A general method for chaos synchronization and parameters estimation between different systems

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.

This paper investigates the hybrid chaos synchronization of identical Liu systems, identical Lu systems, and non-identical Liu and Lu systems by active nonlinear control. Liu system (Liu et al. 2004) and Lu system (Lu and Chen, 2002) are important models of three-dimensional chaotic systems. Hybrid synchronization of the three-dimensional chaotic systems considered in this paper are achieved through the synchronization of the first and last pairs of states and anti-synchronization of the middle pairs of the two systems. Sufficient conditions for hybrid synchronization of identical Liu, identical Lu, and non-identical Liu and Lu systems are derived using active nonlinear control and Lyapunov stability theory. Since the Lyapunov exponents are not needed for these calculations, the active nonlinear control is an effective and convenient method for the hybrid synchronization of the chaotic systems addressed in this paper. Numerical simulations are shown to illustrate the effectiveness of the proposed synchronization schemes.

In the era of big data, vast amounts of digital images are transmitted over the Internet. To safeguard the confidentiality of these images, this paper proposes a multiple image encryption algorithm (MIEA) based on 3D non-equilateral Arnold transformation (3D-NEAT) and a chaotic system, capable of encrypting any number of images simultaneously. First, the K plaintext images are superimposed to form a plaintext cube image. Next, 3D-NEAT is applied to scramble the 3D cube plaintext image at the bit level. Finally, chaotic sequences generated by Chen’s hyper-chaotic system (CHCS) are used to further confuse and diffuse the scrambled 3D cube image, resulting in a 3D cube encrypted image. Experimental simulations and security analyses demonstrate that the 3D ciphertext image produced by the proposed algorithm can withstand various security and statistical analyses and shows strong robustness against occlusion and noise attacks. These results indicate that the algorithm is both secure and efficient, making it suitable for practical applications. Mathematics Subject Classi cation (2020) MSC code1 · MSC code2 · more

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.

Chaos synchronization and anti-synchronization of two chaotic systems with different structures are investigated. When the parameters are known in advance, the synchronization and anti-synchronization are realized by the active control. The method of designing controllers with active control is generalized. When the parameters are completely unknown, the adaptive controllers and the adaptive laws of parameters are given based on the Lyapunov stability theory and adaptive control method, which realizes adaptive synchronization and anti- synchronization of two different chaotic systems, and identifies the unknown parameters. The drive system and response system are interchanged during the controller designing. The relationship of the controllers and the adaptive laws before and after the interchangement are investigated. The numerical simulation results verify the effectiveness of the proposed scheme.

The article aims to study the reduced-order anti-synchronization between projections of fractional order hyperchaotic and chaotic systems using active control method. The technique is successfully applied for the pair of systems viz., fractional order hyperchaotic Lorenz system and fractional order chaotic Genesio-Tesi system. The sufficient conditions for achieving anti-synchronization between these two systems are derived via the Laplace transformation theory. The fractional derivative is described in Caputo sense. Applying the fractional calculus theory and computer simulation technique, it is found that hyperchaos and chaos exists in the fractional order Lorenz system and fractional order Genesio-Tesi system with order less than 4 and 3 respectively. The lowest fractional orders of hyperchaotic Lorenz system and chaotic Genesio-Tesi system are 3.92 and 2.79 respectively. Numerical simulation results which are carried out using Adams-Bashforth-Moulton method, shows that the method is reliable and effective for reduced order anti-synchronization.

Adaptive hybrid synchronization of two different chaotic systems with unknown parameters is investigated. Based on Lyapunov satiability theory, an adaptive hybrid synchronization controller and parameter update laws are designed analytically. Our scheme includes complete synchronization, anti-synchronization and projective synchronization, simultaneously. Numerical simulations on the hyper-chaotic Chen system and hyper-chaotic Lü system are presented to verify the effectiveness of the proposed scheme.

In this paper, we present the Takagi–Sugeno (T–S) fuzzy observed based design and synchronization of chaotic and hyper chaotic dynamical systems. Using T–S fuzzy observed based design, the fuzzy logic controllers for chaotic synchronization are constructed via linear matrix inequality (LMI). Lyapunov exponents of 3D and 4D Chen systems have been calculated. This represents the chaoticity and hyper chaoticity of Chen systems. Analytical and computational studies of chaotic and hyper chaotic Chen systems have been performed by using LMI toolbox. The qualitative and simulated results show the validity and the performance of the proposed observer in an excellent agreement.

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.

HYPERCHAOS IN THE FRACTIONAL-ORDER NONAUTONOMOUS CHEN'S SYSTEM AND ITS SYNCHRONIZATION

Secured Compressive sensing of the images becomes one of the essential issues in multimedia applications. In the recent times, encryption in compressive sensing is achieved via the use of multiple one dimensional chaotic system (1D chaotic) and hyper-chaotic system (HC). However, the security of the system still needs to be improved. To solve this issue, novel encrypted compressive sensing of images based on Fractional order hyper chaotic Chen system and DNA operations is proposed in this paper. The basic idea is to introduce a new algorithm which provides compression rate below the Nyquist rate along with increased security by jointly using Fractional order hyper chaotic chen system and DNA operations for image encryption. The encryption algorithm combines Fractional order hyperchaotic chen system and DNA operations. Fractional order hyperchaotic chen system has high randomness and rich dynamic phenomena which enhance the encryption and the algorithm efficiency is further increased by DNA operations.4-Dimensioanal Fractional order hyper chaotic chen system is used to generate the measurement matrix. Compressive sensing measurements are converted to a stream of binary digits and the correlation between the adjacent bits is further reduced through global scrambling. The DNA operations are performed on the scrambled binary sequences and hyper chaotic sequences, which increases the algorithm efficiency. The results of the experiments accomplish that the proposed encryption method is extremely sensitive to small changes in secret keys, shows good performance in histogram analysis, correlation analysis, Information Entropy and is very sensitive to a bit change in an input image. The Block Compressive sensing (BCS) reconstruction algorithms are used to validate the proposed encryption method. In the proposed method, experimental analysis are performed on different images of size 512 × 512 which are divided in to blocks of size 32 X 32. The reconstruction analysis is performed with different subsampling rates of 0.1, 0.2, 0.3, 0.4 and 0.5.The results depicted that the proposed method maintains the robustness and reconstruction quality of Compressive sensing with enhanced encryption.

In this paper, a synchronization scheme for six non-identical hyperchaotic systems is investigated. The paper addresses a synchronization scheme of five leader (master) and one follower (slave) non-identical hyperchaotic systems. This synchronization scheme is first of its kind and different from existing schemes as instead of a scaling system a compound scaling signal is used. The presented scheme is termed as ‘multi-switching compound-compound anti-synchronization’ (MCCAS). Asymptotically stable synchronized states are established between leader and follower systems by employing nonlinear control method and Lyapunov stability criteria. Additional leader systems and hyperchaotic systems enhance the complexity of this synchronization scheme which is useful in secure communication. The information signal can be split and loaded with different signals of leader systems to increase the grade of security. To justify the theoretical results, an example is considered where Pang-Liu, Zheng, Chen, Liu and Newton-Leipnik hyperchaotic systems are considered as leader systems and Lorenz hyperchaotic system is considered as a follower system. Theoretical results are confirmed by graphical analysis.

To solve the problem of existing chaotic systems with unknown nonlinearities, enormous parameters and external disturbances, in this paper, a synchronization controller with parameter adaptive laws is proposed based on nested back-propagation neural networks and the adaptive method, where the nested back-propagation neural networks are used to approximate the unknown nonlinearities based on same experiences and the unknown parameters are estimated by the adaptive method. Then the asymptotical synchronization of the drive-response chaotic systems is synthesized via state feedback controllers and updated adaptive laws. Specifically, the nested back-propagation neural networks are developed by grouping and layering the hidden neurons using the principle of partition of unity and the state domain for modularizing the concealed layer. Finally, a numerical example is given to illustrate the effectiveness of this method.

This paper addresses a unified mathematical expression describing a class of chaotic systems, for which the problem of synchronization and anti-synchronization between different chaotic systems with fully uncertain parameters and different structure are studied. Based on the Lyapunov stability theory, a novel, simple, and systemic adaptive synchronization controller is designated, the analytic expression of the controller and the adaptive laws of parameters are developed. Moreover, the proposed scheme can be extended to anti-synchronize a class of chaotic systems. Two chaotic systems with different structure and fully uncertain parameters are employed as the examples to show the effectiveness of the proposed adaptive synchronization and anti-synchronization schemes. Additionally, the robustness and noise immunity of the adaptive synchronization scheme is investigated by measuring the mean squared error of the systems.

Multi-switching combination synchronization of three different chaotic systems via nonlinear control