Generalized synchronization of chaos in identical systems with hidden degrees of freedom

Generalized synchronization on the onset of auxiliary system approach

Generalized synchronization is an emergent functional relationship between the states of the interacting dynamical systems. To analyze the stability of a generalized synchronization state, the auxiliary system technique is a seminal approach that is broadly used nowadays. However, a few controversies have recently arisen concerning the applicability of this method. In this study, we systematically analyze the applicability of the auxiliary system approach for various coupling configurations. We analytically derive the auxiliary system approach for a drive-response coupling configuration from the definition of the generalized synchronization state. Numerically, we show that this technique is not always applicable for two bidirectionally coupled systems. Finally, we analytically derive the inapplicability of this approach for the network of coupled oscillators and also numerically verify it with an appropriate example.

Soft-wired long-term memory in a natural recurrent neuronal network

Recurrent neuronal networks are known to be endowed with fading (short-term) memory, whereas long-term memory is usually considered to be hard-wired in the network connectivity via Hebbian learning, for instance. Here, we use the neuronal network of the roundworm C. elegans to show that recurrent architectures in living organisms can exhibit long-term memory without relying on specific hard-wired modules. We applied a genetic algorithm, using a binary genome that encodes for inhibitory-excitatory connectivity, to solve the unconstrained optimization problem of fitting the experimentally observed dynamics of the worm's neuronal network. Our results show that the network operates in a complex chaotic regime, as measured by the permutation entropy. In that complex regime, the response of the system to repeated presentations of a time-varying stimulus reveals a consistent behavior that can be interpreted as long-term memory. This memory is soft-wired, since it does not require structural changes in the network connectivity, but relies only on the system dynamics for encoding.

Consistency properties of chaotic systems driven by time-delayed feedback

Consistency refers to the property of an externally driven dynamical system to respond in similar ways to similar inputs. In a delay system, the delayed feedback can be considered as an external drive to the undelayed subsystem. We analyze the degree of consistency in a generic chaotic system with delayed feedback by means of the auxiliary system approach. In this scheme an identical copy of the nonlinear node is driven by exactly the same signal as the original, allowing us to verify complete consistency via complete synchronization. In the past, the phenomenon of synchronization in delay-coupled chaotic systems has been widely studied using correlation functions. Here, we analytically derive relationships between characteristic signatures of the correlation functions in such systems and unequivocally relate them to the degree of consistency. The analytical framework is illustrated and supported by numerical calculations of the logistic map with delayed feedback for different replica configurations. We further apply the formalism to time series from an experiment based on a semiconductor laser with a double fiber-optical feedback loop. The experiment constitutes a high-quality replica scheme for studying consistency of the delay-driven laser and confirms the general theoretical results.

Consistency properties of a chaotic semiconductor laser driven by optical feedback

We experimentally study consistency properties of a semiconductor laser in response to a coherent optical drive originating from delayed feedback. The laser is connected to a short and a long optical fiber loop, switched such that only one is providing input to the laser at a time. This way, repeating the exact same optical drive twice, we find consistent or inconsistent responses depending on the pump parameter and we relate the kind of response to strong and weak chaos. Moreover, we are able to experimentally determine the sub-Lyapunov exponent, underlying the consistency properties.

Inapplicability of an auxiliary-system approach to chaotic oscillators with mutual-type coupling and complex networks

The auxiliary system approach being de facto the standard for the study of generalized synchronization in the unidirectionally coupled chaotic oscillators is also widely used to examine the mutually coupled systems and networks of nonlinear elements with the complex topology of links between nodes. In this Brief Report we illustrate by two simple counterexamples that the auxiliary-system approach gives incorrect results for the mutually coupled oscillators and therefore to study the generalized synchronization this approach may be used only for the drive-response configuration of nonlinear oscillators and networks.

Engineering generalized synchronization in chaotic oscillators

We report a method of engineering generalized synchronization (GS) in chaotic oscillators using an open-plus-closed-loop coupling strategy. The coupling is defined in terms of a transformation matrix that maps a chaotic driver onto a response oscillator where the elements of the matrix can be arbitrarily chosen, and thereby allows a precise control of the GS state. We elaborate the scheme with several examples of transformation matrices. The elements of the transformation matrix are chosen as constants, time varying function, state variables of the driver, and state variables of another chaotic oscillator. Numerical results of GS in mismatched Rössler oscillators as well as nonidentical oscillators such as Rössler and Chen oscillators are presented.

The existence of generalized synchronization of chaotic systems in complex networks

The paper studies the existence of generalized synchronization in complex networks, which consist of chaotic systems. When a part of modified nodes are chaotic, and the others have asymptotically stable equilibriums or orbital asymptotically stable periodic solutions, under certain conditions, the existence of generalized synchronization can be turned to the problem of contractive fixed point in the family of Lipschitz functions. In addition, theoretical proofs are proposed to the exponential attractive property of generalized synchronization manifold. Numerical simulations validate the theory.

Chaotic phase synchronization and desynchronization in an oscillator network for object selection

Object selection refers to the mechanism of extracting objects of interest while ignoring other objects and background in a given visual scene. It is a fundamental issue for many computer vision and image analysis techniques and it is still a challenging task to artificial visual systems. Chaotic phase synchronization takes place in cases involving almost identical dynamical systems and it means that the phase difference between the systems is kept bounded over the time, while their amplitudes remain chaotic and may be uncorrelated. Instead of complete synchronization, phase synchronization is believed to be a mechanism for neural integration in brain. In this paper, an object selection model is proposed. Oscillators in the network representing the salient object in a given scene are phase synchronized, while no phase synchronization occurs for background objects. In this way, the salient object can be extracted. In this model, a shift mechanism is also introduced to change attention from one object to another. Computer simulations show that the model produces some results similar to those observed in natural vision systems.

The development of generalized synchronization on complex networks

In this paper, we numerically investigate the development of generalized synchronization (GS) on typical complex networks, such as scale-free networks, small-world networks, random networks, and modular networks. By adopting the auxiliary-system approach to networks, we observe that GS generally takes place in oscillator networks with both heterogeneous and homogeneous degree distributions, regardless of whether the coupled chaotic oscillators are identical or nonidentical. We show that several factors, such as the network topology, the local dynamics, and the specific coupling strategies, can affect the development of GS on complex networks.

First experimental observation of generalized synchronization phenomena in microwave oscillators

In this Letter, we report for the first time on the experimental observation of the generalized synchronization regime in the microwave electronic systems, namely, in the multicavity klystron generators. A new approach devoted to the generalized synchronization detection has been developed. The experimental observations are in the excellent agreement with the results of numerical simulation. The observed phenomena gives a strong potential for new applications requiring microwave chaotic signals.

Dynamics of two laterally coupled semiconductor lasers: strong- and weak-coupling theory

The stability and nonlinear dynamics of two semiconductor lasers coupled side to side via evanescent waves are investigated by using three different models. In the composite-cavity model, the coupling between the lasers is accurately taken into account by calculating electric field profiles (composite-cavity modes) of the whole coupled-laser system. A bifurcation analysis of the composite-cavity model uncovers how different types of dynamics, including stationary phase-locking, periodic, quasiperiodic, and chaotic intensity oscillations, are organized. In the individual-laser model, the coupling between individual lasers is introduced phenomenologically with ad hoc coupling terms. Comparison with the composite-cavity model reveals drastic differences in the dynamics. To identify the causes of these differences, we derive a coupled-laser model with coupling terms which are consistent with the solution of the wave equation and the relevant boundary conditions. This coupled-laser model reproduces the dynamics of the composite-cavity model under weak-coupling conditions.

Scenarios for generalized synchronization with chaotic driving

In chaotically driven nonlinear dynamical systems, weak generalized synchrony can arise through distinct scenarios or routes in a manner similar to the onset of low-dimensional chaos or the creation of strange nonchaotic attractors in quasiperiodically driven systems. The limit sets of the dynamics for weak generalized synchronization are nonchaotic-the Lyapunov exponent is nonpositive-and are geometrically strange. Quantitative measures related to the parameter sensitivity exponent and finite-time Lyapunov exponent distributions can be defined in order to characterize generalized synchronization.

Chaotic dynamics and synchronization in microchip solid-state lasers with optoelectronic feedback

We experimentally observe the dynamics of a two-mode Nd:YVO4 microchip solid-state laser with optoelectronic feedback. The total laser output is detected and fed back to the injection current of the laser diode for pumping. Chaotic oscillations are observed in the microchip laser with optoelectronic self-feedback. We also observe the dynamics of two microchip lasers coupled mutually with optoelectronic link. The output of one laser is detected by a photodiode and the electronic signal converted from the laser output is sent to the pumping of the other laser. Chaotic fluctuation of the laser output is observed when the relaxation oscillation frequency is close to each other between the two microchip lasers. Synchronization of periodic wave form is also obtained when the microchip lasers have a single-longitudinal mode.

Chaotic synchronization through coupling strategies

Usually, complete synchronization (CS) is regarded as the form of synchronization proper of identical chaotic systems, while generalized synchronization (GS) extends CS in nonidentical systems. However, this generally accepted view ignores the role that the coupling plays in determining the type of synchronization. In this work, we show that by choosing appropriate coupling strategies, CS can be observed in coupled chaotic systems with parameter mismatch, and GS can also be achieved in coupled identical systems. Numerical examples are provided to demonstrate these findings. Moreover, experimental verification based on electronic circuits has been carried out to support the numerical results. Our work provides a method to obtain robust CS in synchronization-based chaos communications.

Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling

We theoretically investigate the nonlinear dynamics and synchronization properties between two mutually coupled semiconductor lasers units. Each unit can self-oscillate by means of delayed optoelectronic feedback loops. The mutual optoelectronic interactions between the laser units take into account the finite propagation time of the signals. Under perfectly symmetric conditions, we find different "death by delay" islands that persist for instantaneous coupling. The appearance of (zero lag) isochronous chaotic synchronization, under appropriate driving conditions, is another distinctive feature of the delayed feedback loops in the laser units. For slightly asymmetric operation, we obtain frequency locked bands (Arnold Tongue) whose width periodically changes with the coupling delay time.

Consistency of nonlinear system response to complex drive signals

The consistency of a nonlinear system's response to a repeated complex waveform drive signal is an important consideration in classical and quantum systems as diverse as lasers, neuronal networks, and manufacturing plants. We show from a consideration of different characteristic waveforms that there is typically an optimal drive amplitude for the most consistent response; internal noise sources dominate for small amplitude driving while deterministic system nonlinearity reduces consistency for large amplitudes. We test this general concept and its measurement experimentally and numerically on the specific example of a laser system.

Coupling scheme for complete synchronization of periodically forced chaotic CO2 lasers

We present a way of coupling two nonautonomous, periodically forced, chaotic C O2 lasers in a master-slave configuration in order to achieve complete synchronization. The method consists of modulating the forcing of the slave laser by means of the difference between the intensities of the two lasers, and lends itself to a simple physical implementation. Experimental evidence of complete synchronization induced by a suitable coupling strength is shown, and a numerical model is used to achieve further insight of the synchronization phenomena. Finally, we describe a possible application of the investigated technique to the design of a digital communication system.

Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator

We demonstrate generalized synchronization in a spatiotemporal chaotic system, a liquid crystal spatial light modulator with optoelectronic feedback.