Multilayered tori in a system of two coupled logistic maps

The Letter describes different mechanisms for the formation and destruction of tori that are formed as layered structures of several sets of interlacing manifolds, each with their associated stable and unstable resonance modes. We first illustrate how a three layered torus can arise in a system of two coupled logistic maps through period-doubling or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We hereafter present two different scenarios by which a multilayered torus can be destructed. One scenario involves a cascade of period-doubling bifurcations of both the stable and the saddle cycles, and the second scenario describes a transition in which homoclinic bifurcations destroy first the two outer layers and thereafter also the inner layer of a three-layered torus. It is suggested that the formation of multilayered tori is a generic phenomenon in non-invertible maps.     

Switching phenomenon induced by breakdown of chaotic phase synchronization

We investigate chaotic phase synchronization (CPS) in three-coupled chaotic oscillator systems. According to the coupling strength and mismatches in the frequencies of these oscillators, we can observe complete CPS where all three oscillators exhibit CPS, and partial CPS where only two oscillators exhibit CPS. When the coupling strength is weakened, we observe a phenomenon that complete CPS among the three oscillators is suddenly disrupted without going through partial CPS. In this case oscillators exhibit quasi-CPS where two oscillators appear to exhibit CPS transiently, and the combination of the two oscillators changes with time. We call this phenomenon CPS switching D. It is revealed that phase fluctuation plays an important role in CPS switching D. It is also shown that the amplitude with a specific structure strengthens the degree of CPS switching. In the present paper, we characterize this CPS switching and discuss its mechanism.     

A system of alternately excited coupled non-autonomous oscillators manifesting phenomena intrinsic to complex analytical maps

A feasible model is introduced that manifests phenomena intrinsic to iterative complex analytical maps (such as the Mandelbrot set and Julia sets). The system is composed of two alternately excited coupled oscillators. The idea is based on a turn-by-turn transfer of the excitation from one subsystem to another [S.P. Kuznetsov, Example of a physical system with a hyperbolic attractor of the Smale-Williams type, Phys. Rev. Lett. 95 (2005) 144101] accompanied with appropriate non-linear transformation of the complex amplitude of the oscillations in the course of the process. Analytical and numerical studies are performed. Special attention is paid to an analysis of the violation of the applicability of the slow amplitude method with the decrease in the ratio of the period of the excitation transfer to the basic period of the oscillations. The main effect is the rotation of the Mandelbrot-like set in the complex parameter plane; one more effect is the destruction of subtle small-scale fractal structure of the set due to the presence of non-analytical terms in the complex amplitude equations.     

Statistical independence in nonlinear maps coupled to non-invertible transformations

We investigate the connections between functions of type xn=p(θTzn) and nonlinear maps coupled to non-invertible transformations. These systems can produce unpredictable dynamics. We study the higher-order correlations in the generated sequences. We show that (theoretically) it is possible to construct systems that can generate sequences that constitute a set of statistically independent random variables. We apply the results in the improvement of a two-dimensional coupled map system that has been used in practical applications as e.g. cryptosystems and data compression.     

Message filtering characteristics of semiconductor laser as receiver in optical chaos communication

The message filtering characteristics of the receiver in closed-loop chaotic optical communication system are numerically studied based on laser rate equations. A pair of external cavity semiconductor lasers was employed as the chaotic carrier transmitter and the synchronized chaos receiver. We examined the filtering properties of the semiconductor laser receiver for message encoded with chaos masking. Our results demonstrate that, the lower the message frequency, the more easily the receiver filters out the message from chaotic carrier. We also analyzed the effects of each parameter mismatches between the transmitter and the receiver on the quality of the recovered message. Comparing the synchronization quality with the signal-to-noise ratio affected by parameter mismatches, we find that the quality of the recovered message depends not only on the synchronization quality but also on the filtering characteristics of the receiver. The filtering characteristics of receiver will be playing an important role on the quality of the recovered message in the case of large mismatches.     

A Novel Adaptive Observer-Based Control Scheme for Synchronization and Suppression of a Class of Uncertain Chaotic Systems

A novel adaptive observer-based control scheme is presented for synchronization and suppression of a class of uncertain chaotic system. First, an adaptive observer based on an orthogonal neural network is designed. Subsequently, the sliding mode controllers via the proposed adaptive observer are proposed for synchronization and suppression of the uncertain chaotic systems. Theoretical analysis and numerical simulation show the effectiveness of the proposed scheme.     

Synchronization and control of chaos in coupled chaotic multimode Nd:YAG lasers

In this Letter we numerically investigate the dynamics of a system of two coupled chaotic multimode Nd:YAG lasers with two mode and three mode outputs. Unidirectional and bidirectional coupling schemes are adopted; intensity time series plots, phase space plots and synchronization plots are used for studying the dynamics. Quality of synchronization is measured using correlation index plots. It is found that for laser with two mode output bidirectional direct coupling scheme is found to be effective in achieving complete synchronization, control of chaos and amplification in output intensity. For laser with three mode output, bidirectional difference coupling scheme gives much better chaotic synchronization as compared to unidirectional difference coupling but at the cost of higher coupling strength. We also conclude that the coupling scheme and system properties play an important role in determining the type of synchronization exhibited by the system.     

Adaptive-impulsive synchronization in drive-response networks of continuous systems and its application

Based on the comparison theorem for the stability of impulsive control system, adaptive-impulsive synchronization in drive-response networks of continuous systems with time-delay and non-time-delay is investigated. And the continuous control input, the simple updated laws and a linear impulsive controller are proposed. Moreover, two numerical examples are presented to verify the effectiveness and correctness of the theorem, using the energy resource system and Lü's system as the nodes of the networks.     

High precision fast projective synchronization in chaotic (hyperchaotic) systems

A high precision fast projective synchronization method was proposed through introducing an impact factor. The numerical simulations indicate that the synchronization precision was increased three orders of magnitude higher than available methods in literature, under the same conditions of numerical method and hardware. The synchronization can achieved after first iteration. The synchronization speed has been increased by 13 484 times and 23 000 times when and , respectively, compared to general methods.     

State and parameter estimation in nonlinear systems as an optimal tracking problem

In verifying and validating models of nonlinear processes it is important to incorporate information from observations in an efficient manner. Using the idea of synchronization of nonlinear dynamical systems, we present a framework for connecting a data signal with a model in a way that minimizes the required coupling yet allows the estimation of unknown parameters in the model. The need to evaluate unknown parameters in models of nonlinear physical, biophysical, and engineering systems occurs throughout the development of phenomenological or reduced models of dynamics. Our approach builds on existing work that uses synchronization as a tool for parameter estimation. We address some of the critical issues in that work and provide a practical framework for finding an accurate solution. In particular, we show the equivalence of this problem to that of tracking within an optimal control framework. This equivalence allows the application of powerful numerical methods that provide robust practical tools for model development and validation.     

Impulsive control of projective synchronization in chaotic systems

Scaling factor of projective synchronization in coupled partially linear chaotic systems is hardly predictable. To control projective synchronization of chaotic systems in a preferred way, an impulsive control scheme is introduced to direct the scaling factor onto a desired value. The control approach is derived from the impulsive differential equation theory. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results. Furthermore, some interesting and surprising numerical results are discussed.     

Nonlinear Dynamics of a Sliding Chain in a Periodic Potential

Nonlinear dynamics of the sliding process of a chain driven with a constant velocity at one end in a periodic substrate potential is investigated. The driven chain exhibits distinctly different dynamical characteristics at different velocities. In the low velocity region, the chain moves in a stick-slip manner. When the driving velocity is increased, the stick-slip behaviour is replaced by complicated and regular oscillatory motions. The dependence of the dynamics on the coupling strength is studied and the step-like behaviour is found, where different steps correspond to different dynamical phases.     

Synchronization of chaos in unidirectionally and bidirectionally coupled multiple time delay laser diodes with electro-optical feedback

We report on chaos synchronization in both unidirectionally and bidirectionally coupled multiple time delay laser diodes with electro-optical feedback. We derive existence and sufficient stability conditions for the synchronization regimes. We calculate the Lyapunov exponents, information dimension, and Kolmogorov-Sinai entropy for a single and double delay time lasers to demonstrate that multiple time delay laser system can offer higher complexity than a single time delay laser. We demonstrate that in coupled multiple time delay lasers additional feedback(s) can play a stabilizing role. We compare the synchronization quality for closed loop and open loop receiver laser configurations and find better synchronization quality for partially open loop receiver (when the receiver laser has only one feedback loop), than the open loop receiver configuration (when the receiver contains no feedback loops). We also study the effect of the feedback phase on the correlation coefficient between the interacting laser systems. Analytical results are fully supported by numerical simulations.     

Adaptive cluster synchronization in complex dynamical networks

Cluster synchronization is investigated in different complex dynamical networks. In this Letter, a novel adaptive strategy is proposed to make a complex dynamical network achieve cluster synchronization, where the adaptive strategy of one edge is adjusted only according to its local information. A sufficient condition about the global stability arbitrarily grouped of cluster synchronization is derived. Several numerical simulations show the effectiveness of the adaptive strategy.     

Projective Synchronization in Time-Delayed Chaotic Systems

For the first time, we report on projective synchronization between two time delay chaotic systems with single time delays. It overcomes some limitations of the previous work, where projective synchronization has been investigated only in finite-dimensional chaotic systems, so we can achieve projective synchronization in infinitedimensional chaotic systems. We give a general method with which we can achieve projective synchronization in time-delayed chaotic systems. The method is illustrated using the famous delay-differential equations related to optical bistability. Numerical simulations fully support the analytical approach.     

Semi-passivity and synchronization of diffusively coupled neuronal oscillators

We discuss synchronization in networks of neuronal oscillators which are interconnected via diffusive coupling, i.e. linearly coupled via gap junctions. In particular, we present sufficient conditions for synchronization in these networks using the theory of semi-passive and passive systems. We show that the conductance based neuronal models of Hodgkin-Huxley, Morris-Lecar, and the popular reduced models of FitzHugh-Nagumo and Hindmarsh-Rose all satisfy a semi-passivity property, i.e. that is the state trajectories of such a model remain oscillatory but bounded provided that the supplied (electrical) energy is bounded. As a result, for a wide range of coupling configurations, networks of these oscillators are guaranteed to possess ultimately bounded solutions. Moreover, we demonstrate that when the coupling is strong enough the oscillators become synchronized. Our theoretical conclusions are confirmed by computer simulations with coupled Hindmarsh-Rose and Morris-Lecar oscillators. Finally we discuss possible “instabilities” in networks of oscillators induced by the diffusive coupling.     

Parameter estimation using balanced synchronization

Synchronization between experimental observations and a dynamical model with undetermined parameters can assist in completing the specification of the model parameters. The quality of the synchronization, a cost function to be minimized, typically depends on the difference between the data time series and the model time series. If the coupling between the data and the model is too strong, this cost function is small for any data and any model, and the variation of the cost function with respect to the parameters of interest is too small to permit selection of a value of the parameters. If the coupling is too small, synchronization is lost. We introduce two methods for balancing the competing desires of a small cost function and the numerical ability to determine parameters accurately. One method of ‘balanced’ synchronization adds a requirement that the conditional Lyapunov exponent of the model system, conditioned on being driven by the data, remain negative but small. The other method allows the coupling to vary in time according to the error in synchronization. This second method succeeds because the data and the model exhibit generalized synchronization in the region where the parameters of the model are well determined.     

Pattern Synchronization in a Two-Layer Neuronal Network

Pattern synchronization in a two-layer neuronal network is studied. For a single-layer network of Rulkov map neurons, there are three kinds of patterns induced by noise. Additive noise can induce ordered patterns at some intermediate noise intensities in a resonant way; however, for small and large noise intensities there exist excitable patterns and disordered patterns, respectively. For a neuronal network coupled by two single-layer networks with noise intensity differences between layers, we find that the two-layer network can achieve synchrony as the interlayer coupling strength increases. The synchronous states strongly depend on the interlayer coupling strength and the noise intensity difference between layers.     

Projective cluster synchronization in drive-response dynamical networks

In this paper, we study the projective cluster synchronization in a drive-response dynamical network with 1+N coupled partially linear chaotic systems. Because the scaling factors characterizing the dynamics of projective synchronization remain unpredictable, pinning control ideas are adopted to direct the different scaling factors onto the desired values. It is also shown that the projection cluster synchronization can be realized by controlling only one node in each cluster. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results.     

Impact of modulated multiple optical feedback time delays on laser diode chaos synchronization

By studying laser diodes subject to multiple time-delayed optical feedback, we demonstrate that the signatures of time delays in the autocorrelation function of the laser output can be erased in systems incorporating modulated feedback time delays. This property enhances the suitability of such laser systems highly suitable for secure chaos-based communication systems. Chaos synchronization properties of a laser-based modulated multiple time-delayed system are also reported for the first time.