cortical neural network is modeled by a homogeneous integrate-and-fire network with unreliable inhibitory synapses. For the case of fast synapses, numerical and analytical calculations show that the network relaxes into a stationary state of high attention. The majority of the neurons has a membrane potential just below the threshold; as a consequence the network can react immediately - on the time scale of synaptic transmission- on external pulses.

Synchronization of mutually coupled chaotic lasers in the presence of a shutter.

Two mutually coupled chaotic diode lasers exhibit stable isochronal synchronization in the presence of self-feedback. When the mutual communication between the lasers is discontinued by a shutter and the two uncoupled lasers are subject to self-feedback only, the desynchronization time is found to scale as Adtau, where Ad>1 and tau corresponds to the optical distance between the lasers. Prior to synchronization, when the two lasers are uncorrelated and the shutter between them is opened, the synchronization time is found to be much shorter, though still proportional to tau.

Public-channel cryptography based on mutual chaos pass filters.

We study the mutual coupling of chaotic lasers and observe both experimentally and in numeric simulations that there exists a regime of parameters for which two mutually coupled chaotic lasers establish isochronal synchronization, while a third laser coupled unidirectionally to one of the pair does not synchronize. We then propose a cryptographic scheme, based on the advantage of mutual coupling over unidirectional coupling, where all the parameters of the system are public knowledge. We numerically demonstrate that in such a scheme the two communicating lasers can add a message signal (compressed binary message) to the transmitted coupling signal and recover the message in both directions with high fidelity by using a mutual chaos pass filter procedure.

Stable isochronal synchronization of mutually coupled chaotic lasers.

The dynamics of two mutually coupled chaotic diode lasers are investigated experimentally and numerically. By adding self-feedback to each laser, stable isochronal synchronization is established. This stability, which can be achieved for symmetric operation, is essential for constructing an optical public-channel cryptographic system.

Genetic attack on neural cryptography.

Different scaling properties for the complexity of bidirectional synchronization and unidirectional learning are essential for the security of neural cryptography. Incrementing the synaptic depth of the networks increases the synchronization time only polynomially, but the success of the geometric attack is reduced exponentially and it clearly fails in the limit of infinite synaptic depth. This method is improved by adding a genetic algorithm, which selects the fittest neural networks.