基于共轭Lorenz系统的新四维超混沌系统研究
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摘要
基于共轭Lorenz系统,运用反馈控制技术获得了有2个非线性项的新四维二次超混沌多项式系统;为了更好地理解此系统,研究了系统的局部动力学特性,包括系统耗散性、平衡点个数与稳定性、Lyapunov维数等,证明了系统存在Hopf分岔且给出了此分岔的发生条件;进一步利用系统的相图、Lyapunov指数、分岔等数值分析技术验证了新系统存在复杂动力学性态。
Based on conjugate Lorenz system,a new 4D second-order hyperchaotic multinomial system with two nonlinear terms is obtained by using feedback control technique. In order to better understand this system,the local dynamic properties of this system including dissipativity,the number and stability of the equilibrium point,Lyapunov dimension and so on are studied,the existence of Hopf bifurcation of this system is proved,and the generating condition of this bifurcation is given. Numerical analysis method including phase portrait,bifurcation diagram,Lyapunov exponents and so on are further used to verify that the complex dynamic property exists in the new system.
Based on conjugate Lorenz system,a new 4D second-order hyperchaotic multinomial system with two nonlinear terms is obtained by using feedback control technique. In order to better understand this system,the local dynamic properties of this system including dissipativity,the number and stability of the equilibrium point,Lyapunov dimension and so on are studied,the existence of Hopf bifurcation of this system is proved,and the generating condition of this bifurcation is given. Numerical analysis method including phase portrait,bifurcation diagram,Lyapunov exponents and so on are further used to verify that the complex dynamic property exists in the new system.
引文
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[11]YANG Q G,LIU Y J.A Hyperchaotic System from a Chaotic System with One Saddle and Two Stable Node-foci[J].Journal of Mathematical Analysis and Applications,2009(360):293-306
[12]张艳红,杨启贵.具有唯一平衡点的四维超混沌Lü-like系统的研究[J].重庆工商大学学报(自然科学版),2017,34(3):49-55ZHANG Y H,YANG Q G.Study on a 4D Hyperchaotic Lü-like System Only with One Equilibrium[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2017,34(3):49-55(in Chinese)
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[14]HASSARD B,KAZARINOFF N,WAN Y.Theory and Application of Hopf Bifurcation[M].Cambridge:Cambridge University Press,1982
[2]LORENZ E N.The Predictability of Hydrodynamic Flow[J].Transactions of the New York Academy of Science,1963,25(4):409-432
[3]CHEN G R,UETA T.Yet Another Chaotic Attractor Creation and Chaos[J].International Journal of Bifurcation and Chaos,1999,9(7):1465-1466
[4]CELIKOVSKY S,CHEN G R.On a Generalized Lorenz Canonical Form of Chaotic Systems[J].International Journal of Bifurcation and Chaos,2002,12(8):1789-1812
[5]LU J H,CHEN G R.A New Chaotic Attractor Coined[J].European Journal of Operational Research,2009,197(2):693-700
[6]YANG Q G,CHEN G R,ZHOU T S.A Unified Lorenztype System and Its Canomical Form[J].International Journal of Bifurcation and Chaos,2006,16(10):2855-2871
[7]YANG Q G,CHEN G R,HUANG K F.Chaotic Attractors of the Conjugate Lorenz-type System[J].International Journal of Bifurcation and Chaos,2007,17(7):3929-3949
[8]RSSLER O E.An Equation for Hyperchaos[J].Physics Letters A,1976(71):155-157
[9]MATSUMOTO T,CHUA L O,KOBAYASHI K.Hyperchaos:Laboralory Experiment and Numerical Confirmation[J].IEEE Transactions on Circuits and Systems,1986(11):1143-1147
[10]KAPITANIAK T,CHUA L O,ZHONG G Q.Experimental Hyperchaos in Coupled Chua's Circuits[J].IEEE Transactions on Circuits&Systems I-Fundamental Theory&Application,1994,41(7):499-503
[11]YANG Q G,LIU Y J.A Hyperchaotic System from a Chaotic System with One Saddle and Two Stable Node-foci[J].Journal of Mathematical Analysis and Applications,2009(360):293-306
[12]张艳红,杨启贵.具有唯一平衡点的四维超混沌Lü-like系统的研究[J].重庆工商大学学报(自然科学版),2017,34(3):49-55ZHANG Y H,YANG Q G.Study on a 4D Hyperchaotic Lü-like System Only with One Equilibrium[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2017,34(3):49-55(in Chinese)
[13]WOLF A,SWIFT J B,SWINNEY H L,et al.Determining Lyapunov Exponents from a Time Series[J].Physica D,1985(16):285-317
[14]HASSARD B,KAZARINOFF N,WAN Y.Theory and Application of Hopf Bifurcation[M].Cambridge:Cambridge University Press,1982