含慢变参数的非线性系统时变分岔问题研究综述
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- 英文论文题名：The summary of the bifurcation problems of the nonlinear time-varying system with a slowly varying parameter
- 论文作者：曹树谦 ; 马新东 ; 赵峰 ; 郭虎伦
- 英文论文作者：Ma Xindong ; Cao Shuqian ; Zhao Feng ; Guo Hulun ; Department of Mechanics ; Tianjin University ; Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control ; State Key Laboratory of Engines ; Tianjin University
- 年：2016
- 作者机构：天津大学机械工程学院力学系;天津市非线性动力学与混沌控制重点实验室;内燃机燃烧学国家重点实验室;
- 论文关键词：慢变参数 ; 非线性系统 ; 时变分岔 ; 综述
- 英文论文关键词：Slowly varying bifurcation parameter ; nonlinear dynamic system ; time-varying bifurcation ; summary
- 会议召开时间：2016-08-12
- 会议录名称：第十六届全国模态分析与试验学术会议论文集
- 语种：中文
- 分类号：O231.2
- 学会代码：ZGZH
- 会议名称：第十六届全国模态分析与试验学术会议
- 会议地点：中国天津
- 主办单位：中国振动工程学会模态分析与试验专业委员会
- 学会名称：中国振动工程学会
- 页数：15
- 文件大小：753k
- 原文格式：D
- 会议级别：全国
摘要
时变分岔是指非线性系统的分岔参数随时间慢变并穿越其定常系统分岔点时,系统的解或者轨道出现不平凡的现象。本文在关键参数对响应的影响和反馈控制两方面对时变分岔问题进行分类,总结了几类典型时变分岔研究问题的主要特征,给出了常用的研究方法,并提出了时变分岔下一步的研究方向。
Time-varying bifurcation is the phenomenon that the solution or track of systems are nontrivial if the bifurcation parameter of the nonlinear system slowly varies with time and passes through the bifurcation point of the time-independent equation. This paper gives several kinds of time-varying bifurcation problems and classical features according to the influence of key parameters on response and the feedback control. Some research methods are also addressed. The following direction of research is proposed.
Time-varying bifurcation is the phenomenon that the solution or track of systems are nontrivial if the bifurcation parameter of the nonlinear system slowly varies with time and passes through the bifurcation point of the time-independent equation. This paper gives several kinds of time-varying bifurcation problems and classical features according to the influence of key parameters on response and the feedback control. Some research methods are also addressed. The following direction of research is proposed.
引文
[1]Erneux.T.and Mandel.P.,Temporal aspects of absorptive optical bistability[J],Phys.Rev.A,1983,28(2):896~909.
[2]Mandel.P.and Erneux.T.,Laser-Lorenz equations with a time-dependent parameter[J].Phys.Rev.Lett.,1984,53(19):1818~1820.
[3]Erneux.T.and Mandel.P.,Stationary,harmonic,and pulsed operations of an optical bistable laser with saturable absorberⅡ[J],Phys.Rev.A,1984,30(4):1902~1909.
[4]Erneux.T.and Laplante.J.P.,Jump transition due to a time-dependent bifurcation parameter in the bistable iodate-arsenous acid reaction[J],J.Chem.Phys.,1989,90(11):6120~6134.
[5]Laplante.J.P.,Erneux.T.and Georgiou.M.,Jump transition due to a time-dependent bifurcation parameter:An experimental,numerical,and analytical study of the bistable iodate-arsenous acid reaction[J],J.Chem.Phys.,1991,94(1):371~378.
[6]Baer.S.M,Erneux.T.,and Rinzel.J.,The slow passage through a Hopf bifurcation:delay,memory effects and resonance.SIAM J.APPL.MATH.,1989,49(1):55~71.
[7]Erneux.T.and Mandel.P.,Slow passage through the laser first threshold[J],Phys.Rev.A,1989,39(10):5179~5188.
[8]李以农,郑玲,闻邦椿,一类具有参数慢变的非线性振动系统,重庆大学学报(自然科学版),2000,23(6).
[9]化存才.非线性时变动力系统的分岔及其应用[D].北京:北京航空航天大学理学院,2000.
[10]Manella.R.,McC lintock P.V.E.,and F.Moss,Phys.Lett.A,1987,120,11.
[11]Holden.L and Erneux.T,SIAM J.Appl.Math.,1993,53,1045.
[12]Nayfeh AH and Asfar KR,Non-stationary parametric oscillations.Journal of Sound and Vibration,1988,124(3):529–537.
[13]Neal HL and Nayfeh AH,Response of a single-degreeof-freedom system to a non-stationary principal parametric excitation.International Journal of Non-Linear Mechanics,1990,25(2/3):275–284.
[14]Ng.L.,Rand.R.,and O’Neil.M.,Slow passage through resonance in Mathieu’s equation,Journal of Vibration and Control,2003,9:685-707.
[15]Haberman.R.,Slowly varying jump and transition phenomena associated with algebraic bifurcation problems,SIAM J.Appl.Math.,1979,37:69–109.
[16]Erneux.T.and Mandel.P.,Imperfect bifurcation with a slowly varying control parameter[J],SIAM J.Appl.Math.,1986,46(1):1~15.
[17]Mandel.P.,Erneux.T.,The slow passage through a steady bifurcation Delay and memory effects[J],J.Statist.Phy.,1987,48(5/6):1059~1070.
[18]化存才,陆启韶,吸收型光学双稳态方程的时变分岔与动力学行为[J],物理学报,2000,49(4):733~740.
[19]化存才,陆启韶,抽运参数随时间慢变所诱发Laser-Lorenz系统的分岔与光学双稳态[J],物理学报,1999,48(3):408~415.
[20]Maree.G.J.M.,Sudden exchange in a second-order nonlinear system with a slowly-varying parameter,Internat.J.Non-Linear Mech.,1993,28(4):406~426.
[21]Rinzel J,Baer M.Threshold for repetitive activity for a slow stimulus ramp:a memory effect and its dependence on fluctuations.Biophys J 1988;54:551–555.
[22]Baer.S.M.and Erneux.T.,Singular Hopf bifurcation to relaxation oscillations,SIAM J.Appl.Math.,461986:721-739.
[23]Diminnie.D.C.,Haberman.R.,Slow passage through a saddle-center bifurcation,J.Nonlinear Sci.2000,10:197-221.
[24]Haberman.R.,Slow passage through a transcritical bifurcation for Hamiltonian systems and the change in action due to a nonhyperbolic homoclinic orbit,Chaos,2000,10:641-648.
[25]Diminnie.D.C.,Haberman.R.,Action and period of homoclinic and periodic orbits for the unfolding of a saddle-center bifurcation,submitted for publication.
[26]Benoit.E.,Equations diff′erentielles:relation entr′ee-sortie Comptes rendus de l’Acad′emie des Sciences de Paris,293,s′erie I,293-296(1981).
[27]Benoit.E.,Callot J.L.,Diener.F.and M.Diener,Chasse au canard.Collectanea Mathematica,Barcelone,1981,.31(1-3);37-119.
[28]化存才,刘延柱,陆启韶,具有鞍结分岔的二次系统的同宿和时变分岔,上海交通大学学报,2002,36(3).
[29]化存才,陆启韶,时变参数系统的非完全分岔及其在Duffing方程中的应用,应用数学和力学,2000,21(9).
[30]化存才,陆启韶,一类时变动力系统的高余维分岔及其控制,高校应用数学学报A辑,2000,16(2):154~162.
[31]Erneux.T.,Reise.E.L.,Holden.L.J.and Georgiou.M.,Slow passage through bifurcation and limit points.Asymptotic theory and applications,Dynamic Bifurcations Lecture Notes in Mathematics,1991,1493:14-28.
[32]Younghae Do,Juan M.Lopez,Slow passage through multiple bifurcation points,Discrete and continuous dynamical systems series B,2013,18(1):95~107.
[33]陆启韶,分岔与奇异性[M],上海科技教育出版社,上海,1995.
[34]田春红,一类具有反馈控制参数的时变分岔,郑州大学学报(理学版),2007,39(3).
[2]Mandel.P.and Erneux.T.,Laser-Lorenz equations with a time-dependent parameter[J].Phys.Rev.Lett.,1984,53(19):1818~1820.
[3]Erneux.T.and Mandel.P.,Stationary,harmonic,and pulsed operations of an optical bistable laser with saturable absorberⅡ[J],Phys.Rev.A,1984,30(4):1902~1909.
[4]Erneux.T.and Laplante.J.P.,Jump transition due to a time-dependent bifurcation parameter in the bistable iodate-arsenous acid reaction[J],J.Chem.Phys.,1989,90(11):6120~6134.
[5]Laplante.J.P.,Erneux.T.and Georgiou.M.,Jump transition due to a time-dependent bifurcation parameter:An experimental,numerical,and analytical study of the bistable iodate-arsenous acid reaction[J],J.Chem.Phys.,1991,94(1):371~378.
[6]Baer.S.M,Erneux.T.,and Rinzel.J.,The slow passage through a Hopf bifurcation:delay,memory effects and resonance.SIAM J.APPL.MATH.,1989,49(1):55~71.
[7]Erneux.T.and Mandel.P.,Slow passage through the laser first threshold[J],Phys.Rev.A,1989,39(10):5179~5188.
[8]李以农,郑玲,闻邦椿,一类具有参数慢变的非线性振动系统,重庆大学学报(自然科学版),2000,23(6).
[9]化存才.非线性时变动力系统的分岔及其应用[D].北京:北京航空航天大学理学院,2000.
[10]Manella.R.,McC lintock P.V.E.,and F.Moss,Phys.Lett.A,1987,120,11.
[11]Holden.L and Erneux.T,SIAM J.Appl.Math.,1993,53,1045.
[12]Nayfeh AH and Asfar KR,Non-stationary parametric oscillations.Journal of Sound and Vibration,1988,124(3):529–537.
[13]Neal HL and Nayfeh AH,Response of a single-degreeof-freedom system to a non-stationary principal parametric excitation.International Journal of Non-Linear Mechanics,1990,25(2/3):275–284.
[14]Ng.L.,Rand.R.,and O’Neil.M.,Slow passage through resonance in Mathieu’s equation,Journal of Vibration and Control,2003,9:685-707.
[15]Haberman.R.,Slowly varying jump and transition phenomena associated with algebraic bifurcation problems,SIAM J.Appl.Math.,1979,37:69–109.
[16]Erneux.T.and Mandel.P.,Imperfect bifurcation with a slowly varying control parameter[J],SIAM J.Appl.Math.,1986,46(1):1~15.
[17]Mandel.P.,Erneux.T.,The slow passage through a steady bifurcation Delay and memory effects[J],J.Statist.Phy.,1987,48(5/6):1059~1070.
[18]化存才,陆启韶,吸收型光学双稳态方程的时变分岔与动力学行为[J],物理学报,2000,49(4):733~740.
[19]化存才,陆启韶,抽运参数随时间慢变所诱发Laser-Lorenz系统的分岔与光学双稳态[J],物理学报,1999,48(3):408~415.
[20]Maree.G.J.M.,Sudden exchange in a second-order nonlinear system with a slowly-varying parameter,Internat.J.Non-Linear Mech.,1993,28(4):406~426.
[21]Rinzel J,Baer M.Threshold for repetitive activity for a slow stimulus ramp:a memory effect and its dependence on fluctuations.Biophys J 1988;54:551–555.
[22]Baer.S.M.and Erneux.T.,Singular Hopf bifurcation to relaxation oscillations,SIAM J.Appl.Math.,461986:721-739.
[23]Diminnie.D.C.,Haberman.R.,Slow passage through a saddle-center bifurcation,J.Nonlinear Sci.2000,10:197-221.
[24]Haberman.R.,Slow passage through a transcritical bifurcation for Hamiltonian systems and the change in action due to a nonhyperbolic homoclinic orbit,Chaos,2000,10:641-648.
[25]Diminnie.D.C.,Haberman.R.,Action and period of homoclinic and periodic orbits for the unfolding of a saddle-center bifurcation,submitted for publication.
[26]Benoit.E.,Equations diff′erentielles:relation entr′ee-sortie Comptes rendus de l’Acad′emie des Sciences de Paris,293,s′erie I,293-296(1981).
[27]Benoit.E.,Callot J.L.,Diener.F.and M.Diener,Chasse au canard.Collectanea Mathematica,Barcelone,1981,.31(1-3);37-119.
[28]化存才,刘延柱,陆启韶,具有鞍结分岔的二次系统的同宿和时变分岔,上海交通大学学报,2002,36(3).
[29]化存才,陆启韶,时变参数系统的非完全分岔及其在Duffing方程中的应用,应用数学和力学,2000,21(9).
[30]化存才,陆启韶,一类时变动力系统的高余维分岔及其控制,高校应用数学学报A辑,2000,16(2):154~162.
[31]Erneux.T.,Reise.E.L.,Holden.L.J.and Georgiou.M.,Slow passage through bifurcation and limit points.Asymptotic theory and applications,Dynamic Bifurcations Lecture Notes in Mathematics,1991,1493:14-28.
[32]Younghae Do,Juan M.Lopez,Slow passage through multiple bifurcation points,Discrete and continuous dynamical systems series B,2013,18(1):95~107.
[33]陆启韶,分岔与奇异性[M],上海科技教育出版社,上海,1995.
[34]田春红,一类具有反馈控制参数的时变分岔,郑州大学学报(理学版),2007,39(3).