M. Zigzag

Synchronization of unidirectional time delay chaotic networks and the greatest common divisor

We present the interplay between synchronization of unidirectional coupled chaotic nodes with heterogeneous delays and the greatest common divisor (GCD) of loops composing the oriented graph. In the weak chaos region and for GCD=1 the network is in chaotic zero-lag synchronization, whereas for GCD=m>1 synchronization of m-sublattices emerges. Complete synchronization can be achieved when all chaotic nodes are influenced by an identical set of delays and in particular for the limiting case of homogeneous delays. Results are supported by simulations of chaotic systems, self-consistent and mixing arguments, as well as analytical solutions of Bernoulli maps.

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.

Nonlocal mechanism for cluster synchronization in neural circuits

The interplay between the topology of cortical circuits and synchronized activity modes in distinct cortical areas is a key enigma in neuroscience. We present a new nonlocal mechanism governing the periodic activity mode: the greatest common divisor (GCD) of network loops. For a stimulus to one node, the network splits into GCD-clusters in which cluster neurons are in zero-lag synchronization. For complex external stimuli, the number of clusters can be any common divisor. The synchronized mode and the transients to synchronization pinpoint the type of external stimuli. The findings, supported by an information mixing argument and simulations of Hodgkin-Huxley population dynamic networks with unidirectional connectivity and synaptic noise, call for reexamining sources of correlated activity in cortex and shorter information processing time scales.

Synchronization in small networks of time-delay coupled chaotic diode lasers

Topologies of two, three and four time-delay-coupled chaotic semiconductor lasers are experimentally and theoretically found to show new types of synchronization. Generalized zero-lag synchronization is observed for two lasers separated by long distances even when their self-feedback delays are not equal. Generalized sub-lattice synchronization is observed for quadrilateral geometries while the equilateral triangle is zero-lag synchronized. Generalized zero-lag synchronization, without the limitation of precisely matched delays, opens possibilities for advanced multi-user communication protocols.

Zero-lag synchronization of chaotic units with time-delayed couplings

Zero-lag synchronization (ZLS) is achieved in a very restricted mutually coupled chaotic systems, where the delays of the self-coupling and the mutual coupling are identical or fulfil some restricted ratios. Using a set of multiple self-feedbacks and mutual couplings we demonstrate both analytically and numerically that ZLS is achieved for a wide range of mutual delays. It indicates that ZLS can be achieved without the knowledge of the mutual distance between the communicating partners and has an important implication in the possible use of ZLS in communications networks as well as in the understanding of the emergence of such synchronization in the neuronal activities.

Coexistence of exponentially many chaotic spin-glass attractors.

A chaotic network of size N with delayed interactions which resembles a pseudoinverse associative memory neural network is investigated. For a load α = P/N < 1, where P stands for the number of stored patterns, the chaotic network functions as an associative memory of 2P attractors with macroscopic basin of attractions which decrease with α. At finite α, a chaotic spin-glass phase exists, where the number of distinct chaotic attractors scales exponentially with N. Each attractor is characterized by a coexistence of chaotic behavior and freezing of each one of the N chaotic units or freezing with respect to the P patterns. Results are supported by large scale simulations of networks composed of Bernoulli map units and Mackey-Glass time delay differential equations.