Transitions from partial to complete generalized synchronizations in bidirectionally coupled chaotic oscillators

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.

Intermittent route to generalized synchronization in bidirectionally coupled chaotic oscillators

The type of transition from asynchronous behavior to the generalized synchronization regime in mutually coupled chaotic oscillators has been studied. To separate the epochs of the synchronous and asynchronous motion in time series of mutually coupled chaotic oscillators, a method based on the local Lyapunov exponent calculation has been proposed. The efficiency of the method has been testified using the examples of unidirectionally coupled dynamical systems for which the type of transition is well known. The transition to generalized synchronization regime in mutually coupled systems has been shown to be an on-off intermittency as well as in the case of the unidirectional coupling.

Discontinuity-induced intermittent synchronization transitions in coupled non-smooth systems

The synchronization transition in coupled non-smooth systems is studied for increasing coupling strength. The average order parameter is calculated to diagnose synchronization of coupled non-smooth systems. It is found that the coupled non-smooth system exhibits an intermittent synchronization transition from the cluster synchronization state to the complete synchronization state, depending on the coupling strength and initial conditions. Detailed numerical analyses reveal that the discontinuity always plays an important role in the synchronization transition of the coupled non-smooth system. In addition, it is found that increasing the coupling strength leads to the coexistence of periodic cluster states. Detailed research illustrates that the periodic clusters consist of two or more coexisting periodic attractors. Their periodic trajectory passes from one region to another region that is divided by discontinuous boundaries in the phase space. The mutual interactions of the local nonlinearity and the spatial coupling ultimately result in a stable periodic trajectory.

Transitions from chimeras to coherence: An analytical approach by means of the coherent stability function

Chimera states have been a vibrant subject of research in the recent past, but the analytical treatment of transitions from chimeras to coherent states remains a challenge. Here we analytically derive the necessary conditions for this transition by means of the coherent stability function approach, which is akin to the master stability function approach that is traditionally used to study the stability of synchronization in coupled oscillators. In chimera states, there is typically at least one group of oscillators that evolves in a drifting, random manner, while other groups of oscillators follow a smoother, more coherent profile. In the coherent state, there thus exists a smooth functional relationship between the oscillators. This lays the foundation for the coherent stability function approach, where we determine the stability of the coherent state. We subsequently test the analytical prediction numerically by calculating the strength of incoherence during the transition point. We use leech neurons, which exhibit a coexistence of chaotic and periodic tonic spiking depending on initial conditions, coupled via nonlocal electrical synapses, to demonstrate our approach. We systematically explore various dynamical states with the focus on the transitions between chimeras and coherence, fully confirming the validity of the coherent stability function. We also observe complete synchronization for higher values of the coupling strength, which we verify by the master stability function approach.

Generalized synchronization in a conservative and nearly conservative systems of star network

We report the coexistence of synchronized and unsynchronized states in a mutually coupled star network of nearly conservative non-identical oscillators. Generalized synchronization is observed between the central oscillator with the peripherals, and phase synchronization is found among the peripherals in weakly dissipative systems. However, the basin size of the synchronization region decreases as dissipation strength is increased. We have demonstrated these phenomena with the help of Duffing and Lorenz84 oscillators with conservative, nearly conservative, and dissipative properties. The observed results are robust against the network size.

Consistency between functional and structural networks of coupled nonlinear oscillators

In data-based reconstruction of complex networks, dynamical information can be measured and exploited to generate a functional network, but is it a true representation of the actual (structural) network? That is, when do the functional and structural networks match and is a perfect matching possible? To address these questions, we use coupled nonlinear oscillator networks and investigate the transition in the synchronization dynamics to identify the conditions under which the functional and structural networks are best matched. We find that, as the coupling strength is increased in the weak-coupling regime, the consistency between the two networks first increases and then decreases, reaching maximum in an optimal coupling regime. Moreover, by changing the network structure, we find that both the optimal regime and the maximum consistency will be affected. In particular, the consistency for heterogeneous networks is generally weaker than that for homogeneous networks. Based on the stability of the functional network, we propose further an efficient method to identify the optimal coupling regime in realistic situations where the detailed information about the network structure, such as the network size and the number of edges, is not available. Two real-world examples are given: corticocortical network of cat brain and the Nepal power grid. Our results provide new insights not only into the fundamental interplay between network structure and dynamics but also into the development of methodologies to reconstruct complex networks from data.

Synchronization transition in networked chaotic oscillators: the viewpoint from partial synchronization

Synchronization transition in networks of nonlocally coupled chaotic oscillators is investigated. It is found that in reaching the state of global synchronization the networks can stay in various states of partial synchronization. The stability of the partial synchronization states is analyzed by the method of eigenvalue analysis, in which the important roles of the network topological symmetry on synchronization transition are identified. Moreover, for networks possessing multiple topological symmetries, it is found that the synchronization transition can be divided into different stages, with each stage characterized by a unique synchronous pattern of the oscillators. Synchronization transitions in networks of nonsymmetric topology and nonidentical oscillators are also investigated, where the partial synchronization states, although unstable, are found to be still playing important roles in the transitions.

Clustering versus non-clustering phase synchronizations

Clustering phase synchronization (CPS) is a common scenario to the global phase synchronization of coupled dynamical systems. In this work, a novel scenario, the non-clustering phase synchronization (NPS), is reported. It is found that coupled systems do not transit to the global synchronization until a certain sufficiently large coupling is attained, and there is no clustering prior to the global synchronization. To reveal the relationship between CPS and NPS, we further analyze the noise effect on coupled phase oscillators and find that the coupled oscillator system can change from CPS to NPS with the increase of noise intensity or system disorder. These findings are expected to shed light on the mechanism of various intriguing self-organized behaviors in coupled systems.

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.

Nature of weak generalized synchronization in chaotically driven maps

Weak generalized synchrony in a drive-response system occurs when the response dynamics is a unique but nondifferentiable function of the drive, in a manner that is similar to the formation of strange nonchaotic attractors in quasiperiodically driven dynamical systems. We consider a chaotically driven monotone map and examine the geometry of the limit set formed in the regime of weak generalized synchronization. The fractal dimension of the set of zeros is studied both analytically and numerically. We further examine the stable and unstable sets formed and measure the regularity of the coupling function. The stability index as well as the dimension spectrum of the equilibrium measure can be computed analytically.

Synchronization and quorum sensing in an ensemble of indirectly coupled chaotic oscillators

The fact that the elements in some realistic systems are influenced by each other indirectly through a common environment has stimulated a new surge of studies on the collective behavior of coupled oscillators. Most of the previous studies, however, consider only the case of coupled periodic oscillators, and it remains unknown whether and to what extent the findings can be applied to the case of coupled chaotic oscillators. Here, using the population density and coupling strength as the tuning parameters, we explore the synchronization and quorum sensing behaviors in an ensemble of chaotic oscillators coupled through a common medium, in which some interesting phenomena are observed, including the appearance of the phase synchronization in the process of progressive synchronization, the various periodic oscillations close to the quorum sensing transition, and the crossover of the critical population density at the transition. These phenomena, which have not been reported for indirectly coupled periodic oscillators, reveal a corner of the rich dynamics inherent in indirectly coupled chaotic oscillators, and are believed to have important implications to the performance and functionality of some realistic systems.

Generalized synchronization of complex dynamical networks via impulsive control

This paper investigates the generalized synchronization (GS) of two typical complex dynamical networks, small-world networks and scale-free networks, in terms of impulsive control strategy. By applying the auxiliary-system approach to networks, we demonstrate theoretically that for any given coupling strength, GS can take place in complex dynamical networks consisting of nonidentical systems. Particularly, for Barabasi-Albert scale-free networks, we look into the relations between GS error and topological parameter m, which denotes the number of edges linking to a new node at each time step, and find out that GS speeds up with increasing m. And for Newman-Watts small-world networks, the time needed to achieve GS decreases as the probability of adding random edges increases. We further reveal how node dynamics affects GS speed on both small-world and scale-free networks. Finally, we analyze how the development of GS depends on impulsive control gains. Some abnormal but interesting phenomena regarding the GS process are also found in simulations.

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.

Hölder continuity of two types of generalized synchronization manifold

This paper studies the existence of Hölder continuity of generalized synchronization (GS). Based on the modified system approach, GS is classified into three types: equilibrium GS, periodic GS, and C-GS, when the modified system has an asymptotically stable equilibrium, asymptotically stable limit cycles, and chaotic attractors, respectively. The existence of the first two types of Hölder continuous GS inertial manifolds are strictly theoretically proved.

Paths to globally generalized synchronization in scale-free networks

We apply the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators, including Hénon maps, logistic maps, and Lorentz oscillators. As the coupling strength epsilon between nodes of the network is increased, transitions from partially to globally generalized synchronization and intermittent behaviors near the synchronization thresholds, are found. The generalized synchronization starts from the hubs of the network and then spreads throughout the whole network with the increase of epsilon . Our result is useful for understanding the synchronization process in complex networks.

Synchronization properties of bidirectionally coupled semiconductor lasers under asymmetric operating conditions

We study, both experimentally and numerically, a system of two coupled semiconductor lasers in an asymmetric configuration. A laser subject to optical feedback is bidirectionally coupled to a free running laser. While maintaining the coupling strength, we change the feedback rate and observe a transition from highly correlated low-frequency fluctuations to episodic synchronization between dropouts and jump-ups. Our results resemble those obtained recently in a unidirectionally coupled system [Buldú, Phys. Rev. Lett. 96, 024102 (2006)].

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.

Effect of noise on generalized chaotic synchronization

When two characteristically different chaotic oscillators are coupled, generalized synchronization can occur. Motivated by the phenomena that common noise can induce and enhance complete synchronization or phase synchronization in chaotic systems, we investigate the effect of noise on generalized chaotic synchronization. We develop a phase-space analysis, which suggests that the effect can be system dependent in that common noise can either induce/enhance or destroy generalized synchronization. A prototype model consisting of a Lorenz oscillator coupled with a dynamo system is used to illustrate these phenomena.

Bifurcation and chaos in coupled ratchets exhibiting synchronized dynamics

The bifurcation and chaotic behavior of unidirectionally coupled deterministic ratchets is studied as a function of the driving force amplitude and frequency . A classification of the various types of bifurcations likely to be encountered in this system was done by examining the stability of the steady state in linear response as well as constructing a two-parameter phase diagram in the plane. Numerical explorations revealed varieties of bifurcation sequences including quasiperiodic route to chaos. Besides, the familiar period-doubling and crises route to chaos exhibited by the one-dimensional ratchet were also found. In addition, the coupled ratchets display symmetry-breaking, saddle-nodes and bubbles of bifurcations. Chaotic behavior is characterized by using the Lyapunov exponent spectrum; while a perusal of the phase space projected in the Poincaré cross section confirms some of the striking features.

Public-key encryption based on generalized synchronization of coupled map lattices

Currently used public-key cryptosystems are based on difficulties in solving certain numeric theoretic problems, in which the way to predict the private key from the knowledge of the public key is computationally infeasible. Here we propose a method of constructing public-key cryptosystems by generalized synchronization of coupled map lattices, in which the difficulty in predicting the synchronous function is used as the trap-door function to deduce the private key from the public key. In specific, we implement this idea on the method of "Merkle's puzzles," and find that, incorporated with the chaotic dynamics, this traditional method is equipped with some new features and can be practical in certain situations.

Phase synchronization in the perturbed Chua circuit

We show experimental and numerical results of phase synchronization between the chaotic Chua circuit and a small sinusoidal perturbation. Experimental real-time phase synchronized states can be detected with oscilloscope visualization of the attractor, using specific sampling rates. Arnold tongues demonstrate robust phase synchronized states for perturbation frequencies close to the characteristic frequency of the unperturbed Chua.

Analysis of generalized synchronization in directionally coupled chaotic phase-coherent oscillators by local minimal fluctuations

The method of local-minimum fluctuation is proposed to analyze generalized synchronization in directionally coupled chaotic phase-coherent oscillators. It is shown that the emergence of generalized synchronization is manifested by the qualitative changes in the statistic of local minimum fluctuations of the receiver oscillator.