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Recently, a new hyperchaos generator, obtained by controlling a three-dimensional autonomous chaotic system — Chen's system — with a periodic driving signal, has been found. In this letter, we formulate and study the hyperchaotic behaviors in the corresponding fractional-order hyperchaotic Chen's system. Through numerical simulations, we found that hyperchaos exists in the fractional-order hyperchaotic Chen's system with order less than 4. The lowest order we found to have hyperchaos in this system is 3.4. Finally, we study the synchronization problem of two fractional-order hyperchaotic Chen's systems.

This paper presents a new hyperchaotic system, obtained by adding a controller to the second equation of the three-dimensional autonomous Chen's chaotic system. The hyper-chaos system undergoes a change from hyperchaos to limit cycle when the parameter varies. The system is not only demonstrated by computer simulations but also verified with bifurcation analysis.

A new four-dimensional continuous autonomous hyperchaotic system is considered. It possesses two parameters, and each equation of it has one quadratic cross product term. Some basic properties of it are studied. The dynamic behaviors of it are analyzed by the Lyapunov exponent (LE) spectrum, bifurcation diagrams, phase portraits, and Poincaré sections. The system has larger hyperchaotic region. When it is hyperchaotic, the two positive LE are both large and they are both larger than 1 if the system parameters are taken appropriately.

This paper formulates a new hyperchaotic system for particle motion. The continuous dependence on initial conditions of the system’s solution and the equilibrium stability, bifurcation, energy function of the system are analyzed. The hyperchaotic behaviors in the motion of the particle on a horizontal smooth plane are also investigated. It shows that the rich dynamic behaviors of the system, including the degenerate Hopf bifurcations and nondegenerate Hopf bifurcations at multiple equilibrium points, the irregular variation of Hamiltonian energy, and the hyperchaotic attractors. These results generalize and improve some known results about the particle motion system. Furthermore, the constraint of hyperchaos control is obtained by applying Lagrange’s method and the constraint change the system from a hyperchaotic state to asymptotically state. The numerical simulations are carried out to verify theoretical analyses and to exhibit the rich hyperchaotic behaviors.

In this paper, a new hyperchaotic system is formulated by introducing an additional state into the third-order unified system. Some of its basic dynamical properties, such as Lyapunov exponent, bifurcation diagram and the Poincáre section are investigated. It was found that the system is hyperchaotic in several different regions of the parameters. The analysis of equilibrium points and stability are also given. Two different methods, i.e., nonlinear hyperbolic function feedback control and tracking control methods, are used to control hyperchaos in the new hyperchaotic system. Based on the Routh–Hurwitz criteria, the conditions suppressing hyperchaos to unstable equilibrium point are discussed. A tracking control method is proposed. It is also proved that the strategy can make the system approach any desired smooth orbit at an exponential rate. Numerical results have shown the effectiveness of the control methods.

This paper presents a four-dimensional hyperchaos Qi system, obtained by adding linear term and nonlinear term of nonlinear controller to Qi chaos system. The hyperchaos Qi system is studied by bifurcation diagram, Lyapunov exponent spectrum and phase diagram. Numerical simulations show that the new system's behavior can be periodic, chaotic and hyperchaotic as the parameter varies.

This paper studies the global synchronization of a new hyperchaotic Lorenz system proposed by Wang et al. Based on the Lyapunov stability theory, the coupled control matrix is discussed when either knowing or unknowing the system boundary, respectively. The analysis of theory and numerical simulations show that the synchronization of hyperchaos Lorenz system can be realized effectively with the methods.

In this paper, periodic parametric perturbations are used to control chaos in Liu system. By changing the frequency of the perturbation signal, Liu system can be guided to not only periodic motion but also hyperchaos. Phase diagram, bifurcation diagram, Lyapunov exponents spectrum and Poincáre map are used to analyze the dynamic behavior of the controlled system in the numerical simulations, which show the effectiveness of our method.

This paper reports a novel four-dimensional hyperchaos generated from Qi system, obtained by adding nonlinear controller to Qi chaos system. The novel hyperchaos is studied by bifurcation diagram, Lyapunov exponent spectrum and phase diagram. Numerical simulations show that the new system's behavior can be periodic, chaotic and hyperchaotic as the parameter varies. Based on the time-domain approach, a simple observer for the hyperchaotic is proposed to guarantee the global exponential stability of the resulting error system. The scheme is easy to implement and different from the other observer design since it does not need to transmit all signals of the dynamical system.

In this paper, two different kinds of methods are adopted to control Liu system — feedback method and nonfeedback method. On the one hand, direct feedback and adaptive time-delayed feedback are taken as examples for the study of feedback control. In the direct feedback method, Liu system can be stabilized at one equilibrium point or a limit cycle surrounding its equilibrium. In the adaptive time-delayed feedback method, feedback coefficient and delay time can be adjusted adaptively to stabilize Liu system at its original unstable periodic orbit. On the other hand, periodic parametric perturbation is used to control chaos in Liu system as a typical nonfeedback method. By changing the frequency of the perturbation signal, Liu system can be guided to not only periodic motion but also hyperchaos. Numerical simulations show the effectiveness of our methods.

In this paper, a systematic design is proposed to generate multi-scroll attractors with hyperchaotic behavior using fractional-order systems, in which switched SC-CNN is triggered with error function. Sprott Systems Case H is reconstructed with fractional-order switched SC-CNN system. Herein, the goal is to increase the complexity of chaotic signals, hence providing safer and reliable communication by generating multi-scroll attractors with hyperchaotic behavior using fractional-order systems. Theoretical analysis of the proposed system’s dynamical behaviors is scrutinized, while numerical investigations are carried out with equilibrium points, Lyapunov exponent, bifurcation diagrams, Poincaré mapping and 0/1 test methods. Numerical results are validated through simulations and on an FPAA platform.

We show that coupled Kerr and Duffing oscillators with small nonlinearities and strong external pumping can generate chaotic and hyperchaotic beats. The appearance of chaos within beats depends strongly on the type of interactions between the nonlinear oscillators. To indicate chaotic behavior of the system we make use of the Lyapunov exponents. The structure of chaotic beats can be qualitatively different — the envelope function can be smooth if the system is undamped or can give the impression of noise structure in the presence of strong damping and nonlinear interactions between the individual oscillators. The systems considered can be used, in practice, as generators of chaotic beats with chaotically modulated envelopes and frequencies.

Shape memory and pseudoelastic effects are thermomechanical phenomena associated with martensitic phase transformations, presented by shape memory alloys. The dynamical analysis of intelligent systems that use shape memory actuators involves a multi-degree of freedom system. This contribution concerns with the chaotic response of shape memory systems. Two different systems are considered: a single and a two-degree of freedom oscillator. Equations of motion are formulated assuming a polynomial constitutive model to describe the restitution force of oscillators. Since equations of motion of the two-degree of freedom oscillator are associated with a five-dimensional system, the analysis is performed considering two oscillators, both with single-degree of freedom, connected by a spring-dashpot system. With this assumption, it is possible to analyze the transmissibility of motion between two oscillators. Results show some relation between the transmissibility of order, chaos and hyperchaos with temperature.

In this paper, the open-plus-closed-loop control strategy is adapted to synchronize discrete chaos. Two synchronization problems of chaos are studied: one is to drive a chaotic map with the aim of obtaining desirable chaotic dynamics; the other is to identify chaotic behaviors of a nonlinear map for different initial conditions. It is shown that in the latter case the needed control signal can be arbitrarily small. Two numerical examples, namely, the Gaussian map and a hyperchaotic map, are investigated in detail for demonstration of the effectiveness of the approach. The results show that synchronization of discrete chaos can be realized if the control is activated in the basin of entrainment.

The generalized Hénon maps (GHM) are discrete-time systems with given finite dimension, which show chaotic and hyperchaotic behavior for certain parameter values and initial conditions. A study of these maps is given where particularly higher-dimensional cases are considered.

In this paper, we demonstrate that some hyperchaotic circuits can be synchronized by using only one state variable. We applied three kinds of synchronization schemes, a continuous synchronization, an impulsive synchronization, and a selective synchronization to these hyperchaotic circuits. Their performance is examined from the viewpoint of synchronization stability and convergence time.

In this paper an approach for generating new hyperchaotic attractors from coupled Chua circuits is proposed. The technique, which exploits sine functions as nonlinearities, enables n×m-scroll attractors to be generated. In particular, it is shown that n×m-scroll dynamics can be easily designed by modifying four parameters related to the circuit nonlinearities. Simulation results are reported to illustrate the capability of the proposed approach.

In this paper, we present the hyperchaos dynamics of a modified canonical Chua's electrical circuit. This circuit, which is capable of realizing the behavior of every member of the Chua's family, consists of just five linear elements (resistors, inductors and capacitors), a negative conductor and a piecewise linear resistor. The route followed is a transition from regular behavior to chaos and then to hyperchaos through border-collision bifurcation, as the system parameter is varied. The hyperchaos dynamics, characterized by two positive Lyapunov exponents, is described by a set of four coupled first-order ordinary differential equations. This has been investigated extensively using laboratory experiments, Pspice simulation and numerical analysis.

We reinvestigate the dynamical behavior of a first order scalar nonlinear delay differential equation with piecewise linearity and identify several interesting features in the nature of the associated bifurcations and chaos as a function of delay time and external forcing parameters. In particular, we point out that the fixed point solution exhibits a stability island in the two parameter space of time delay and strength of nonlinearity. The significant role played by transients in attaining steady state solutions is pointed out. Various routes to chaos and existence of hyperchaos even for low values of time delay evidenced by multiple positive Lyapunov exponents are brought out. The study is extended to the case of two coupled systems, one with delay and the other one without delay.

In this letter, a simple nonlinear state feedback controller is designed for generating hyperchaos from a three-dimensional autonomous chaotic system. The hyperchaotic system is not only demonstrated by computer simulations but also verified with bifurcation analysis, and is implemented experimentally via an electronic circuit.