A New Six-Dimensional Hyper-Chaotic 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 an eight-term novel highly chaotic system with three quadratic nonlinearities. The phase portraits of the novel highly chaotic system are illustrated and the dynamic properties of the highly chaotic system are discussed. The novel highly chaotic system has three unstable equilibrium points. We show that the equilibrium point at the origin is a saddle point, while the other two equilibrium points are a saddle-focus and a critical point. The novel highly chaotic system has rotation symmetry about the \(x_3\) axis. The Lyapunov exponents of the novel highly chaotic system are obtained as \(L_1 = 7.6557\), \(L_2 = 0\) and \(L_3 = -24.6796\), while the Kaplan–Yorke dimension of the novel chaotic system is obtained as \(D_{KY} = 2.3102\). Since the maximal Lyapunov exponent of the novel chaotic system has a high value, viz. \(L_1 = 7.6557\), the novel chaotic system is highly chaotic. Since the sum of the Lyapunov exponents is negative, the novel chaotic system is dissipative. Next, we derive new results for the global chaos control of the novel highly chaotic system with unknown parameters using adaptive control method. We also derive new results for the global chaos synchronization of the identical novel highly chaotic systems with unknown parameters using adaptive control method. The main control results are established using Lyapunov stability theory. MATLAB simulations are depicted to illustrate the phase portraits of the novel highly chaotic system and also the adaptive control results derived in this work.

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.

Integer and fractional order analysis of a 3D system and generalization of synchronization for a class of chaotic systems

This paper presents the development of a new hyperchaotic system, created by the extension of the established Lü system to the four-dimension. The complex dynamics of this new system are explored through both theoretical analysis and numerical simulations. The study focuses on the system’s dynamic behavior, equilibrium points, Lyapunov exponents, and Poincaré sections, as well as bifurcation diagrams and coexisting attractors to thoroughly characterize its properties. Moreover, two main methods were investigated to control the hyperchaos: linear feedback control and adaptive control. These approaches aim to stabilize the hyperchaotic system at unstable equilibrium points, even when system parameters are either known or unknown. Numerical simulations are performed to illustrate the effectiveness of the proposed controllers.

Chaotic analysis and combination-combination synchronization of a novel hyperchaotic system without any equilibria

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.

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.

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 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.

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.

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).

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.

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.

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.