Lorenz系统族的全局吸引集和正向不变集的研究及其应用
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摘要
混沌系统的最终界在混沌系统的定性行为的研究中有着重要的作用,若我们可以找到一个混沌系统的全局吸引集,则可以断定在这个全局吸引集之外不会存在该系统其它的平衡位置、周期解、概周期解、游荡回复解和其它任何吸引子,这可以简化对一个系统基本的动力学性质的分析和研究,同样在混沌系统的控制和同步中有着广泛的用途。
利用广义Lyapunov函数理论和优化理论研究了三个混沌系统的全局吸引集和正向不变集。
对于一个新混沌系统,分析了该系统的基本的混沌动力学行为,得到了一个三维椭球形式的全局吸引集和正向不变集,从数学理论上给予了严格的证明,且估计了该系统关于x ? z和y ? z在二维中的有界性。
对于Qi混沌系统和Lorenz-Stenflo混沌系统,我们采用了同样的方法得到了不同形式的一系列的椭球型估计式,然后在将得到的结果应用到同步上,实现了系统的全局同步,最后利用Matlab进行仿真,验证了理论的有效性。
The ultimate bound of a chaotic system is important for the study of the qualitative behavior of a chaotic system. If we can show that a system under consideration has a globally attractive set, then we know that the system cannot have equilibrium points, periodic solutions, quasi-periodic solutions, or other chaotic attractors outside the globally attractive set. This greatly simplifies the analysis of the dynamical properties of the system. The ultimate bound also plays an important role in designing scheme for chaos control and chaos synchronization.
This paper investigates the ultimate bound and positively invariant set for three chaotic systems via the generalized Lyapunov function and optimization.
Some basic dynamical properties are investigated for a new chaotic system, we derive a three-dimensional ellipsoidal ultimate bound and positively invariant set and prove the existence in mathematical theory. And the two-dimensional bound with respect to x ? z and y ? zare established.
With the help of the same method, we get the many different ellipsoidal-like formulas for the Qi chaotic system and the Lorenz-Stenflo chaotic system. Then the result is applied to the study of chaos synchronization and the globally synchronization of the two systems is achieved. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme
利用广义Lyapunov函数理论和优化理论研究了三个混沌系统的全局吸引集和正向不变集。
对于一个新混沌系统,分析了该系统的基本的混沌动力学行为,得到了一个三维椭球形式的全局吸引集和正向不变集,从数学理论上给予了严格的证明,且估计了该系统关于x ? z和y ? z在二维中的有界性。
对于Qi混沌系统和Lorenz-Stenflo混沌系统,我们采用了同样的方法得到了不同形式的一系列的椭球型估计式,然后在将得到的结果应用到同步上,实现了系统的全局同步,最后利用Matlab进行仿真,验证了理论的有效性。
The ultimate bound of a chaotic system is important for the study of the qualitative behavior of a chaotic system. If we can show that a system under consideration has a globally attractive set, then we know that the system cannot have equilibrium points, periodic solutions, quasi-periodic solutions, or other chaotic attractors outside the globally attractive set. This greatly simplifies the analysis of the dynamical properties of the system. The ultimate bound also plays an important role in designing scheme for chaos control and chaos synchronization.
This paper investigates the ultimate bound and positively invariant set for three chaotic systems via the generalized Lyapunov function and optimization.
Some basic dynamical properties are investigated for a new chaotic system, we derive a three-dimensional ellipsoidal ultimate bound and positively invariant set and prove the existence in mathematical theory. And the two-dimensional bound with respect to x ? z and y ? zare established.
With the help of the same method, we get the many different ellipsoidal-like formulas for the Qi chaotic system and the Lorenz-Stenflo chaotic system. Then the result is applied to the study of chaos synchronization and the globally synchronization of the two systems is achieved. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme
引文
[1] E.N.Lorenz.混沌的本质[M].北京:气象出版社,1997.
[2] E.N.Lorenz. Deterministic non-periods flows[J]. J Atmos Sci,1963,20:130-141.
[3] D.Ruelle, F.Takens. On nature of turbulauce[J]. Commun. Math. Phy.,1971,20:167-193.
[4] T.Y. Li, J.A.Yorke. Period three implies chaos[J]. Am. Math.Momthy, 1975,82:185-192.
[5] A.Hubler. Adaptive control of chaotic system[J]. Helv. Phys.Acta,1989,62:343-346.
[6] E.Ott, C.Grebogi, J.A.Yorke. Controlling Chaos[J]. Phys. Rev Lett,1990,64:1196-1199.
[7] O.E.R?sser. An equation for continuous chaos[J]. Phys. Lett A,1976,57:397-398.
[8] G.Chen, G.T.Ueta. Yet another chaotic attractor[J]. Int J Bifurcat Chaos,1999,9:1465-6.
[9] J.Lü, G.Chen. A new chaotic attractor coined[J]. Int J Bifurcat Chaos,2002,12:659–61.
[10] J.Lü, G.Chen, D.Cheng, ?elikovsy S. Bridge the gap between the Lorenz system and the Chen system [J]. Int J Bifurcat Chaos,2002,12:2917–26.
[11] T.Matsumoto. A chaotic attractor from Chua's circuit[J]. IEEE Trans Circ and Syst, 1984,31:1055-1058.
[12] S.?elikovsy, G.Chen. On a generalized Lorenz canonical form of chaotic systems[J]. Int J Bifurcat Chaos,2002,12:1789–812.
[13] T.Ueta, G.Chen. Bifurcation analysis of Chen's equation[J]. Int J Bifurcat Chaos, 2000,10:1917–31.
[14]陈关荣,吕金虎. Lorenz系统族的动力学分析、控制与同步[M].北京:科学出版社,2003.
[15]王兴元.复杂非线性系统中的混沌[M].北京:电子工业出版社,2003.
[16]张化光,王智良,黄伟.混沌系统的控制理论[M].沈阳:东北大学出版社,2003.
[17]张琪昌,王洪礼,竺致文等.分岔与混沌理论及应用[M].天津:天津大学出版社,2005.
[18]廖晓昕.稳定的数学理论与应用[M].武汉:华中师范大学出版社,2001.
[19]胡岗,萧井华,郑志刚.混沌控制[M].上海:上海科技教育出版社,2000.
[20] G.Chen. Optimal control of chaotic system[J]. Int J Bifurcat Chaos,1994,4:461-463.
[21] G.Chen. Feedback anticontrol of chaos[J]. Int J Bifurcat Chaos,1998,8:461-463.
[22] T.Warwick. Arigorous ODE solver and smale 14th problem[J]. Found Comput. Math, 2002,2:53-117.
[23] G.Lenovo, V.Retimann. Attraktoreingrenzuny fur Nichtlineare system[J]. Teubner-Verlag. Leipzing, 1987.
[24] G.Lenovo, A.Bunin, N.Koksch. Attractor localization of the Lorenz system[J]. ZAMM, 1987, 67: 649-656.
[25] Y.Pogromsky, G.Santoboni, H.Nijmeijer. An ultimate bound on the trajectories of the Lorenz system and its applications[J]. Nonlinearity,2003,16:1597-605.
[26] X.X.Liao, Y.L.Fu, S.L.Xie. On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization[J]. Science in China Ser. F Information Sciences 2005,48:304-321.
[27] D.Li, J.Lu, X.Wu, G.Chen. Estimating the bounds for the Lorenz family of chaotic systems[J]. Chaos, Solitons & Fractals ,2005,23:529-534.
[28] D.Li, J.Lu, X.Wu, G.Chen. Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system[J]. J. Math. Anal. Appl,2006,323: 844-853.
[29] P.Yu, X.X.Liao. Globally attractive and positively invariant set of the Lorenz system[J]. Int J Bifurcat Chaos,2006,16:757-64.
[30] P.Yu, X.X.Liao. New estimations for globally attractive and positively invariant set of the family of the lorenz systems[J]. Int J Bifurcat Chaos,2006,11:3383–3390.
[31]廖晓昕,罗海庚,傅予力等.论Lorenz系统族的全局指数吸引集和正向不变集[J].中国科学,2007,37:757-769.
[32] W.X.Qin, G.Chen. On the boundedness of solutions of the Chen system[J]. J. Math. Anal. Appl, 2007,329:445-451.
[33] Y.L.Shu, Y.H.Zhang. Estimating the globally attractive set and positively invariant set of a unified chaotic system[J]. Journal of Chongqing University,2008,3:216-220.
[34] P.Yu, X.X.Liao. On the study of globally exponentially attractive set of a general chaotic system[J]. Communications in Nonlinear Science and Numerical Simulation, 2008, 13:1495–1507.
[35] X.X.Liao, P.Yu, S.L.Xie. Study on the global property of the smooth Chua’s system[J]. Int J Bifurcat Chaos,2006,16:2815–2841.
[36] D.Li, J.Lu, X.Wu. Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz–Haken system[J]. Chaos, Solitons & Fractals,doi:10.1016/j chaos, 2007,06,038.
[37] Y.J.Sun. Solution bounds of generalized Lorenz chaotic systems[J]. Chaos, Solitons & Fractals,doi:10.1016/j.chaos.2007.08.015.
[38] Y.L.Shu, Y.H.Zhang. Estimating of the ultimate bound and positively invariant set for a generalized Lorenz system[J], Journal of Chongqing University,2008,7:151-154.
[39]马知恩,周义仓.常微分稳定性和定性方法[M].北京:科学出版社,2001.
[40]褚衍东,李险峰,张建刚等.一个新混沌系统的的计算机仿真与电路模拟[J].四川大学学报(自然科学版),2007;44:596-602.
[41]刘晓君,李险峰,张建刚.一个新自治混沌系统的混沌同步控制[J].复杂系统与复杂性科学,2007,4:51-57.
[42] Y.L.Shu, H.X.Xu, Y.H.Zhao. Estimating the ultimate bound and positively invariant set for a new chaotic system and its application in chaos synchronization[J]. Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2009.04.003.
[43] G.Y.Qi, G.Chen. S.Z.Du. Analysis of a new chaotic system[J]. Physica A, 2005,352:295-308.
[44]胥红星,舒永录,原冠秀.一个混沌系统的最终有界集及其在同步中的应用[J].重庆工商大学学报(自然科学版),2008,25;564-568.
[45] L.Stenflo. Generalized Lorenz equations for Acoustic-Gravity waves in the atmosphere[J]. Phy.Scr,1996,329:83-84.
[46] C.Zhou. Bifurcation behaviour of the generalized Lorenaz equations at large rotation numbers[J]. J Math Phys ,1997,38:5525-39.
[47] Z.Liu. The first integral of nonlinear acoustic gravity wave equation[J]. Phy.Scr, 2000,61:526.
[48] S.Banerjee, P.Saha, A.R.Chowdhury. Chaotic scenario in the Stenflo equations[J]. Phy.Scr, 2001,63:177-180.
[49] S.Banerjee, P.Saha, A.R.Chowdhury. On the application of adaptive control and phase synchronization in nonlinear fluid dynamics[J]. Int J nonlinear Mech, 2004,39:25-31.
[50] U.E.Vincent. Synchronization of idential and non-inentical 4-D chaotic systems using active control[J]. Chaos, Solitons & Fractals, 2008,37:1065-1075.
[51] J.A.Laoye, U.E.Vincent. Chaos control of 4D chaotic systems using recursive backstepping nonlinear controller[J]. Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2007.04.020.
[52] Y.L.Shu, H.X.Xu, Y.H.Zhao. The ultimate bound of a generalized Lorenz chaotic system and its application[J].工程数学学报.(已投).
[53] S.Lefchetz. Differential Equations: Geometric Theory[M]. New York: Interscience Publishers, 1963.
[2] E.N.Lorenz. Deterministic non-periods flows[J]. J Atmos Sci,1963,20:130-141.
[3] D.Ruelle, F.Takens. On nature of turbulauce[J]. Commun. Math. Phy.,1971,20:167-193.
[4] T.Y. Li, J.A.Yorke. Period three implies chaos[J]. Am. Math.Momthy, 1975,82:185-192.
[5] A.Hubler. Adaptive control of chaotic system[J]. Helv. Phys.Acta,1989,62:343-346.
[6] E.Ott, C.Grebogi, J.A.Yorke. Controlling Chaos[J]. Phys. Rev Lett,1990,64:1196-1199.
[7] O.E.R?sser. An equation for continuous chaos[J]. Phys. Lett A,1976,57:397-398.
[8] G.Chen, G.T.Ueta. Yet another chaotic attractor[J]. Int J Bifurcat Chaos,1999,9:1465-6.
[9] J.Lü, G.Chen. A new chaotic attractor coined[J]. Int J Bifurcat Chaos,2002,12:659–61.
[10] J.Lü, G.Chen, D.Cheng, ?elikovsy S. Bridge the gap between the Lorenz system and the Chen system [J]. Int J Bifurcat Chaos,2002,12:2917–26.
[11] T.Matsumoto. A chaotic attractor from Chua's circuit[J]. IEEE Trans Circ and Syst, 1984,31:1055-1058.
[12] S.?elikovsy, G.Chen. On a generalized Lorenz canonical form of chaotic systems[J]. Int J Bifurcat Chaos,2002,12:1789–812.
[13] T.Ueta, G.Chen. Bifurcation analysis of Chen's equation[J]. Int J Bifurcat Chaos, 2000,10:1917–31.
[14]陈关荣,吕金虎. Lorenz系统族的动力学分析、控制与同步[M].北京:科学出版社,2003.
[15]王兴元.复杂非线性系统中的混沌[M].北京:电子工业出版社,2003.
[16]张化光,王智良,黄伟.混沌系统的控制理论[M].沈阳:东北大学出版社,2003.
[17]张琪昌,王洪礼,竺致文等.分岔与混沌理论及应用[M].天津:天津大学出版社,2005.
[18]廖晓昕.稳定的数学理论与应用[M].武汉:华中师范大学出版社,2001.
[19]胡岗,萧井华,郑志刚.混沌控制[M].上海:上海科技教育出版社,2000.
[20] G.Chen. Optimal control of chaotic system[J]. Int J Bifurcat Chaos,1994,4:461-463.
[21] G.Chen. Feedback anticontrol of chaos[J]. Int J Bifurcat Chaos,1998,8:461-463.
[22] T.Warwick. Arigorous ODE solver and smale 14th problem[J]. Found Comput. Math, 2002,2:53-117.
[23] G.Lenovo, V.Retimann. Attraktoreingrenzuny fur Nichtlineare system[J]. Teubner-Verlag. Leipzing, 1987.
[24] G.Lenovo, A.Bunin, N.Koksch. Attractor localization of the Lorenz system[J]. ZAMM, 1987, 67: 649-656.
[25] Y.Pogromsky, G.Santoboni, H.Nijmeijer. An ultimate bound on the trajectories of the Lorenz system and its applications[J]. Nonlinearity,2003,16:1597-605.
[26] X.X.Liao, Y.L.Fu, S.L.Xie. On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization[J]. Science in China Ser. F Information Sciences 2005,48:304-321.
[27] D.Li, J.Lu, X.Wu, G.Chen. Estimating the bounds for the Lorenz family of chaotic systems[J]. Chaos, Solitons & Fractals ,2005,23:529-534.
[28] D.Li, J.Lu, X.Wu, G.Chen. Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system[J]. J. Math. Anal. Appl,2006,323: 844-853.
[29] P.Yu, X.X.Liao. Globally attractive and positively invariant set of the Lorenz system[J]. Int J Bifurcat Chaos,2006,16:757-64.
[30] P.Yu, X.X.Liao. New estimations for globally attractive and positively invariant set of the family of the lorenz systems[J]. Int J Bifurcat Chaos,2006,11:3383–3390.
[31]廖晓昕,罗海庚,傅予力等.论Lorenz系统族的全局指数吸引集和正向不变集[J].中国科学,2007,37:757-769.
[32] W.X.Qin, G.Chen. On the boundedness of solutions of the Chen system[J]. J. Math. Anal. Appl, 2007,329:445-451.
[33] Y.L.Shu, Y.H.Zhang. Estimating the globally attractive set and positively invariant set of a unified chaotic system[J]. Journal of Chongqing University,2008,3:216-220.
[34] P.Yu, X.X.Liao. On the study of globally exponentially attractive set of a general chaotic system[J]. Communications in Nonlinear Science and Numerical Simulation, 2008, 13:1495–1507.
[35] X.X.Liao, P.Yu, S.L.Xie. Study on the global property of the smooth Chua’s system[J]. Int J Bifurcat Chaos,2006,16:2815–2841.
[36] D.Li, J.Lu, X.Wu. Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz–Haken system[J]. Chaos, Solitons & Fractals,doi:10.1016/j chaos, 2007,06,038.
[37] Y.J.Sun. Solution bounds of generalized Lorenz chaotic systems[J]. Chaos, Solitons & Fractals,doi:10.1016/j.chaos.2007.08.015.
[38] Y.L.Shu, Y.H.Zhang. Estimating of the ultimate bound and positively invariant set for a generalized Lorenz system[J], Journal of Chongqing University,2008,7:151-154.
[39]马知恩,周义仓.常微分稳定性和定性方法[M].北京:科学出版社,2001.
[40]褚衍东,李险峰,张建刚等.一个新混沌系统的的计算机仿真与电路模拟[J].四川大学学报(自然科学版),2007;44:596-602.
[41]刘晓君,李险峰,张建刚.一个新自治混沌系统的混沌同步控制[J].复杂系统与复杂性科学,2007,4:51-57.
[42] Y.L.Shu, H.X.Xu, Y.H.Zhao. Estimating the ultimate bound and positively invariant set for a new chaotic system and its application in chaos synchronization[J]. Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2009.04.003.
[43] G.Y.Qi, G.Chen. S.Z.Du. Analysis of a new chaotic system[J]. Physica A, 2005,352:295-308.
[44]胥红星,舒永录,原冠秀.一个混沌系统的最终有界集及其在同步中的应用[J].重庆工商大学学报(自然科学版),2008,25;564-568.
[45] L.Stenflo. Generalized Lorenz equations for Acoustic-Gravity waves in the atmosphere[J]. Phy.Scr,1996,329:83-84.
[46] C.Zhou. Bifurcation behaviour of the generalized Lorenaz equations at large rotation numbers[J]. J Math Phys ,1997,38:5525-39.
[47] Z.Liu. The first integral of nonlinear acoustic gravity wave equation[J]. Phy.Scr, 2000,61:526.
[48] S.Banerjee, P.Saha, A.R.Chowdhury. Chaotic scenario in the Stenflo equations[J]. Phy.Scr, 2001,63:177-180.
[49] S.Banerjee, P.Saha, A.R.Chowdhury. On the application of adaptive control and phase synchronization in nonlinear fluid dynamics[J]. Int J nonlinear Mech, 2004,39:25-31.
[50] U.E.Vincent. Synchronization of idential and non-inentical 4-D chaotic systems using active control[J]. Chaos, Solitons & Fractals, 2008,37:1065-1075.
[51] J.A.Laoye, U.E.Vincent. Chaos control of 4D chaotic systems using recursive backstepping nonlinear controller[J]. Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2007.04.020.
[52] Y.L.Shu, H.X.Xu, Y.H.Zhao. The ultimate bound of a generalized Lorenz chaotic system and its application[J].工程数学学报.(已投).
[53] S.Lefchetz. Differential Equations: Geometric Theory[M]. New York: Interscience Publishers, 1963.