Synchronizing chaotic systems up to an arbitrary scaling matrix via a single signal

This paper introduces a novel type of synchronization, where two chaotic systems synchronize up to an arbitrary scaling matrix. In particular, each drive system state synchronizes with a linear combination of response system states by using a single synchronizing signal. The proposed observer-based method exploits a theorem that assures asymptotic synchronization for a wide class of continuous-time chaotic (hyperchaotic) systems. Two examples, involving Rössler’s system and a hyperchaotic oscillator, show that the proposed technique is a general framework to achieve any type of synchronization defined to date.     

Generalized synchronization of strictly different systems: Partial-state synchrony

Generalized synchronization (GS) occurs when the states of one system, through a functional mapping are equal to states of another. Since for many physical systems only some state variables are observable, it seems convenient to extend the theoretical framework of synchronization to consider such situations. In this contribution, we investigate two variants of GS which appear between strictly different chaotic systems. We consider that for both the drive and response systems only one observable is available. For the case when both systems can be taken to a complete triangular form, a GS can be achieved where the functional mapping between drive and response is found directly from their Lie-algebra based transformations. Then, for systems that have dynamics associated to uncontrolled and unobservable states, called internal dynamics, where only a partial triangular form is possible via coordinate transformations, for this situation, a GS is achieved for which the coordinate transformations describe the functional mapping of only a few state variables. As such, we propose definitions for complete and partial-state GS. These particular forms of GS are illustrated with numerical simulations of well-known chaotic benchmark systems.     

Chaos May Be an Optimal Plan

An interpretation of some chaotic systems as the result of optimal decisions is presented. First, a generalized discrete-time two-person game is introduced that may be solved by use of dynamic programming. Then, a specific game of this type is formulated whose optimal solution transforms an originally linear discrete-time system into a well-known discrete-time chaotic system. Finally, a particular continuous-time optimal control problem is formulated, whose optimal feedback solution transforms an originally linear continuous-time system into a well-known continuous-time chaotic system.     

Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal

In this paper, a drive-response synchronization method with linear output error feedback is presented for synchronizing a class of fractional-order chaotic systems via a scalar transmitted signal. Based on stability theory of fractional-order systems and linear system theory, a necessary and sufficient condition for the existence of the feedback gain vector such that global synchronization between the fractional-order drive system and response system can be achieved and its design method are given. This synchronization approach that is simple, global and theoretically rigorous enables synchronization of fractional-order chaotic systems be achieved in a systematic way and does not require the computation of the conditional Lyapunov exponents. An example is used to illustrate the effectiveness of the proposed synchronization method.     

Exponential synchronization for fractional‐order chaotic systems with mixed uncertainties

This article focuses on the problem of exponential synchronization for fractional‐order chaotic systems via a nonfragile controller. A criterion for α‐exponential stability of an error system is obtained using the drive‐response synchronization concept together with the Lyapunov stability theory and linear matrix inequalities approach. The uncertainty in system is considered with polytopic form together with structured form. The sufficient conditions are derived for two kinds of structured uncertainty, namely, (1) norm bounded one and (2) linear fractional transformation one. Finally, numerical examples are presented by taking the fractional‐order chaotic Lorenz system and fractional‐order chaotic Newton–Leipnik system to illustrate the applicability of the obtained theory. © 2014 Wiley Periodicals, Inc. Complexity 21: 114–125, 2015     

Generalized synchronization of chaos based on suitable separation

In this Letter, generalized synchronization with a kind of function relationship between the states of drive and response chaotic systems is achieved. From matrix measure theory, some sufficient conditions for generalized synchronization are derived through suitable separation by decomposing the system as the linear part and the nonlinear one. Simulation results are provided for illustration and verification of the proposed method.     

Robust digital controllers for uncertain chaotic systems: A digital redesign approach

In this paper, a new and systematic method for designing robust digital controllers for uncertain nonlinear systems with structured uncertainties is presented. In the proposed method, a controller is designed in terms of the optimal linear model representation of the nominal system around each operating point of the trajectory, while the uncertainties are decomposed such that the uncertain nonlinear system can be rewritten as a set of local linear models with disturbed inputs. Applying conventional robust control techniques, continuous-time robust controllers are first designed to eliminate the effects of the uncertainties on the underlying system. Then, a robust digital controller is obtained as the result of a digital redesign of the designed continuous-time robust controller using the state-matching technique. The effectiveness of the proposed controller design method is illustrated through some numerical examples on complex nonlinear systems––chaotic systems.     

Hölder continuity of three types of generalized synchronization manifolds of non-autonomous systems

This paper studies the existence of Hölder continuity of the generalized synchronization (GS) manifold. When the modified response system has an asymptotically stable equilibrium, periodic or quasi-periodic orbit, and chaotic attractor, GS is classified into four types accordingly. The first three types of GS are considered, and based on the Schauder fixed point theorem, sufficient conditions for Hölder continuous GS in the coupled non-autonomous systems are derived and theoretically proved.     

Hölder continuity of three types of generalized synchronization manifolds of non-autonomous systems

This paper studies the existence of Hölder continuity of the generalized synchronization (GS) manifold. When the modified response system has an asymptotically stable equilibrium, periodic or quasi-periodic orbit, and chaotic attractor, GS is classified into four types accordingly. The first three types of GS are considered, and based on the Schauder fixed point theorem, sufficient conditions for Hölder continuous GS in the coupled non-autonomous systems are derived and theoretically proved.     

Anti-control of continuous-time dynamical systems

Based on two basic characteristics of continuous-time autonomous chaotic systems, namely being globally bounded while having a positive Lyapunov exponent, this paper develops a universal and practical anti-control approach to design a general continuous-time autonomous chaotic system via Lyapunov exponent placement. This self-unified approach is verified by mathematical analysis and validated by several typical systems designs with simulations. Compared to the common trial-and-error methods, this approach is semi-analytical with feasible guidelines for design and implementation. Finally, using the Shilnikov criteria, it is proved that the new approach yields a heteroclinic orbit in a three-dimensional autonomous system, therefore the resulting system is indeed chaotic in the sense of Shilnikov.     

Adaptive synchronization of T–S fuzzy chaotic systems with unknown parameters

This paper presents a fuzzy model-based adaptive approach for synchronization of chaotic systems which consist of the drive and response systems. Takagi–Sugeno (T–S) fuzzy model is employed to represent the chaotic drive and response systems. Since the parameters of the drive system are assumed unknown, we design the response system that estimates the parameters of the drive system by adaptive strategy. The adaptive law is derived to estimate the unknown parameters and its stability is guaranteed by Lyapunov stability theory. In addition, the controller in the response system contains two parts: one part that can stabilize the synchronization error dynamics and the other part that estimates the unknown parameters. Numerical examples, including Duffing oscillator and Lorenz attractor, are given to demonstrate the validity of the proposed adaptive synchronization approach.     

On the fractional-order extended Kalman filter and its application to chaotic cryptography in noisy environment

In this paper via a novel method of discretized continuous-time Kalman filter, the problem of synchronization and cryptography in fractional-order systems has been investigated in presence of noisy environment for process and output signals. The fractional-order Kalman filter equation, applicable for linear systems, and its extension called the extended Kalman filter, which can be used for nonlinear systems, are derived. The result is utilized for chaos synchronization with the aim of cryptography while the transmitter system is fractional-order, and both the transmitter and transmission channel are noisy. The fractional-order stochastic chaotic Chen system is then presented to apply the proposed method for chaotic signal cryptography. The results show the effectiveness of the proposed method.     

Master–slave synchronization of continuously and intermittently coupled sampled-data chaotic oscillators

In this paper, we consider the problem of synchronizing a master–slave chaotic system in the sampled-data setting. We consider both the intermittent coupling and continuous coupling cases. We use an Euler approximation technique to discretize a continuous-time chaotic oscillator containing a continuous nonlinear function. Next, we formulate the problem of global asymptotic synchronization of the sampled-data master–slave chaotic system as equivalent to the states of a corresponding error system asymptotically converging to zero for arbitrary initial conditions. We begin by developing a pulse-based intermittent control strategy for chaos synchronization. Using the discrete-time Lyapunov stability theory and the linear matrix inequality (LMI) framework, we construct a state feedback periodic pulse control law which yields global asymptotic synchronization of the sampled-data master–slave chaotic system for arbitrary initial conditions. We obtain a continuously coupled sampled-data feedback control law as a special case of the pulse-based feedback control. Finally, we provide experimental validation of our results by implementing, on a set of microcontrollers endowed with RF communication capability, a sampled-data master–slave chaotic system based on Chua’s circuit.     

Adaptive synchronization of Cohen-Grossberg neural networks with unknown parameters and mixed time-varying delays

In this paper, we investigate the synchronization problem of chaotic Cohen-Grossberg neural networks with unknown parameters and mixed time-varying delays. An adaptive linear feedback controller is designed to guarantee that the response system can be synchronized with a drive system by utilizing Lyapunov stability theory and parameter identification. Our synchronization criteria are easily verified and do not need to solve any linear matrix inequality. These results generalize a few previous known results and remove some restrictions on amplification function and time delay. This research also demonstrates the effectiveness of application in secure communication. Numerical simulations are carried out to illustrate the main results.     

Arbitrary observer scaling of all chaotic drive system states via a scalar synchronizing signal

This paper introduces a straightforward method to arbitrarily scale a drive system attractor using a synchronized linear observer. This scaling is controlled by a single observer parameter. Theoretical and simulation results for both continuous- and discrete-time systems demonstrate that linear observers can duplicate all chaotic system states in any desired scale using only a scalar synchronizing signal.     

Numerical stability of chaotic synchronization using a nonlinear coupling function

Nonlinear coupling has been used to synchronize some chaotic systems. The difference evolutional equation between coupled systems, determined via the linear approximation, can be used to analyze the stability of the synchronization between drive and response systems. According to the stability criteria the coupled chaotic systems are asymptotically synchronized, if all eigenvalues of the matrix found in this linear approximation have negative real parts. There is no synchronization, if at least one of these eigenvalues has positive real part. Nevertheless, in this paper we have considered some cases on which there is at least one zero eigenvalue for the matrix in the linear approximation. Such cases demonstrate synchronization-like behavior between coupled chaotic systems if all other eigenvalues have negative real parts.     

Observer-based synchronization of uncertain chaotic system and its application to secure communications

Within the drive-response configuration, this paper considers the synchronization of uncertain chaotic systems based on observers and chaos-based secure communication. Even if there are unknown disturbances and parameters in the drive system, a robust adaptive observer can be used as response system to realize chaotic synchronization. The proposed method is then applied to secure communication. The transmitter is constructed by injecting the information into the drive system with proper manner and one of the transmitting signal is the sum of one of the output and the information signal. The Lur’e chaotic system is considered as an illustrative example to demonstrate the effectiveness of the proposed approaches.     

Synchronization Criterion for Lur’e Systems via Delayed PD Controller

In this paper, the effects of a time varying delay on a chaotic drive-response synchronization are considered. Using a delayed feedback proportional-derivative (PD) controller scheme, a delay-dependent synchronization criterion is derived for chaotic systems represented by the Lur’e system with sector and slope restricted nonlinearities. The derived criterion is a sufficient condition for the absolute stability of the error dynamics between the drive and the response systems. By the use of a convex representation of the nonlinearity and the discretized Lyapunov-Krasovskii functional, stability condition is obtained via the LMI formulation. The condition represented in the terms of linear matrix inequalities (LMIs) can be solved by the application of convex optimization algorithms. The effectiveness of the work is verified through numerical examples.     

Projective synchronization of time‒delayed chaotic systems with unknown parameters using adaptive control method

The present article aims to study the projective synchronization between two identical and non?identical time?delayed chaotic systems with fully unknown parameters. Here the asymptotical and global synchronization are achieved by means of adaptive control approach based on Lyapunov–Krasovskii functional theory. The proposed technique is successfully applied to investigate the projective synchronization for the pairs of time?delayed chaotic systems amongst advanced Lorenz system as drive system with multiple delay Rössler system and time?delayed Chua's oscillator as response system. An adaptive controller and parameter update laws for unknown parameters are designed so that the drive system is controlled to be the response system. Numerical simulation results, depicted graphically, are carried out using Runge–Kutta Method for delay?differential equations, showing that the design of controller and the adaptive parameter laws are very effective and reliable and can be applied for synchronization of time?delayed chaotic systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Generalizations of the concept of marginal synchronization of chaos

Generalizations of the concept of marginal synchronization between chaotic systems, i.e. synchronization with zero largest conditional Lyapunov exponent, are considered. Generalized marginal synchronization in drive–response systems is defined, for which the function between points of attractors of different systems is given up to a constant. Auxiliary system approach is shown to be able to detect this synchronization. Marginal synchronization in mutually coupled systems which can be viewed as drive–response systems with the response system influencing the drive system dynamics is also considered, and an example from solid-state physics is analyzed. Stability of these kinds of synchronization against changes of system parameters and noise is investigated. In drive–response systems generalized marginal synchronization is shown to be rather sensitive to the changes of parameters and may disappear either due to the loss of stability of the response system, or as a result of the blowout bifurcation. Nonlinear coupling of the drive system to the response system can stabilize marginal synchronization.