Research On Complexity And Synchronization Of A New Five-dimensional Hyperchaotic System

Ever since the American meteorologist, E. N. Lorenz, found the first strange at-tractor, chaos,as a main subject of theoretical research, has developed enormously inmany fields. The investigation of hyperchaos is a new subject based on chaotic system.Compared with chaotic system, hyperchaotic system has much more complex dynami-cal behavior with at least two positive Lyapunov exponents, System's randomness andindeterminacy has enhanced much. Hence hyperchaos has much more extensive valuein engineering application. Such as hyperchaos-based encryption and secure communica-tion. There does not seem to be any systematic methodology for purposefully designing ahyperchaotic system to date. In particular, purposefully designing a hyperchaotic systemfrom an originally chaotic system with some simple means. So it has great applicationsignificance to design, implement and analyze hyperchaotic systems with at least twopositive Lyapunov exponents.Based on the research results of chaos and hyperchaos in theory, this paper has madeanalysis of hyperchaos and chaos synchronization. The paper is organized as follows:Firstly, the definition, basic characteristic of chaotic and hyperchaotic systems arerecommended. Basic means and conception of chaos and hyperchaos has also been men-tioned. Meanwhile, the characteristic of some classic chaotic and hyperchaotic systemshave also been studied.Secondly, using feedback control, a new five-dimensional hyperchaotic system withthree positive Lyapunov exponents is introduced. The stability properties of equilib-ria,the conditions of Hopf bifurcation and the expression of periodic solution are roundlystudied by means of the norm form theory,the Hopf bifurcation theorem and softwareMathematica. Furthermore ,using software Matlab , some complicated dynamical behav-iors of the corresponding system is numerically displayed by phase portrait, spectrum ofthe Lyapunov exponents, program of bifurcation figure and so on.Thirdly, the self-adaptive synchronization of the new hyperchaotic system are re-searched. Simple proper controllers are designed for synchronizing response system anddrive system. The numerical simulation implies the e?ectiveness of this method .