Synchronization of Chaotic Systems without Direct Connections Using Reinforcement Learning

This paper investigates the adaptive - synchronization of the fractional-order chaotic systems with nonidentical structures. Based on the stability of fractional-order systems and adaptive control technique, a general formula for designing the controller and parameters update law is proposed to achieve adaptive - synchronization between two different chaotic systems with different structures. The effective scheme parameters identification and - synchronization of chaotic systems can be realized simultaneously. Furthermore, two typical illustrative numerical simulations are given to demonstrate the effectiveness of the proposed scheme, for each case, we design the controller and parameter update laws in detail. The numerical simulations are performed to verify the effectiveness of the theoretical results.

This paper proposes a new finite-time controller that realizes multi-switching synchronization of chaotic systems with bounded disturbances using the drive and response system synchronization arrangement. The finite-time controller derives the synchronization error to zero within a specified time. The proposed controller consists of three basic terms; each of them accomplishes a distinct objective: (1) stability of the control loop, (2) smooth and fast convergence behavior of the synchronization error, and (3) disturbance rejection. This study also devises a methodology for designing the finite-time controller and describes a general approach that furnishes a systematic procedure for the analysis of the closed loop. The smooth behavior terminology introduced in (2) refers to the over-damped convergence of the synchronization error signals to zero and the synthesis of chattering-free control effort. The analysis, which assures the global stability of the closed loop, uses the second stability theorem of the Lyapunov, while the finite-time stability technique determines the finite-time convergence. This paper includes simulations of two numerical examples for the validation of the theoretical findings and discusses the comparative analysis. The proposed methodology is suitable to design controllers for a wide range of hyper(chaotic) systems. The contributions of the paper are: (1) describe an innovative generalize analytical methodology for the multi-switching combination synchronization of chaotic systems and (2) propose a novel controller design that insures the finite-time synchronization.

The coexistence of anti-synchronization and synchronization in chaotic systems is investigated. A novel algorithm is proposed to determine the variables of the master system that should anti-synchronize with corresponding variables of the slave system. Control strategies that guarantee the coexistence of synchronization and anti-synchronization in the unified chaotic system are presented; while numerical simulations are employed to validate and illustrate the effectiveness of the proposed method.

Presented in this paper, is an adaptive control scheme and a parameters update law based upon the Lyapunov stability theorem. It is intended for the synchronization of a new chaotic system with unknown parameters. Feed back control system for synchronization of chaotic system is proposed by using complete synchronization approach. In the proposed method, each system is constructed with MATLAB simulink blocks, and then Simulink designs are translated into System Generator design so that bitstream file used to program FPGA is obtained. Finally, the design is implemented into FPGA by downloading bitstream file into FPGA. As an application of FPGAs the synchronization of chaotic systems has been achieved. Additionally, we show the effectiveness of proposed method by the use of numerical solution.

Tracking control for the output using an observer-based H ∞ fuzzy synchronization of time-varying delayed discrete- and continuous-time chaotic systems is proposed in this paper. First, from a practical point of view, the chaotic systems here consider the influence of time-varying delays, disturbances, and immeasurable states. Then, to facilitate a uniform control design approach for both discrete- and continuous-time chaotic systems, the dynamic models along with time-varying delays and disturbances are reformulated using the T-S (Takagi–Sugeno) fuzzy representation. For control design considering immeasurable states, a fuzzy observer achieves master-slave synchronization. Third, combining both a fuzzy observer for state estimation and a controller (solved from generalized kinematic constraints) output tracking can be achieved. To make the design more practical, we also consider differences of antecedent variables between the plant, observer, and controller. Finally, using Lyapunov’s stability approach, the results are sufficient conditions represented as LMIs (linear matrix inequalities). The contributions of the method proposed are threefold: (i) systemic and unified problem formulation of master-slave synchronization and tracking control for both discrete and continuous chaotic systems; (ii) practical consideration of time-varying delay, immeasurable state, different antecedent variables (of plant, observer, and controller), and disturbance in the control problem; and (iii) sufficient conditions from Lyapunov’s stability analysis represented as LMIs which are numerically solvable observer and controller gains from LMIs. We carry out numerical simulations on a chaotic three-dimensional discrete-time system and continuous-time Chua’s circuit. Satisfactory numerical results further show the validity of the theoretical derivations.

We examine the effect of additive noise and drift, in the dynamics of a chaotic driving signal, on the synchronization of chaotic response systems. Simple scaling laws associated with the synchronization deviation level under these types of contamination are presented. Time series used as the driving signals are experimentally measured from an electronic circuit and a mechanical system (a vibrating wire). The response systems are models that were obtained by fitting an ordinary differential equation to time series data. The possible relevance of this work to nondestructive testing, system identification, and communications is discussed. Finally, we present some results regarding the relationship between the synchronization deviation level and the coupling strength.

Sector-condition-based results for adaptive control and synchronization of chaotic systems under input saturation

In this paper a topic regarding the synchronization of chaotic systems is dealt with: the case of separation and synchronization of many chaotic signals generated by different chaotic circuits and combined together is examined. In particular, an observer based strategy has been adopted, and an approach for the simultaneous stabilization of many Luenberger observers has been investigated to face the problem of separation and synchronization. The design strategy is based on linear matrix inequalities (LMIs). Indeed, the LMI problem is referred to have a solution if a dual optimization problem admits a solution. In our case the feasibility condition, if it does exist, allows us to establish that the separation and synchronization problem for the chosen circuit admits a solution. Some numerical simulations are reported. Further results refer to an experimental circuit showing the suitability of the approach. Furthermore, the use of the proposed scheme to transmit two or more information masked into two or more multiplexed chaotic signals and the design of suitable parameters through the introduced technique based on LMIs are discussed.

Chaos has important applications in physics, chemistry, biology, ecology, secure communications, cryptosystems and many scientific branches. Control and synchronization of chaotic systems are important research problems in chaos theory. Sliding mode control is an important method used to solve various problems in control systems engineering. In robust control systems, the sliding mode control is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as insensitivity to parameter uncertainties and disturbance. In this work, we first investigate research in the dynamic analysis and properties of the enzymes-substrates reactions system with ferroelectric behaviour in the brain waves which was studied by Enjieu Kadji et al. (Chaos Solitons Fractals 32:862–882, 2007, [10]). Next, we apply multivariable super-twisting sliding mode control to globally stabilize all the trajectories of the enzymes-substrates biological chaotic system. Furthermore, we use multivariable super-twisting sliding mode control for the global chaos synchronization of identical enzymes-substrates biological chaotic systems. Super-twisting sliding mode control is very useful for the global stabilization and synchronization of the enzymes-substrates reaction system as it achieves finite time stability for the system. Numerical simulations using MATLAB are shown to illustrate all the main results derived in this work.

Chaotic neural networks consisting of a great number of chaotic neurons are able to reproduce the rich dynamics observed in biological nervous systems. In recent years, the memristor has attracted much interest in the efficient implementation of artificial synapses and neurons. This work addresses adaptive synchronization of a class of memristor-based neural chaotic systems using a novel adaptive backstepping approach. A systematic design procedure is presented. Simulation results have demonstrated the effectiveness of the proposed adaptive synchronization method and its potential in practical application of memristive chaotic oscillators in secure communication.

In recent years, a large amount of work on chaos-based cryptosystems has been published (Kocarev (2001); Millerioux et al. (2008)). A general methodology for designing chaotic and hyperchaotic cryptosystems has been developed using the control systems theory (Grassi et al. (1999); Liao et al. (1999); Yang et al. (1997a;b)). The chaotic communication system is closely related to the concept of chaos synchronization. An overview of chaotic secure communication systems can be found in (Yang (2004)). He classified the continuous-time chaotic secure communication systems into four generations. In the third generation, the combination of the classical cryptographic technique and chaotic synchronization is used to enhance the degree of security. Specifically, Yang et al. proposed a new chaos-based secure communication scheme in an attempt to thwart the attacks (Yang et al. (1997a;b)). They have combined both conventional cryptographic method and synchronization of chaotic systems. Their cryptographic method consists of an encryption function (the multi-shift cipher), a decryption function (the inverse of the encryption function), a chaotic encrypter that generates the key signal for the encryption function, and a decrypter that estimates the key signal. The approach has a limitation since the cryptosystem design may fail if different chaotic circuits are utilized. So far, this generation has the highest security in all the chaotic communication systems had been proposed and has not yet broken. From the control theoretic perspective, the transmitter and the receiver in the chaotic communication system can be considered as the nonlinear plant and its observer, respectively. Grassi et al. proposed a nonlinear-observer-based decrypter to reconstruct the state of the encrypter (Grassi et al. (1999); Liao et al. (1999)). They extended the Chua’s oscillator to the observer-based decrypter. The cryptosystem does not require initial conditions of the encrypter and the decrypter belonging to the same basin of attraction. If we can design a decrypter without the knowledge of the parameters of the encrypter, the chaos-based secure communication systems are not secure, because the parameters of the encrypter is selected as static secret keys in the cryptosystem. Parameter identification and adaptive synchronization methods may be effective for intruders in building reconstruction mechanisms, even when a synchronizing system is not available. Therefore, it is important for secure issues to investigate whether adaptive identifiers without the system information of encrypter can be constructed or not. We have recently designed an observer-based chaotic communication system combining the cryptosystems proposed by Grassi et al. (Grassi et al. (1999)) and by Liao et al. (Liao et al. (1999)) that allows us to assign the relative degree and the zeros of its encrypter system (Matsuo et al. (2004)). Specifically, we constructed three cryptosystems based on a Chua’s circuit by assigning its relative degree and zeros. The cryptosystem consists of 20

In this research work, we announce a six-term novel 3-D dissipative chaotic system with two quadratic nonlinearities. First, this work describes the dynamic equations and qualitative properties of the novel chaotic system. We show that the novel chaotic system has three unstable equilibrium points. We also show that the novel chaotic system has a rotation symmetry about the \(x_3\) axis. The Lyapunov exponents of the novel chaotic system are obtained as \(L_1 = 1.2334, L_2 = 0\) and \(L_3 = -4.7329\). Since the sum of the Lyapunov exponents is negative, the novel chaotic system is dissipative. Also, the Kaplan-Yorke dimension of the novel chaotic system is derived as \(D_{KY} = 2.2606\). Next, this work describes the active synchronization of identical novel chaotic systems with known parameters. Furthermore, this work describes the adaptive synchronization of identical novel chaotic systems with unknown parameters. Both the active and adaptive synchronization results are established using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this work for the six-term novel 3-D novel chaotic system.

This paper investigates the problem of fuzzy impulsive control to synchronize two chaotic systems using a novel time-dependent Lyapunov function approach. Compared with the existing time-independent Lyapunov methods, the proposed method enables us to exploit more information on the impulsive intervals. Initially, using the Lyapunov technique and two parameterized linear matrix inequality (LMI) techniques, some less conservative synchronization criteria via a fuzzy impulsive controller using the states of both drive and response chaotic systems are derived. Subsequently, an LMI approach to designing such a fuzzy impulsive controller is developed to realize the synchronization. Finally, the proposed method is applied to the chaotic Lorenz system and Rosler system to illustrate its effectiveness.

This paper investigates the adaptive chaos synchronization of a new chaotic system with uncertain parameters. Based on the passive control, a passivity-based adaptive control law is presented, which transforms the error dynamical system between the drive system and the response system into an equivalent passive system. Then it is proved that the equivalent system can be global asymptotically stabilized, which means the synchronization of the new chaotic system is achieved. Numerical simulations are presented to verify the analytical results.

A number of methods have been proposed for synchronizing chaotic systems. The most widely used methods are continuous synchronization schemes. In a continuous synchronization scheme, chaotic systems are coupled to each other continuously such that synchronization errors converge to zero. In this paper, chaos synchronization in coupled discrete-time dynamical systems is presented. Simulation results finally demonstrate the effectiveness of the method. Experimental results show that chaotic and hyperchaotic systems can be synchronized by impulses sampled from one or two state variables. The impulsive synchronization can be applied to almost all chaotic and hyperchaotic systems even in the case when continuous synchronization systems fail to work. In this paper the concept of chaotic communication is based on digital chaotic systems where dual receivers are used for evaluation of information. The main advantage of digital chaotic systems is simple reproduction of devices, simple parameter setting and parameter stability. The dual receiver is used for improved function and error detection.