Adaptive hybrid synchronization of two different chaotic systems with unknown parameters

A new adaptive iterative learning control (AILC) scheme is proposed to solve the finite-time hybrid function projective synchronization (HFPS) problem of chaotic systems with unknown periodic time-varying parameters. Fourier series expansion (FSE) is introduced to deal with the problem of uncertain time-varying parameters. The bound of the expanded remaining items is unknown. A typical convergent series is used to deal with the unknown bound in the design process of the controller. The adaptive iterative learning synchronization controller and parameter update laws are designed. Two different chaotic systems are synchronized asymptotically according to different proportional functions on a finite time interval by Lyapunov stability analysis. The simulation example proves the feasibility and effectiveness of the proposed method.

This paper mainly investigates adaptive hybrid projective synchronization of two different hyperchaotic systems with fully unknown parameters. Based on Lyapunov stability theory, adaptive synchronization controllers and parameter update laws are designed such that the states of the drive system and the response system are ultimately asymptotically hybrid projective synchronized. Simulation results are presented to show the feasibility and effectiveness of the proposed method.

Adaptive control and synchronization of a new hyperchaotic system with unknown parameters

In this paper, we proposed the FPGA implementation of complete synchronization methodology for a novel chaotic systems using adaptive control scheme. Chaotic systems are non linear dynamical system having at least one positive Lyapunov exponents. In this paper, first we derived a new adaptive controller and parameter update law to synchronize the identical novel chaotic system using adaptive control theory and Lyapunov stability theorem. Then the novel chaotic system, adaptive controllers and the synchronized chaotic system are implemented in FPGA vertex 6 using Xilinx System Generator technology. After implementation, the RTL schematic diagram, the source consumed by the novel chaotic system and the synchronized systems are obtained. The Isim smulation results are also given to show the effectiveness of our proposed synchronization methodology.

To solve the problem of indeterminate synchronization time in different chaotic systems, this paper presents a time-controllable synchronization scheme. A general synchronization controller and parameter update laws are proposed to stabilize the error system, thus the drive and response systems could be synchronized up to a given scaling matrix at a pre-specified exponential convergence rate. The synchronization time formula is strictly deduced, which suggests that the speed of synchronization is determined by several parameters, such as exponential rate, initial system value and other parameters brought in by the controller. By adjusting these parameters, the performance of the synchronization can be effectively improved. In numerical simulation, two nonidentical 3D autonomous chaotic systems are chosen to verify this method. The error system can be rapidly stabilized, and unknown parameters are also identi?ed correctly. Firally, two groups of time-controllable parameters are given to verify the theory, wherein synchronization of both cases can be obtained quickly and each result of the synchronization is consistent with the theoretical calculation. The synchronization scheme is characterized by high safety and efficiency, and has its potential value in secure communication.

This paper addresses an adaptive fuzzy feedback controller design problem for finite-time Mittag-Leffler synchronization (FTMLS) of fractional-order quaternion-valued reaction-diffusion T-S fuzzy molecular modeling of delayed neural networks. A novel approach is proposed to effectively deal with the joint effects from fuzzy rules and reaction-diffusion terms for the class of T-S fuzzy fractional-order reaction-diffusion delayed quaternion-valued neural networks (FORDDQVNNs) under consideration. By employing Lyapunov stability theory, Caputo fractional derivative, several algebraic criteria are established to guarantee the FTMLS of T-S fuzzy FORDDQVNNs via designed fuzzy feedback controller. Moreover, the adaptive controller and parameter update laws are designed via adaptive control methods. Compared with existing results in the literature, we also show that our results are less conservative than existing ones with these illustrative T-S fuzzy FORDDQVNNs. A numerical example is presented to verify the analysis results and illustrate the effectiveness of the proposed FTMLS conditions.

A lot of progress has been made in the research of hybrid function projective synchronization (HFPS) for chaotic real nonlinear systems, while the HFPS of two different chaotic complex nonlinear systems with nonidentical dimensions is seldom reported in the literatures. So this paper discusses the HFPS of general chaotic complex system described by a unified mathematical expression with different dimensions and fully unknown parameters. Based on the Lyapunov stability theory, the adaptive controller is designed to synchronize two general uncertain chaotic complex systems with different orders in the sense of HFPS and the parameter update laws for estimating unknown parameters of chaotic complex systems are also given. Moreover, the control coefficients can be automatically adapted to updated laws. Finally, the HFPS between hyperchaotic complex Lorenz system and complex Chen system and that between chaotic complex Lorenz system and hyperchaotic complex Lü are taken as two examples to demonstrate the effectiveness and feasibility of the proposed HFPS scheme.

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.

The stability of a linear generalized synchronization approach between different chaot- ic systems with unknown parameters is investigated in this paper. An adaptive controller and some parameter update laws are proposed to achieve stable linear generalized synchronization and parame- ter identification of different chaotic systems. This novel synchronization method effectively unifies complete synchronization, anti-synchronization, projective synchronization, rotation synchronization and function synchronization. The main results are illustrated by two examples: the rotation synchro- nization between Lorenz and Chen system; the modified projective synchronization between Lorenz system and Lu system.

To explore the impact of unknown terms and parameters on chaotic characteristics in chaotic systems, this paper examines the effects of state variables and unknown parameters. The study focuses on different combinations of linear, nonlinear, and constant terms It primarily investigates the role of multi-order state variables and their application to chaotic system models of varying dimensions. Firstly, by simulating a three-dimensional chaotic system, the paper analyzes how different combinations of nonlinear terms and initial conditions affect the system's chaotic behavior. Secondly, it evaluates the chaotic characteristics of a four-dimensional system, combining nonlinear terms with unknown parameters, using tools such as Lyapunov index diagrams, sample entropy, and dynamic trajectory plots. Finally, the paper integrates the constructed chaotic system with chaotic mapping to develop a two-level key chaotic image encryption system, thoroughly assessing its security and resistance to interference.

Function projective synchronization (FPS) of two novel hyperchaotic systems with four-scroll attractors which have been found up to the present is investigated. Adaptive control is employed in the situation that system parameters are unknown. Based on Lyapunov stability theory, an adaptive controller and a parameter update law are designed so that the two systems can be synchronized asymptotically by FPS. Numerical simulation is provided to show the effectiveness of the proposed adaptive controller and the parameter update law.

Adaptive synchronization between two different noise-perturbed chaotic systems with fully unknown parameters

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 studies the robust adaptive full state hybrid projective synchronization (FSHPS) scheme for a class of chaotic complex systems with uncertain parameters and external disturbances. By introducing a compensator and using nonlinear control and adaptive control, the robust adaptive FSHPS scheme is derived, which can eliminate the influence of uncertainties effectively and achieve adaptive FSHPS of the chaotic (hyperchaotic) complex systems asymptotically with a small error bound. The adaptive laws of the unknown parameters are given, and the sufficient conditions of realizing FSHPS are derived as well. Moreover, we also discuss the case that parameters of chaotic complex system are complex. Finally, the complex Chen system and Lu system, and the hyperchaotic complex Lorenz system are taken as two examples and the numerical simulations are provided to verify the effectiveness and robustness of the proposed control scheme.

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