Identical Chaotic Systems Research Articles

Chaotic dynamics from a pseudo-linear system

Abstract Investigating the possibility of applying techniques from linear systems theory to the setting of non-linear systems has been the focus of many papers. The pseudo-linear (PL) form representation of non-linear dynamical systems has led to the concept of non-linear eigenvalues (NEValues) and non-linear eigenvectors (NEVectors). When the NEVectors do not depend on the state vector of the system, then the NEValues determine the global qualitative behaviour of a non-linear system throughout the state space. The aim of this paper is to use this fact to construct a non-linear dynamical system of which the trajectories of the system show continual stretching and folding. We first prove that the system is globally bounded. Next we analyse the system numerically by studying bifurcations of equilibria and periodic orbits. Chaos arises due to a period doubling cascade of periodic attractors. Chaotic attractors are presumably of Hénon-like type, which means that they are the closure of the unstable manifold of a saddle periodic orbit. We also show how PL forms can be used to control the chaotic system and to synchronize two identical chaotic systems.

Synchronisation of fractional-order complex systems and its application

In this paper, a passive control scheme based on the fractional-order calculus is proposed. We study the modified complex projective synchronisation between two identical fractional-order complex chaotic systems, and its application in the secure communication. The fractional-order complex chaotic Lorenz system is employed to encrypt the emitted signal. In the transmitter module, the information signal is modulated into one parameter of the Lorenz system. It is assumed that the same parameter is unknown in the receiver module. In order to synchronise two systems with different initial conditions, the controllers and an appropriate parameter update rule are designed. Theoretical analysis and numerical simulations show that this method is feasible and robust to some extent in the presence of channel noise.

Fuzzy‐model‐based sampled‐data chaotic synchronisation under the input constraints consideration

In this study, the authors propose a novel sampled-data fuzzy chaotic synchronisation scheme under the input constraints consideration. The sampled-data chaos synchronisation controller feedbacks the synchronisation error between the state vectors of both the drive chaotic system and the response chaotic system at a constant sampling period. The chaotic synchronisation is achieved by stabilising the synchronisation error dynamics based on the H-infinity criterion. Linear matrix inequality-based sufficient conditions for synchronising two identical chaotic systems are derived based on the newly proposed time-dependent fuzzy Lyapunov–Krasovskii functional. Unlike previous approaches, the modelling error term is fully addressed so as to enhance the synchronisation performance. Finally, the effectiveness of the proposed method is validated through a numerical example.

Exponential Synchronization of Chaotic Xian System Using Linear Feedback Control

In this paper, a new linear feedback controller for synchronization of two identical chaotic systems in a master‐slave configuration is presented. This controller requires knowing a priori Lipschitz constant of the nonlinear function of the chaotic system on its attractor. The controller development is based on an algebraic Riccati equation. If the gain matrix and the matrices of Riccati equation are selected in such a way that a unique positive definite solution is obtained for this equation, then, with respect to previous works, a stronger result can be guaranteed here: the exponential convergence to zero of the synchronization error. Additionally, the nonideal case is also studied, that is, when unmodeled dynamics and/or disturbances are present in both master system and slave system. On this new condition, the synchronization error does not converge to zero anymore. However, it is still possible to guarantee the exponential convergence to a bounded zone. Numerical simulation confirms the satisfactory performance of the suggested approach.

Chaotic financial system with uncertain parameters - its control and synchronisation

This paper examines the control and synchronisation of the chaotic financial system with uncertain parameters as well as known parameters using adaptive control technique. The designed adaptive controller control and globally synchronises two identical chaotic financial systems evolving from different initial conditions. The designed controller is capable of stabilising the financial system at any position as well as controlling it to track any trajectory that is a smooth function of time. Numerical simulations are presented to demonstrate the feasibility of the proposed schemes.

Complete synchronisation of supply chain system using adaptive integral sliding mode control method

In this paper, the synchronisation problem of supply chain system (SCS) is addressed with active and adaptive integral sliding mode controlling method. Active integral sliding mode synchronisation is performed for two identical supply chain systems by assuming that all parameters of the systems are known. When the internal and external distortion parameters of the system are considered unknown, an appropriate feedback controller is developed based on the adaptive integral sliding mode control mechanism to synchronise two identical supply chain chaotic systems and to estimate the unknown parameters of the systems. The stability evaluation of the synchronisation methods is performed by the Lyapunov stability theorem. In addition, the performance evaluation of the designed controllers and the theoretical analysis are verified by some illustrative numerical simulations. Simulation results indicate excellent convergence from both speed and accuracy points of view.

Complete synchronisation of supply chain system using adaptive integral sliding mode control method

In this paper, the synchronisation problem of supply chain system (SCS) is addressed with active and adaptive integral sliding mode controlling method. Active integral sliding mode synchronisation is performed for two identical supply chain systems by assuming that all parameters of the systems are known. When the internal and external distortion parameters of the system are considered unknown, an appropriate feedback controller is developed based on the adaptive integral sliding mode control mechanism to synchronise two identical supply chain chaotic systems and to estimate the unknown parameters of the systems. The stability evaluation of the synchronisation methods is performed by the Lyapunov stability theorem. In addition, the performance evaluation of the designed controllers and the theoretical analysis are verified by some illustrative numerical simulations. Simulation results indicate excellent convergence from both speed and accuracy points of view.

A new three-dimensional chaotic system: its adaptive control and circuit design

A new three-dimensional chaotic system with two nonlinearities is proposed in this research work. Specifically, the two nonlinearities considered in the construction of the new chaotic system are a quadratic nonlinearity and a quartic nonlinearity. A systematic study of the three-dimensional chaotic system has been made including phase portraits, dissipativity, rest points and their stability, Lyapunov chaos exponents, Kaplan-Yorke fractal dimension, etc. As main control results, we design feedback control laws using adaptive control theory to achieve global stabilisation of the new chaotic system and also global synchronisation of identical chaotic systems with unknown parameters. Finally, an electronic circuit design of the new chaotic system using electronic work bench (EWB) is described in detail to confirm the feasibility of the theoretical chaotic model.

On the Discretization of Continuous-Time Chaotic Systems for Digital Implementations

Recently, many continuous-time chaotic systems were synthesized using microcontrollers and FPGAs. This requires applying mathematical discretization to convert integration into recursion. Depending on the approximation algorithms, the speed of the numerical processors, and the number of bits used to represent data, different accuracies and stabilities could be obtained. This article explores the conditions necessary to faithfully generate signals that reflect the true behavior of the chaotic systems, while maintaining the same values for their Lya-punov exponents. This is very important for chaos control and synchronization, especially for applications in secure communication that rely on digital cryptography. The Lorenz system and the Duffing oscillator are investigated to illustrate the effect of having autonomous versus non-autonomous structures. In addition, the Nosé-Hoover dynamical model is investigated to detect the relationship between agility and numerical accuracy. The obtained results prove that identical deterministic chaotic systems can behave differently, for the same set of initial conditions, depending on the discretization algorithm. This added sensitivity necessitates careful design of the mathematical models required for digital implementations of chaotic systems. The article concludes with useful recommendations for best practices in designing, synthesizing, and implementing digital chaotic systems, while commenting on the best compromise between mathematical complexity and numerical accuracy.

Improved synchronization criteria of Lur’e systems under sampled-data control

In this paper, we address the issue of $$\mathcal {H}_{\infty }$$ synchronization for two identical chaotic systems with time-varying delays and uncertainties under sampled-data control. By developing some new terms, an improved piecewise Lyapunov–Krasovskii functional (LKF) is constructed to take full advantage of characteristic about real sampling and nonlinear function vector. Furthermore, some relaxed matrices constructed in LKF are not necessarily positive definite. By using the LKF and free-matrix-based integral inequality, some sufficient criteria can be obtained to ensure the stability of error systems and reduce the influence of external disturbances with an $$\mathcal {H}_{\infty }$$ norm bound. The sampled-data controller can be synthesized by solving a group of linear matrix inequalities with the maximal sampling interval. Finally, the numerical examples are considered and analyzed by the proposed approach so as to show the benefit and the superiority of the proposed approach.

On multi switching compound synchronization of non identical chaotic systems

Multi switching compound synchronization (MSCOS) between four non identical chaotic systems is investigated. Chen system is considered as scaling drive system. Lorenz and Lü systems are taken as base drive systems. The compound of the multi drive system is then synchronized with Rössler response system using multi switching synchronization scheme. In MSCOS, the state variables of three drive systems synchronize with different state variables of response system, simultaneously. For suitable choice of scaling factors, switched modified function projective synchronization is obtained as a special case of MSCOS among chaotic systems. To achieve the desired synchronization among four non identical chaotic systems, sufficient condition is obtained using a nonlinear controller and Lyapunov stability theory, and corresponding theoretical proofs are given. The effectiveness of the proposed controllers is verified by numerical simulations using MATLAB.

Robust predictive synchronization of uncertain fractional-order time-delayed chaotic systems

In this paper, a novel robust predictive control strategy is proposed for the synchronization of fractional-order time-delay chaotic systems. A recurrent non-singleton type-2 fuzzy neural network (RNT2FNN) is used for the estimation of the unknown functions. Additionally, another RNT2FNN is used for the modeling of the tracking error. A nonlinear model-based predictive controller is then designed based on the proposed fuzzy model. The asymptotic stability of the approach is derived based on the Lyapunov stability theorem. A number of simulation examples are presented to verify the effectiveness of the proposed control method for the synchronization of two uncertain fractional-order time-delay identical and nonidentical chaotic systems. The proposed control strategy is also employed for high-performance position control of a hydraulic actuator. In this example, the nonlinear mechanical model of the hydraulic actuator, instead of a mathematical model, is simulated. The example demonstrates that the proposed control strategy can be applied to a wide class of nonlinear systems.

Adaptive neural network synchronization for uncertain strick-feedback chaotic systems subject to dead-zone input

In this paper, an adaptive neural network (NN) synchronization controller is designed for two identical strict-feedback chaotic systems (SFCSs) subject to dead-zone input. The dead-zone models together with the system uncertainties are approximated by NNs. The dynamic surface control (DSC) approach is applied in the synchronization controller design, and the traditional problem of “explosion of complexity” that usually occurs in the backstepping design can be avoided. The proposed synchronization method guarantees the synchronization errors tend to an arbitrarily small region. Finally, this paper presents two simulation examples to confirm the effectiveness and the robustness of the proposed control method.

Global Finite-Time Multi-Switching Synchronization of Externally Perturbed Chaotic Oscillators

Quick recovery of the information signals in secure communications restricts the hacking duration. The short synchronization convergence time is a crucial parameter for faster recovery. This paper develops a novel nonlinear finite-time synchronization control algorithm. This controller accomplishes the global finite-time multi-switching synchronization between two externally perturbed chaotic systems in the drive–response system synchronization scheme. The proposed controller assures the global convergence of the error dynamics in finite-time based on the Lyapunov theory. This implicitly guarantees the global stability of the closed loop. This paper considers Lp(0 < p < 1) norm inequality for the construction of the Lyapunov function. This method provides a means to determine the parameters of the proposed finite-time controller. This paper also studies the significance of structural components of the proposed controller that are responsible for the finite-time synchronization convergence. Computer simulation results of ‘two identical chaotic Lorenz systems’ and ‘chaotic Lorenz and Chen systems’ validate the theoretical findings. The paper discusses the simulation results and compares them with peer works as well.

Robust synchronization of uncertain fractional-order chaotic systems with time-varying delay

This paper presents a new technique using a recurrent non-singleton type-2 sequential fuzzy neural network (RNT2SFNN) for synchronization of the fractional-order chaotic systems with time-varying delay and uncertain dynamics. The consequent parameters of the proposed RNT2SFNN are learned based on the Lyapunov–Krasovskii stability analysis. The proposed control method is used to synchronize two non-identical and identical fractional-order chaotic systems, with time-varying delay. Also, to demonstrate the performance of the proposed control method, in the other practical applications, the proposed controller is applied to synchronize the master–slave bilateral teleoperation problem with time-varying delay. Simulation results show that the proposed control scenario results in good performance in the presence of external disturbance, unknown functions in the dynamics of the system and also time-varying delay in the control signal and the dynamics of system. Finally, the effectiveness of proposed RNT2SFNN is verified by a nonlinear identification problem and its performance is compared with other well-known neural networks.

Difference Synchronization of Identical and Nonidentical Chaotic and Hyperchaotic Systems of Different Orders Using Active Backstepping Design

This paper introduces a new type of synchronization scheme, referred to as difference synchronization scheme, wherein the difference between the state variables of two master [slave] systems synchronizes with the state variable of a single slave [master] system. Using the Lyapunov stability theory and the active backstepping technique, controllers are derived to achieve the difference synchronization of three identical hyperchaotic Liu systems evolving from different initial conditions, as well as the difference synchronization of three nonidentical systems of different orders, comprising the 3D Lorenz chaotic system, 3D Chen chaotic system, and the 4D hyperchaotic Liu system. Numerical simulations are presented to demonstrate the validity and feasibility of the theoretical analysis. The development of difference synchronization scheme has increases the number of existing chaos synchronization scheme.

Stability of dynamical behavior of a new hyper chaotic system in certain range and its hybrid projective synchronization behavior

This article gives an insight for important tools to study the stability of dynamical behavior of newly constructed hyper chaotic systems within some certain parameter range. Further, hybrid projective synchronization (HPS) of two identical new hyper chaotic systems is defined and scheme of HPS is developed by using tracking control method. A new hyper chaotic system has been constructed and then response system. Numerical simulations verify the effectiveness of this scheme, which has been performed by mathematica and MATLAB.

A Novel Adaptive Active Control Projective Synchronization of Chaotic Systems

This paper investigates adaptive active control projective synchronization scheme. A general synchronization controller and parameter update laws are proposed to stabilize the error system for the identical structural chaotic systems. It is the first time that the active synchronization, the projective synchronization, and the adaptive synchronization are combined to achieve the synchronization of chaotic systems, which extend the control capability of achieving chaotic synchronization. By using a constant diagonal matrix, the active control is developed. Especially, when designing the controller, we just need to ensure that the diagonal elements of the diagonal matrix are less than or equal 0. So, the synchronization of chaotic systems can be realized more easily. Furthermore, by proposing an active controller, in combination with several different control schemes, we lower the complexity of the design process of the controller. More importantly, the larger the absolute value of product of the diagonal elements of diagonal matrix is, the smoother the curve of chaotic synchronization is and the shorter the time of chaotic synchronization is. In our paper, we take Lorenz system as an example to verify the effectiveness of the proposed synchronization scheme. Theoretical analysis and numerical simulations demonstrate the feasibility of this control method.

Comparative performances of synchronisation between different classes of chaotic systems using three control techniques

This paper puts forward the comparative performances for synchronisation between (i) systems from different chaotic system families, (ii) systems from the Unified Chaotic System (UCS) family, (iii) a hyperchaotic and a chaotic systems and (iv) identical chaotic systems. Three different well-known control techniques, i.e. Nonlinear Active Control (NAC), Sliding Mode Control (SMC) and Adaptive Control (AC) are used for synchronisation between various pairs of chaotic and/or hyperchaotic systems. Performances of NAC, SMC and AC techniques are investigated and compared with synchronisation of different pairs of chaotic systems based on the error dynamics and required control inputs. The integral square error and required control energy measures are considered for comparison. Finally, a generalised view on the use of the control techniques for synchronisation is finally proposed. Moreover, a new chaotic system is proposed and its qualitative analysis is done to illustrate the chaotic behaviour of the system. The new system is used as an example of synchronisation. MATLAB simulation results are presented which reflect the successful achievement of the objectives.

A novel four-dimensional conservative chaotic system without linear term, its analysis and adaptive control via integral sliding mode control

In this work, a novel conservative four-dimensional chaotic system without linear term is proposed. The fundamental qualitative properties of the novel chaotic system are described with details of volume preserving property, equilibrium points, symmetry, Lyapunov exponents and Kaplan-Yorke dimension. We show that the novel four-dimensional chaotic system has two planes and one line of equilibrium points. Next, an adaptive integral sliding mode controller is designed to stabilise the novel chaotic system with an unknown system parameter. Moreover, an adaptive integral sliding mode controller is designed to achieve global chaos synchronisation of the identical novel chaotic systems with an unknown system parameter. The adaptive control mechanism helps the control design by estimating the unknown parameter. Numerical simulations using MATLAB are shown to illustrate all the main results derived in this work.