（2+1）维Boussinesq方程的混沌行为与控制
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摘要
从1963年洛伦兹发表《决定论非周期流》至今,非线性系统科学得到了飞速的发展,于是更深一步揭示了非线性系统共同的性质、基本特征和运动规律.非线性科学于近二十多年可以得到迅猛发展,其中一个非常重要的原因就是:在描述很多大自然现象时,不管是线性还是非线性动力系统中都存在混沌运动.可以说混沌运动无处不在,许多物理学、力学中的问题均能化为带有周期小扰动项的具有同宿轨或者异宿轨的二阶常微分方程.对于此类系统,判断其混沌不变集在Smale马蹄变换意义下是否存在,一般可以用Melnikov方法.
本文利用Melnikov方法研究了带扰动项的(2+1)维Boussinesq方程及相关问题解的分支、周期性态、同宿运动和混沌行为,并用正反馈和耦合反馈方法对混沌进行了控制.此系统混沌现象的存在可由次谐不稳定性与同宿横截点的存在得到,而该系统的周期分岔与分岔的Melnikov序列亦进一步获得.不断完善发展以Boussinesq方程为基础的浅水波模型,可使其更加准确、有效地模拟浅水域的各种自然现象.
Since Lorenz published "Decision on the non-periodic flow" in 1963, non-linear system scientific access to the rapid development, which further reveals the common nature of basic characteristics, the law of sport non-linear system. The found of chaotic motion was extremely important in describing the various types of natural phenomena in power systems, thus nonlinear science can be development rapidly in the last twenty years. That is chaos everywhere, many physical, mechanical problems in the can into a small disturbance with a period with homoclinic orbit or heteroclinic orbit of the second order ordinary differential equations. For such systems can usually be handled by Melnikov method to determine whether there is in the sense of Smale horseshoes in chaotic invariant set.
Bifurcation, periodic, homoclinic and chaotic behaviour for (2+1)-Dmensional Boussinesq equation and related issues are studied under perturbation using the Melnikov method, and use positive feedback and coupling feedback method of chaos to control. Chaos arising from subharmonic instability and homoclinic crossings are observed. Both period-doubling bifurcation and the Melnikov sequence of subharmonic bifurcation are found and lead chaotic behaviour. Continuously improve the development of Boussinesq quation-based model with shallow water wave to make it more accurate and effective simulation of natural phenomena in shallow waters.
本文利用Melnikov方法研究了带扰动项的(2+1)维Boussinesq方程及相关问题解的分支、周期性态、同宿运动和混沌行为,并用正反馈和耦合反馈方法对混沌进行了控制.此系统混沌现象的存在可由次谐不稳定性与同宿横截点的存在得到,而该系统的周期分岔与分岔的Melnikov序列亦进一步获得.不断完善发展以Boussinesq方程为基础的浅水波模型,可使其更加准确、有效地模拟浅水域的各种自然现象.
Since Lorenz published "Decision on the non-periodic flow" in 1963, non-linear system scientific access to the rapid development, which further reveals the common nature of basic characteristics, the law of sport non-linear system. The found of chaotic motion was extremely important in describing the various types of natural phenomena in power systems, thus nonlinear science can be development rapidly in the last twenty years. That is chaos everywhere, many physical, mechanical problems in the can into a small disturbance with a period with homoclinic orbit or heteroclinic orbit of the second order ordinary differential equations. For such systems can usually be handled by Melnikov method to determine whether there is in the sense of Smale horseshoes in chaotic invariant set.
Bifurcation, periodic, homoclinic and chaotic behaviour for (2+1)-Dmensional Boussinesq equation and related issues are studied under perturbation using the Melnikov method, and use positive feedback and coupling feedback method of chaos to control. Chaos arising from subharmonic instability and homoclinic crossings are observed. Both period-doubling bifurcation and the Melnikov sequence of subharmonic bifurcation are found and lead chaotic behaviour. Continuously improve the development of Boussinesq quation-based model with shallow water wave to make it more accurate and effective simulation of natural phenomena in shallow waters.
引文
[1]何英高.非线性时延系统分叉、混沌及同步研究:[硕士学位论文][D].南京:东南大学,2001.
[2] Boussinesq J. Theory of waves and swells propagated in long horizontal rectangular canal and imparting to the liquid contained in this canal[J].France: Journal de Mathematiques Pures et Appliquees,1872,17(2):55-108.
[3] Peregrine D H.Long waves on a beach[J].J Fluid Mech. Great Britain:1967,27:815-827.
[4] Madsen P A, Banijamali B, Schaffer H A. Boussinesq Type Equation With High Accuracy in Dispersion and Nonlinearity[C].Proc 25th Intl Conf Coastal Eng. New York: asce,1996:95-108.
[5] Ranjit S. Dhaliwal, Saleem M. Khan. The axisymmetric boussinesq problem for a semi-space in micropolar theory[J].International Journal of Engineering Science,1976,14(8):769-788.
[6] Clarkson P.A, Kruskal M.D. New similarity reductions of the boussinesq equation[J].J Math Phys,1989,30(10):2201-2213.
[7] Peter A. Clarkson. Nonclassical symmetry reductions of the Boussinesq equation[J].Chaos, Solitons & Fractals.1995,12(5):2261-2301
[8] G.M.A.Allen and G.Rowlands. On the transverse instabilities of solitary waves[J].Phys. Lett.A, 1997,235,145-149.
[9] J.L.Bona, R.L. Sachs. Global existence of smooth solutions and solitary waves for a generalized Boussinesq equation[J].Comm. Math.Phys,1998,118:15-29.
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[16]洪丽莉,张凯,张然.线性Bossinesq方程的多辛Runge-Kutta Nystr?m算法[J].长春:吉林大学学报(理学版),2007,45(6):892-899.
[17] Yong Chen, Zhenya Yan, Honging Zhang. New explicit solitary wave solutions for (2+1)- dimensional Boussinesq equation and (3+1)- dimensional KP equation[J].Physics Letters A, 2003,307:107-113.
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[24]葛晓明.有界噪声参激下Duffing振子混沌运动的解析方法[J].苏州:苏州大学学报(工科版), 2002, 20(2):57-59.
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[34]刘曾荣.混沌研究中的解析方法[M].上海:上海大学出版,2002.
[35]黄润生.混沌及其应用[M].武汉:武汉大学出版社, 2000.
[36]陈立群,刘廷柱.一类复杂力场中磁性刚体航天器的混沌姿态运动[J].北京:空间科学学报,2001,21(1): 81-85.
[37]杨翠红,朱思铭.Willis环上脑动脉瘤模型的混沌分析[J].广州:中山大学学报(自然科学版),2003, 42(3):1-3.
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[39]楼京俊,何其伟,朱石坚.多频激励软弹簧型Duffing系统中的混沌[J].重庆:应用数学和力学,2004, 25(12):1299-1304.
[40]须文波,徐振源,蔡朝洪.利用正反馈引导混沌系统到周期解[J].广州:控制理论与应用,2003,20(4): 647-649.
[41]李芳,徐振源.具有异宿轨道Duffing方程的混沌控制[J].无锡:江南大学学报(自然科学版),2005, 4(5):460-464.
[42]李险峰,褚衍东,刘晓君,邓进.控制Mathieu方程中混沌的三种方法分析[J].上海:制动与冲击,2007, 16(1):35-41.
[43] R.Grimshaw and X.Tian. Periodic and chaotic behaviour in a reduction of the perturbed Korteweg-de Vries equation [J].The Royal Society, A 1994,445:1-21.
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[2] Boussinesq J. Theory of waves and swells propagated in long horizontal rectangular canal and imparting to the liquid contained in this canal[J].France: Journal de Mathematiques Pures et Appliquees,1872,17(2):55-108.
[3] Peregrine D H.Long waves on a beach[J].J Fluid Mech. Great Britain:1967,27:815-827.
[4] Madsen P A, Banijamali B, Schaffer H A. Boussinesq Type Equation With High Accuracy in Dispersion and Nonlinearity[C].Proc 25th Intl Conf Coastal Eng. New York: asce,1996:95-108.
[5] Ranjit S. Dhaliwal, Saleem M. Khan. The axisymmetric boussinesq problem for a semi-space in micropolar theory[J].International Journal of Engineering Science,1976,14(8):769-788.
[6] Clarkson P.A, Kruskal M.D. New similarity reductions of the boussinesq equation[J].J Math Phys,1989,30(10):2201-2213.
[7] Peter A. Clarkson. Nonclassical symmetry reductions of the Boussinesq equation[J].Chaos, Solitons & Fractals.1995,12(5):2261-2301
[8] G.M.A.Allen and G.Rowlands. On the transverse instabilities of solitary waves[J].Phys. Lett.A, 1997,235,145-149.
[9] J.L.Bona, R.L. Sachs. Global existence of smooth solutions and solitary waves for a generalized Boussinesq equation[J].Comm. Math.Phys,1998,118:15-29.
[10]周渊.三类非线性方程解的物理结构:[硕士学位论文][D].成都:四川师范大学.2007.
[11]邹志利.高阶Boussinesq的改进[J].北京:中国科学E辑,1999,29(1):87-96.
[12]张宏伟,呼青英.一类广义Boussinesq方程的整体弱解和整体弱吸引子[J].北京:应用数学,2003,16(1): 23-28.
[13] H. Bredmose, H.A. Sch?ffer, P.A. Madsen Boussinesq evolution equations: numerical efficiency, breaking and amplitude dispersion [J]. Coastal Engineering, December 2004, 51,Issues 11-12: 1117-1142.
[14] A.M.Wazwaz. Variants of the two-dimensional Boussinesq equation with compactons, soiltons and periodic solutions, Computers[J].Math,Appl,2005,49,295-301.
[15]张卫国,田卫中,安俊英.广义修正Boussinesq方程椭分式形式的精确周期解[J].北京:应用数学学报,2006,29(6):1009-1110.
[16]洪丽莉,张凯,张然.线性Bossinesq方程的多辛Runge-Kutta Nystr?m算法[J].长春:吉林大学学报(理学版),2007,45(6):892-899.
[17] Yong Chen, Zhenya Yan, Honging Zhang. New explicit solitary wave solutions for (2+1)- dimensional Boussinesq equation and (3+1)- dimensional KP equation[J].Physics Letters A, 2003,307:107-113.
[18]芮伟国,冀小明,谢绍龙.广义Boussinesq方程的光滑孤子解和各种周期行波解[J].重庆:西南民族大学学报(自然科学版),2007,33(6):1229-1235.
[19]郭玉翠,徐淑奖.描述顺流方向可变剪切流动的一类变系数Boussinesq方程的Painlevé分析和相似约化[J].北京:应用数学学报,2006,29(6):1054-1062.
[20]张素香,李瑞杰,罗锋等.正交曲线坐标下Boussinesq方程的初步研究.北京:科技导报,2006,24(02): 38-42.
[21]李孟国,王正林,蒋德才.关于波浪Boussinesq方程的研究[J].青岛:青岛海洋大学学报,2002,32(3):345-354.
[22]邹恩,李祥飞,陈建国.混沌控制及其优化应用[M].长沙:国防科技大学出版社.2002.
[23]刘延柱,陈立群.非线性振动[M].北京:高等教育出版社,2001.
[24]葛晓明.有界噪声参激下Duffing振子混沌运动的解析方法[J].苏州:苏州大学学报(工科版), 2002, 20(2):57-59.
[25] Q.Cao,K.Djidjeli,W.G.Price,E.H.Twizell. Periodic and chaotic behaviour in a reduced form of the perturbed generalized Korteweg-de Vries and Kadomtsev-Petviashvili equation[J].Elsevier Science B.V. Academic Publishers: D 1999,125: 201-221.
[26]褚衍东,李险峰,张建刚.一类特殊的Mathieu方程的分岔及混沌控制[J].青岛:青岛大学学报(自然科学版),2006:19(3),30-33.
[27]雷佑铭,徐伟.有界噪声和谐和激励联合作用下一类非线性系统的混沌研究[J].北京:物理学报,2007, 56(9):5103-5110.
[28] H.T.Moon. Homoclinic crossings and pattern selection[J]. Phys. Rev.Lett,1990,64:412-414.
[29] H.T.Moon. Soliton turbulence and strange attractor[J]. Phys. Fluid A 1991,3:2709-2715.
[30] D.J.Zheng, W.J.Yeh, O.G.Symko. Periodic doubling in a perturbed Sine-Gorden system[J]. Phys. Lett,1989,179:225-228.
[31] F.Kh.Abdullaev. Dynamical chaos of solitons and nonlinear periodic waves[J]. Phys.Rep,1989, 179:1-78.
[32]刘式适,刘式达.物理学中的非线性方程[M].北京:北京大学出版社,2000.
[33]王树禾.微分方程模型与混沌[M].合肥:中国科技大学出版社,1999.
[34]刘曾荣.混沌研究中的解析方法[M].上海:上海大学出版,2002.
[35]黄润生.混沌及其应用[M].武汉:武汉大学出版社, 2000.
[36]陈立群,刘廷柱.一类复杂力场中磁性刚体航天器的混沌姿态运动[J].北京:空间科学学报,2001,21(1): 81-85.
[37]杨翠红,朱思铭.Willis环上脑动脉瘤模型的混沌分析[J].广州:中山大学学报(自然科学版),2003, 42(3):1-3.
[38] I.S.Gradshteyn, I.M.Ryzhik, Table of Integrals, Series, and Products[M]. Academic Press, New York,1980.
[39]楼京俊,何其伟,朱石坚.多频激励软弹簧型Duffing系统中的混沌[J].重庆:应用数学和力学,2004, 25(12):1299-1304.
[40]须文波,徐振源,蔡朝洪.利用正反馈引导混沌系统到周期解[J].广州:控制理论与应用,2003,20(4): 647-649.
[41]李芳,徐振源.具有异宿轨道Duffing方程的混沌控制[J].无锡:江南大学学报(自然科学版),2005, 4(5):460-464.
[42]李险峰,褚衍东,刘晓君,邓进.控制Mathieu方程中混沌的三种方法分析[J].上海:制动与冲击,2007, 16(1):35-41.
[43] R.Grimshaw and X.Tian. Periodic and chaotic behaviour in a reduction of the perturbed Korteweg-de Vries equation [J].The Royal Society, A 1994,445:1-21.
[44]陈陆军,梁昌洪.孤立子理论及其应用—光孤子理论及光孤子通讯[M].西安:西安电子科技大学出版社, 1997.
[45] Paul.F.Byrd, M.D.Friedman. Handbook of Elliptic Integrals for Engineers and Scientists[M]. Springer-Verlag, New York,1971.
[46]刘式适,刘式达.特殊函数[M].北京:气象出版社,2002.
[47]李骊.强非线性振动系统的定性理论和定量方法[M].北京:科学出版社,1996.
[48] Guoguang Lin. Uniqueness results for monopolar fluids[J]. Communications in Nonlinear Scienceand Numerical Simulation, March 2000,5(1):21-23.
[49] Boling Guo, Guoguang Lin. Existence and uniqueness of stationary solutions of non-Newtonian viscous incompressible fluids[J]. Communications in Nonlinear Science and Numerical Simulation,March 1999,4(1):63-68.
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