A novel 5-D hyper-chaotic system with single equilibrium point and designing a controller for it using the optimal generalized back-stepping method

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.

In this paper, a new 4D hyperchaotic nonlinear dynamical system with two positive Lyapunov exponents is presented. Exhaustive dynamic analyses of the novel hyperchaotic model using several dynamical studies are described. The dynamics of the system considered are first investigated analytically and numerically to explore phenomena and the selection of hyperchaotic behavior utilized for designing image cryptosystem. Since the proposed hyperchaotic model has rich dynamics, it displays hidden attractors. It emerges from this dynamic the existence of a single unstable equilibrium point giving rise to self-excited attractors, hysteresis phenomenon, and hyperchaotic behavior strongly recommended for securing information by its character. Furthermore, the feasibility and synchronization of the proposed system are also presented by developing, respectively, Raspberry surveys and an adaptive synchronization approach of two identical hyperchaotic systems. By employing the hyperchaotic behavior of the 4D map, an image encryption scheme is proposed as well. It is one round of a pixel-based permutation and a bit-wise diffusion phase. The secret key of the 4D map is derived from the SHA-256 value of the input image. It acts as the signature of the input image. Hence, the secret key exhibits high sensitivity to single-bit alteration in the image, which makes the cryptosystem robust against chosen/known-plaintext attacks. Performance analyses prove that the proposed cryptosystem provides the best in terms of the performance/complexity trade-off, as compared to some recently published algorithms.

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.

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.

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.

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.

Design of two dimensional hyperchaotic system through optimization benchmark function

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.

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

The chaotic system is widely used in chaotic cryptosystem and chaotic secure communication. In this paper, a universal method for designing the discrete chaotic system with any desired number of positive Lyapunov exponents is proposed to meet the needs of hyperchaotic systems in chaotic cryptosystem and chaotic secure communication, and three examples of eight-dimensional discrete system with chaotic attractors, eight-dimensional discrete system with fixed point attractors and eight-dimensional discrete system with periodic attractors are given to illustrate how the proposed methods control the Lyapunov exponents. Compared to the previous methods, the positive Lyapunov exponents are used to reconstruct a hyperchaotic system.

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.

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.

In the last 20 years, the chaotic system has been widely studied. A new six-dimension hyper chaotic system is introduced in this paper. We used a new chaotic system based on a six-dimension for the purpose of increasing chaos in the system, where the new system has twelve positive parameters, complicated chaotic dynamics behaviors and gives an analysis of the new systems. The basic characteristics and dynamic behavior of this system are investigated with a presence of chaotic attractor, dissipativity, symmetry, equilibrium points, Lyapunov Exponents, Kaplan-Yorke dimension, waveform analysis and sensitivity toward initial conditions. The results of the analysis exhibit that the new system contains three unstable equilibrium points and the six Lyapunov exponents. Maxim non-negative Lyapunov Exponent (MLE) is obtained as 1.04609 and Kaplan-Yorke are obtained as 4.00695, and the new system characteristics with, unstable, high complexity, and unpredictability, The new system dynamics is simulated utilizing MATHEMATICA program. the phase portraits and the qualitative properties of the new hyper chaotic system have been described at the detail.

Memristor-based chaotic and hyperchaotic systems are of great interest in the recent years, and addition of meminductor and memcapacitors to the family has widened the applications. In this paper, we propose a new chaotic system with fractional-order memristor and memcapacitor components. Nonlinear chaotic properties of the proposed system are investigated with equilibrium points, eigenvalues, Lyapunov exponents, bifurcation and bicoherence plots. We show that a small model disturbance can make the system to show self-excited and hidden attractors. We use the Adomian Decomposition method for implementing the proposed system in Field Programmable Gate Arrays.

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.