Analysis and Control of a 4-D Novel Hyperchaotic System

A new 4-D hyperchaotic two-scroll system with three quadratic nonlinearities and a cubic nonlinearity is proposed in this paper. The dynamical properties of the new hyperchaotic system are described in terms of phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, symmetry, dissipativity, etc. We also establish that the new hyperchaotic system has multistability with coexisting attractors. As a control application, we use integral sliding mode control for active self-synchronization of the new hyperchaotic systems as master-slave systems. As an engineering application, an electronic circuit design of the new hyperchaotic two-scroll system is developed in MultiSIM, which confirms the feasibility of the system.

In this work, we describe a twelve-term novel highly hyperchaotic system with four quadratic nonlinearities and an exponential nonlinearity. The phase portraits of the twelve-term novel hyperchaotic system are depicted and the qualitative properties of the novel hyperchaotic system are discussed. The novel hyperchaotic system has two unstable equilibrium points. The Lyapunov exponents of the novel hyperchaotic system are obtained as \(L_1 = 14.5577, L_2 = 0.1225, L_3 = 0\) and \(L_4 = -36.3884\). The maximal Lyapunov exponent of the novel hyperchaotic system has a high value, viz. \(L_1 = 14.5577\). Thus, the novel 4-D system shows highly hyperchaotic behavior. Also, the Kaplan–Yorke dimension of the novel hyperchaotic system is obtained as \(D_{KY} = 3.4045\), which is a high value for a 4-D hyperchaotic system. Since the sum of the Lyapunov exponents is negative, the novel hyperchaotic system is dissipative. Next, an adaptive controller is designed to globally stabilize the highly hyperchaotic system with unknown parameters. Finally, an adaptive controller is also designed to achieve global chaos synchronization of the identical highly hyperchaotic systems with unknown parameters. MATLAB simulations are presented to depict the phase portraits of the novel highly hyperchaotic system and illustrate all the main adaptive control results derived in this work.

Hyperchaos has important applications in many branches of science and engineering. In this work, we propose a new 4-D hyperchaotic system with three quadratic nonlinearities by modifying the dynamics of hyperchaotic Wang system (Wang et al. 2010). The proposed new hyperchaotic system has a unique equilibrium at the origin, which is a saddle point and unstable. Thus, the new hyperchaotic system exhibits self-excited hyperchaotic attractor. We describe qualitative properties of the new hyperchaotic system such as symmetry, Lyapunov exponents, Kaplan-Yorke dimension, etc. Furthermore, an active control method is derived for the synchronization of two identical new hyperchaotic systems. The circuit experimental results of the new hyperchaotic system show agreement with the numerical simulations.

This work announces an eleven-term novel 4-D hyperchaotic system with two quadratic nonlinearities. A qualitative analysis of the properties of the novel 4-D hyperchaotic system is presented. A special feature of our novel hyperchaotic system is that it has three equilibrium points of which two are unstable and one is locally asymptotically stable. The Lyapunov exponents of the novel hyperchaotic system are obtained as L1 = 1.5146, L2 = 0.2527, L3 = 0 and L4 = -12.7626. The Kaplan-Yorke dimension of the novel hyperchaotic system is derived as DKY. = 3.1385. Next, this work describes the design of an adaptive controller for the global hyperchaos synchronisation of identical novel hyperchaotic systems with unknown parameters. MATLAB simulations are shown to describe all the main results derived in this work.

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 four-dimensional hyperchaotic system with three quadratic nonlinearities is proposed in this paper. The dynamical properties of the new hyperchaotic system are explored in terms of phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, etc. Also, a detailed dynamical analysis of the new hyperchaotic system has been carried out with bifurcation diagram and Lyapunov exponents. As an engineering application, an electronic circuit realization of the new hyperchaotic system is designed via MultiSIM to confirm the feasibility of the theoretical hyperchaotic model.

In this manuscript,a new 4-D hyper-chaotic system is proposed. Followed by optimal and adaptive control theory. Firstly, we analyze the chaotic properties of new 4-D hyper-chaotic system such as dissipation, equilibrium, stability, time series, phase portraits, Lyapunov exponents, bifurcation and Poincare maps. Next, we study the optimal control for the new 4-D hyper-chaotic system which is based on the Pontryagin minimum principle. Further, Lyapunov stability theory is used for adaptive control approach and a parameter estimation update law is given for the new 4-D hyper-chaotic system with completely unknown parameters. Finally, to demonstrate the effectiveness of the proposed method we use MATLAB bvp4c and ode45 for numerical simulation which illustrate the stabilized behaviour of states and control functions for different equilibrium points. The plots displaying the time history of states functions and the parameters estimates have been drawn for the different values of equilibrium points.

We propose a new hyperchaotic complex system. The hyperchaotic complex system has a distinctive chaotic attractor, which is called butterfly-like attractor. We find that the state of the system has a complex evolution process with the change of parameters, and the system will produce at least two kinds of attractors, which we call parametric attractors. We systematically study the mechanics, symmetry, dissipation, Lyapunov exponents and Poincaré maps of the hyperchaotic complex system. We extend the hyperchaotic complex system to fractional order, and discuss the influence of order and parameters’ change on fractional order new hyperchaotic system, and compare it with integer order. What is interesting is that there exist parametric attractors in both integer order and fractional order of the new system. Based on the integer order and fractional order, we further analyze the new hyperchaotic complex system of hybrid order, propose the definition of complex chaotic family and carry out mechanical analysis.

On fractional-order hyperchaotic complex systems and their generalized function projective combination synchronization

A dual-parameter hyperchaotic system with constant Lyapunov exponent and its circuit emulation

This paper investigates the hybrid synchronization of identical hyperbolic Qi systems and hybrid synchronization of hyperchaotic Qi and Lu systems. The hyperchaotic Qi system (2008) and hyperchaotic Lu system (2006) are important models of hyperchaotic systems. Hybrid synchronization of the hyperchaotic systems is achieved through synchronization of two pairs of states and anti-synchronization of the other two pairs of states of the two hyperchaotic systems. Nonlinear control is the method used for the hybrid synchronization of identical hyperbolic Qi systems and hybrid synchronization of hyperchaotic Qi and Lu systems. Since the Lyapunov exponents are not required for these calculations, this method is effective and convenient to achieve hybrid synchronization of the two hyperchaotic systems. Numerical simulations are shown to verify the results.

A novel robust fuzzy controller design method is proposed to stabilize a class of chaotic (hyperchaotic) systems with uncertain parameters based on their equivalent T-S fuzzy models. In this method, the interval system theory is applied to deal with the parametric uncertainty firstly, and then a fuzzy state feedback controller is designed to stabilize the equilibrium of the uncertain chaotic (hyperchaotic) systems robustly based on Exact Linearization (EL) theory and Parallel Distributed Compensation (PDC) technique. Moreover, the sufficient criterion for the robust stability of the equilibrium of the controlled uncertain chaotic (hyperchaotic) systems is derived by Lyapunov stability theory and LMI technique. The designed controller with simple structure and rapid response can stabilize many kinds of uncertain chaotic or hyperchaotic systems. Simulation results are presented to demonstrate the effectiveness of the proposed method.

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.

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 work, we describe the model of a new 5-D hyperchaotic system with three positive Lyapunov exponents. Since the maximum positive Lyapunov exponent of the proposed hyperchaotic system is larger than twelve, the new hyperchaotic system is highly hyperchaotic. We also show that the new 5-D hyperchaotic system exhibits multistability with coexisting attractors. Using Multisim, we design an electronic circuit for the new 5-D hyperchaotic system. The hardware implementation of the new 5-D hyperchaotic system is done by applying two numerical methods. From the experimental results of the FPGA-based implementation, we show that the attractors observed in a Lecroy oscilloscope are in good agreement with numerical simulations. To prove the reliability of the proposed system for cybersecurity purposes, we present a new image cryptosystem using our hyperchaotic system. Experimental outcomes show the efficiency and the reliability of our cryptosystem based on the proposed hyperchaotic system.