Robust fuzzy control for chaotic (hyperchaotic) systems with parametric uncertainties based on LMI technique

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.

Controlling chaotic and hyperchaotic systems via energy regulation

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.

Hyperchaotic systems are defined as chaotic systems with more than one positive Lyapunov exponent. Combined with one null Lyapunov exponent along the flow and one negative Lyapunov exponent to ensure boundedness of the solution, the minimal dimension for a continuous hyperchaotic system is four. The hyperchaotic systems are known to have important applications in secure communications and cryptosystems. First, this work describes an eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities. The qualitative properties of the novel hyperchaotic system are described in detail. The Lyapunov exponents of the system are obtained as \( L_{1} = 0.7781,L_{2} = 0.2299,L_{3} = 0 \) and \( L_{4} = - 12.5062 \). The maximal Lyapunov exponent of the system (MLE) is \( L_{1} = 0.7781 \). The Lyapunov dimension of the novel hyperchaotic system is obtained as \( D_{L} = 3.0806 \). Next, the work describes an adaptive controller design for the global chaos control of the novel hyperchaotic system. The main result for the adaptive controller design has been proved using Lyapunov stability theory. MATLAB simulations are described in detail for all the main results derived in this work for the eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities.

This paper presents a novel seven-dimensional nonlinear hyperchaotic system characterized by a minimal number of nonlinear terms and variables, yet exhibiting high complexity. Standard nonlinear analysis is conducted to unveil the system’s intricacies, emphasizing its notable feature of possessing four to five Lyapunov exponents in certain intervals, signifying its volatility and complexity. Hyperchaotic synchronization is explored using a novel nonsingular terminal sliding control design, effectively achieving synchronization between two hyperchaotic master systems and a hyperchaotic slave system within finite time while mitigating the chattering phenomenon. Practical evaluations through orbital analysis, numerical simulations, and practical implementations further substantiate the efficacy and performance of the proposed system. This study contributes to the advancement of chaotic and hyperchaotic systems, particularly those with dimensions exceeding 5D, offering insights into synchronization techniques and practical applications in engineering and other scientific disciplines.

Synchronization of nonlinear dynamical systems with complex variables has attracted much more attention in various fields of science and engineering. In this paper, the problem of parameter identification and adaptive impulsive synchronization for a class of chaotic (hyperchaotic) complex nonlinear systems with uncertain parameters is investigated. Based on the theories of adaptive control and impulsive control, a synchronization scheme is designed to make a class of chaotic and hyperchaotic complex systems asymptotically synchronized, and uncertain parameters are identified simultaneously in the process of synchronization. Particularly, the proposed adaptive–impulsive control laws for synchronization are simple and can be readily applied in practical applications. The synchronization of two identical chaotic complex Chen systems and two identical hyperchaotic complex Lu systems are taken as two examples to verify the feasibility and effectiveness of the proposed controllers and identifiers.

This article presents a hyperchaotic system of five-dimensional autonomous ODEs that has five cross-product nonlinearities. Under certain parametric conditions, it exhibits three different types of hyperchaotic and chaotic systems which correspond to six hyperchaotic attractors with a non-hyperbolic equilibrium line, four chaotic attractors with seventeen hyperbolic equilibria, and four chaotic attractors with only one hyperbolic equilibrium, respectively. The fundamental dynamics are analyzed theoretically and numerically, such as the onset of hyperchaos and chaos, routes to chaos, persistence of chaos, coexistence of attractors, periodic windows and bifurcations. It is particularly shown that the coexisting attractors of the 5D system inside the hypercone are symmetric. Some dynamical characteristics of these attractors are illustrated.

We present a mechanism for the synchronization of one-way coupled nonlinear systems in which the coupling uses a variable delay, that is reset at finite intervals. Here the delay varies in the same way as the system in time and so the coupling function remains constant for the reset interval at the end of which it is reset to the value at that time. This leads to a novel and discrete error dynamics and the resulting general stability analysis is applicable to chaotic or hyperchaotic systems. We apply this method to standard chaotic systems and hyperchaotic time delay systems. The results of the detailed numerical analysis agree with the results from stability analysis in both cases. This method has the advantage that it is cost-effective since information from the driving system is needed only at intervals of reset. Further, in the context of time delay systems, optimization among the different time-scales depending upon the application is possible due to the flexibility among the four different time-scales in our method, viz. delay in the driving system, anticipation in the response system, system delay time and reset time. We suggest a bi-channel scheme for implementing this method in communication field with enhanced security.

DELAYED FUZZY CONTROLLER DESIGN FOR HYPERCHAOS WITH APPLICATION TO HYPERCHAOTIC CHEN'S SYSTEM

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

This paper reports a novel five Dimensional (5-D) Hyper-chaotic system with two positive Lyapunov exponents which is generated by adding two other dimensions to the 3-D chaotic system which was presented in 2016, by Tuna and Can. It modified generalized Lorenz system. This hyper-chaotic system has very simple algebraic structure but can show a very complex behavior. It has just one equilibrium point. In five dimensional hyper-chaotic systems obtaining a single equilibrium point very rare and interesting. Furthermore, in this letter presents a new numerical calculation method for the Lyapunov exponents of a hyper-chaotic system. After all of them in this research, designed a controller for the above system using the optimal generalized Back-Stepping method (OGBM). This method is an extension of the traditional Back-Stepping method and it optimized with the Shuffled Frog Leaping Algorithm (SFLA).

A new hyperchaotic system with coexisting attractors based on Sprott B chaotic system is proposed in this work. A novel feature of this new hyperchaotic system under investigation is that it has two-wing and fourwing coexisting attractors for two sets of different initial conditions. Thus, the new hyperchaotic system has hidden attractors. Interestingly, the proposed designed control function u (t) i using adaptive control method was able to control and globally synchronizes two identical new hyperchaotic systems evolving from different initial conditions with uncertain parameters. The adaptive synchronization scheme was applied to secure communication. Finally, the numerical simulation results presented demonstrated the effectiveness of the analytical results of the designed 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.

In this paper, finite-time synchronization (FTS) problems in an array of unified chaotic and hyperchaotic systems are mainly studied using some important control methods. First, FTS and finite-time transmission synchronization (FTTS) for N systems are realized in a finite time by designing the effective control schemes, and the stability criteria can be given. The corresponding examples are presented to show the effectiveness and feasibility of the controllers proposed by considering three chaotic and hyperchaotic systems.

Ultimate bound estimation of chaotic systems is a difficult yet interesting mathematical question. At present, explicit ultimate bound sets can be analytically obtained only for some special chaotic systems, and few results are known for hyper-chaotic ones. In this paper, through the Lagrange multiplier method and set operations, one derives two kinds of explicit ultimate bound sets for a novel hyperchaotic system. Based on the estimated result and optimization method, one further estimates the Hausdorff dimension of the hyperchaotic attractor. Numerical simulations show the effectiveness and correctness of the conclusions. The investigations enrich the related results for the hyperchaotic systems, and provide support for the assertion: hyperchaotic systems are ultimately bounded.