A Novel Four-Dimensional Hyperchaotic System and Its Dynamic Analysis

This paper reports a new hyperchaotic system by adding an additional state variable into a three-dimensional chaotic dynamical system, studies 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 behaviours by varying parameter κ. Furthermore, effective linear feedback control method is used to suppress hyperchaos to unstable equilibrium, periodic orbits and quasi-periodic orbits. Numerical simulations are presented to show these results.

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 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 paper, a four-dimensional hyperchaotic system is constructed by adding a state feedback controller into a chaotic system. Numerical simulations and theoretical analysis show that this four-dimensional system will take on periodic,complex periodic, quasi-periodic chaotic hyperchaotic dynamical behaviours as parameters vary. Moreover, an electronic circuit diagram is designed for demonstrating the existence of the hyperchaos. Finally, an adaptive tracking controller is proposed to acquire adaptive tracking control for this hyperchaotic system.

A new hyperchaotic system and its generalized synchronization

Based on elementary quadratic chaotic flows with no equilibria, this paper proposes a novel four-dimensional hyperchaotic hidden attractor system. Compared with traditional chaotic systems, this system has more complex dynamic behaviors and higher degrees of freedom. Developing this novel system can further expand the research scope of chaos theory and reveal the intrinsic laws and characteristics of chaotic behaviors in higher dimensions.Through nonlinear dynamical analysis, the system exhibits diverse dynamic behaviors with varying parameters and possesses multiple hidden attractors, demonstrating the property of multistability. Subsequently, an analog circuit is constructed using Multisim 14.0 to further validate the system. Finally, image encryption is performed by employing different DNA encoding rules and corresponding operation rules based on chaotic sequences for diffusion. The cipher images are analyzed in terms of histograms, information entropy, and correlation. Experimental results indicate that the proposed algorithm achieves effective encryption and can withstand common attacks.

In this paper, some basic dynamical properties of a four-dimensional autonomous hyperchaotic system are investigated by means of Poincaré mapping, Lyapunov exponents and bifurcation diagram. The dynamical behaviours of this new hyperchaotic system are proved not only by performing numerical simulation and brief theoretical analysis but also by conducting an electronic circuit experiment. An efficient approaching is developed for global asymptotic stabilization of this four-dimensional hyperchaotic system. Based on the method of inverse optimal control for nonlinear systems, a linear state feedback is electronically implemented. It is remarkably simple as compared with other chaos control ways, like nonlinear state feedback.

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.

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.

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.

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

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

A five-dimensional multi-scroll chaotic system is presented by introducing two memristive elements into a three-dimensional chaotic system. The resulting model generates multi-scroll attractors whose scroll count can be regulated by tuning the memristors' internal parameters. We analyze the equilibria and then quantify the dynamic behaviors using phase portraits, Poincaré sections, bifurcation diagrams, and Lyapunov exponents. Coexisting multi-scroll attractors are observed, and their attraction basins are mapped to visualize the corresponding spatial domains. Parameter-driven adjustment of local amplitude is also demonstrated, enabling flexible modulation of the system output. A DSP-based implementation is further provided to validate the realizability of the proposed design. The study advances memristor-assisted multi-scroll construction and supports engineering-oriented hardware realization of high-dimensional chaotic systems.

Mirror symmetry multi-wing attractors generated from a novel four-dimensional hyperchaotic system

This paper presents a new four-dimensional non-Hamiltonian conservative hyperchaotic system. In the absence of equilibrium points in the system, the phase trajectories generated by the system have hidden features. The conservative features that vary with the parameter have been analyzed in detail by Lyapunov exponent spectrum, bifurcation diagram, the sum of Lyapunov exponents, and the fractional dimensions, and during the analysis, multiple quasi-periodic four-dimensional tori as well as hyperchaotic attractors have been observed. The Poincaré sections confirm these dynamic behaviors. Amidst the process of studying the dynamical behavior of the system with initial values, the hidden extreme multistability, and the initial offset boosting behavior, the results have been witnessed for the very first time in a conservative chaotic system. The phase diagram and attraction basin also confirm this assertion, while two complex transient transition behaviors have been observed. Moreover, through the introduction of a spectral entropy algorithm, the complexity analysis of the time sequences generated by the system have been performed and compared with the existing literature. The results show that the system has a high degree of complexity. The design and construction of the analog circuit of the system for simulation, the circuit experimental results are consistent with the numerical simulation, further verifying the physical realizability of the newly proposed system. This lays a good foundation for its practical application in engineering.