In this paper, a new simple four-dimensional continuous-time autonomous hyperchaotic system is introduced, which displays a complicated four-wing attractor. The existence of the hyperchaos is verified by bifurcation analysis, and in the meantime bifurcation routes from period to quasi-period, then to chaos and finally to hyperchaos is determined. Different configurations of the hyperchaotic attractor are illustrated not only by computer simulation but also by electronic circuit realization. 1. INTR ODUCT I O N H yperchaos was first reported by Rossler in 1979 [1]. Since then, it has been studied with increasing interests due to its plentiful applications in applied mathematics and physics, especially in engineering and technology such as chaos-based encryption [2], secure communications [3], lasers [4], Colpitts oscillators [5], biological networks [6], nonlinear circuits [7], coupled map lattices [8], chaos control [9-1 0] and chaos synchronization [11-1 2]. The generation of different hyperchaotic systems has attracted more and more attention in the last few years [1 3 -23]. However, it is now known that generating a hyperchaotic attractor from an originally chaotic but not hyperchaotic system by means of control is a theoretically attractive and yet technically challenging task [1 7]. Some effective methods have been developed, e.g., by adding a simple state-feedback controller [1 7-18] or sinusoidal parametric perturbations [19][20], to some typical chaotic systems, which include the generalized Lorenz systems, Lorenz system, Chen system, Lu system, and a unified chaotic system to generate hyperchaotic attractor [1 7-23].Chaos attractor with four-wing has been introduced in [24-27], however, to the best of our knowledge, there are no reports on four-wing hyperchaotic attractor. In this letter, a simple and yet interesting four-dimensional (4D) continuous-time autonomous hyperchaotic system, with a complicated four-wing attractor, is firstly introduced and analyzed. "T aishan Scholarship" Construction Engineering . bifurcation analysis. Bifurcation routes: from period to quasi-period, then to chaos, and finally to hyperchaos are determined. The new-proposed system is implemented via an electronic circuit in laboratory, which shows perfect agreement with the numerical simulations. II. T HE NEW HYP ER CHA OT I C SY ST EMConsider the following 4D continuous-time autonomous system:where a,b,c,d,k,m E R+ are constant parameters.For system (1 ), one has (1 ) ax ay ai aw VV =-+-+-+-= a-b-c< °(2) ax ay az awHence, for all b + C > a , the system is dissipative. The equilibrium of system (1 ) can be found by letting the right-hand side of (1 ) be equal to zero. Obviously, the system has only one equilibrium point So = (0,0,0,0) .By linearizing system (1 ) at So = (0,0,0,0) , one obtains the Jacobian
Some basic properties of the hyperchaotic Lorenz system were analyzed, and the state feedback control methods were introduced for stabilizing unstable equilibrium point of the hyperchaotic system. The feedback controllers utilized the speed and hyperbolic function, respectively. Based on Routh-Hurwitz theorem, the span of the feedback coefficients was derived. Numerical simulations are given for illustration and verification. These methods are robust against system parametric variations, and can strongly reject external constant disturbances.
In this paper, a new four-dimensional continuoustime autonomous hyperchaotic system, which has a complicated four-wing attractor, is formulated by adding a linear controller to a chaotic system. The existence of the hyperchaos is verified with bifurcation analysis and Lyapunov exponent spectrum. The bifurcation routes from periodic, chaotic to hyperchaotic evolutions are observed simultaneously. The essential dynamical behaviors are demonstrated not only by numerical simulation but also by circuit realization.
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