A new hyperchaotic system and its linear feedback control

This paper reports a new hyperchaotic system by adding an additional state variable into a three-dimensional chaotic dynamical system. Some of its basic dynamical properties, such as the hyperchaotic attractor, Lyapunov exponents, bifurcation diagram and the hyperchaotic attractor evolving into periodic, quasi-periodic dynamical behaviors by varying parameter k are studied. Numerical simulations are presented to show these results.

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

In this paper, a new hyperchaotic finance system which is constructed based on a chaotic finance system by adding an additional state variable is presented. The basic dynamical behaviors of this hyperchaotic finance system are investigated, such as the equilibrium, stability, hyperchaotic attractor, Lyapunov exponents, and bifurcation analysis. Furthermore, effective speed feedback controllers and linear feedback controllers are designed for stabilizing hyperchaos to unstable equilibrium points. Numerical simulations are given to illustrate and verify the results.

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

The investigation of chaotic systems containing hidden and coexisting attractors has attracted extensive attention. This paper presents a four-dimensional (4D) novel hyperchaotic system, evolved by adding a linear state feedback controller to a 3D chaotic system with two stable node-focus points. The proposed system has no equilibrium point or two lines of equilibria, depending on the value of the constant term. Complex dynamical behaviors such as hidden chaotic and hyperchaotic attractors and five types of coexisting attractors of the simple 4D autonomous system are investigated and discussed, and are numerically verified by analyzing phase diagrams, Poincaré maps, the Lyapunov exponent spectrum, and its bifurcation diagram. The short unstable cycles in the hyperchaotic system are systematically explored via the variational method, and symbol codings of the cycles with four letters are realized based on the topological properties of the trajectory projection on the 2D phase space. The bifurcations of the cycles are explored through a homotopy evolution approach. Finally, the novel 4D system is implemented by an analog electronic circuit and is found to be consistent with the numerical simulation results.

This paper reports a new four-dimensional hyperchaotic system obtained by adding a controller to a three-dimensional autonomous chaotic system. The new system has two parameters and each equation of the system has one quadratic cross-product term. Some basic properties of the new system are analysed. The different dynamic behaviours of the new system are studied when the system parameter a or b is varied. The system is hyperchaotic in several different regions of the parameter b. Especially, the two positive Lyapunov exponents are both larger and the hyperchaotic region is also larger when this system is hyperchaotic in the case of varying a. The hyperchaotic system is analysed by Lyapunov-exponents spectrum, bifurcation diagrams and Poincaré sections.

In this article, a novel four dimensional autonomous nonlinear systezm called hyperchaotic Rikitake system is proposed. Basic properties of the new system are investigated and the complex dynamical behaviors, such as time series, bifurcation diagram, and Lyapunov exponents are analyzed by dynamic analysis approaches. To control the new hyperchaotic system, the delayed feedback control is introduced. Regarding the time delay as a bifurcation parameter, stability and bifurcations with respect to time delay are investigated. Conditions assuring the existence of Hopf bifurcation and the distribution of roots to the associated characteristic equation are investigated by utilizing the polynomial theorem. Besides, the Hopf bifurcation is proved to occur when the bifurcation parameter (time delay) crosses through derived critical value. Finally, numerical simulations are provided to prove the consistence with the derived theoretical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 180–193, 2016

In this paper, a novel control method that combines passive, linear feedback, and dislocated feedback control methods is proposed and applied to the control of the four-dimensional hyperchaotic finance system which has been introduced and controlled with the linear feedback and speed feedback control methods by Yu, Cai, and Li (2012). The stability of the hyperchaotic finance system at its equilibrium points is ensured on the basis of a Lyapunov function. Computer simulations are used for verifying all the theoretical analyses visually. In the simulations, the proposed control method is also compared with the speed feedback and linear feedback control methods to observe its effectiveness and the results are discussed. DOI: http://dx.doi.org/10.5755/j01.itc.47.1.16898

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

On the base of Qi system, a novel hyper-chaotic system is obtained by adding a linear feedback control. By analyzing the equilibrium, the system has rich and complex dynamical behaviors. Although this new system has only one equilibrium, it is four-wing hyper-chaotic attractor. The system is also verified to have a four-wing hyper-chaotic attractor by Lyapunov exponents, bifurcation diagrams, poincaré map, and numerical simulations. In addition, the system undergoes Hopf bifurcation when parameter f varies, and the theoretical analysis is verified by numerical simulation.

The global finite-time synchronization control of a type of fourth-order hyperchaotic systems with intersecting nonlinearities was studied. Firstly, the matrix form of this kind of hyperchaotic system was given to construct the master-slave synchronization models based on the generalized linear state error feedback controller. Then, the finite-time synchronization problem was equivalent to the finite-time stability of the error system originating from the master-slave hyperchaotic systems. The global finite-time stability of the error system was proven by the finite-time stability theory, further obtaining the criterion for global finite-time synchronization and the synchronization time estimation in mathematical formula. Subsequently, the theoretical results were applied to the well-known hyperchaotic Lorenz-stenflo system. The finite-time synchronization criteria for hyperchaotic Lorenz-stenflo systems were further proven by using optimization technology, and the corresponding synchronization times were estimated. Finally, the simulation was conducted by the MATLAB software, showing that two hyperchaotic Lorenz-stenflo systems could synchronize in a finite time.

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.

Based on the chaos anti-control theory, a new four-dimensional autonomous hyperchaotic system is formulated by adding a periodic driving signal rsin(cos(¿t)) to the second equation of the three-dimensional autonomous Chen chaotic system in this paper. The basic dynamical behavior of the new hyperchaotic system was not only demonstrated by numerical simulation, but also verified with Lyapunov exponents and bifurcation analysis. The new hyperchaotic system can be periodic, quasi-periodic, chaotic and hyperchaotic with the variation of the parameter ¿. It also has broad and stable hyperchaotic region which makes the new hyperchaotic system has a good application prospect.

In this research work a novel 4-D memristive system is presented. The proposed system belongs to the category of dynamical systems with hidden attractors as it displays a line of equilibrium points. Also, it has an hyperchaotic dynamical behavior in a particular range of its parameters space. System’s behavior is investigated through numerical simulations, by using well-known tools of nonlinear theory, such as phase portrait, bifurcation diagram, Lyapunov exponents and Poincare map. Next, the case of chaos control of the system with unknown parameters using adaptive control method is investigated. Finally, an electronic circuit realization of the novel hyperchaotic system using Spice is presented in detail to confirm the feasibility of the theoretical model.