具有非线性恢复力和外力激励的Duffing-Van der Pol系统的复杂动态
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摘要
本文应用动力系统的分支理论,二阶平均方法,Melnikov方法和混沌理论,研究带非线性恢复力和外力激励的Duffing-Van der Pol方程随系统参数变化的复杂动态行为,我们给出了谐波介和其分支存在的条件,以及在周期扰动下系统产生混沌的准则,也给出了在ω_2=nω_1+εv,n=1,2,3,4,5,7的拟周期扰动下平均系统产生混沌的准则,数值模拟验证了理论结果正确性,而用平均方法不能给出在ω=nw_1+εv,n=6,8,9-15(这里ω_1与v不为有理数)的拟周期扰动下产生混沌的准则,但数值模拟显示了原系统出现混沌。同时,用数值模拟(包括同宿和异宿分支曲面,动态分支图,最大Lyapunov指数图,相图,Poincared映射图)发现了许多新的复杂动态。我们发现周期倍分支和来自周期一和周期三轨的逆周期倍分支到一个混沌吸引子或到两个分离的混沌吸引子,混沌行为和周期窗口的交替出现,混沌的突然出现和突然收敛到周期二轨,带有复杂周期窗口和不带周期窗口的大范围混沌区域,不带周期窗口的不变环,具有复杂不变环的混沌区域以及内部危机。本文研究具有非线性恢复力和外力激励的Duffing-Van der Pol方程是前人未研究过的,这研究将丰富动力系统的内容,并具有一定的应用前景。
全文共分三章。第一章是关于连续动力系统的分支理论、二阶平均方法、Melnikov方法与混沌理论的预备知识。
第二章简单介绍了Duffing-Van der Pol方程的一些历史背景知识。
第三章应用二阶平均方法和Melnikov理论研究具有非线性恢复力和外力激励的Duffing-Van der Pol方程,给出了谐波介和其分支存在条件以及周期扰动下系统产生混沌的准则,也给出了在ω=nω_1+εv,n=1,2,3,4,5,7的拟周期扰动下平均系统产生混沌的准则,而不能给出在ω=nω_1+εv,n=6,8,9-15的拟周期扰动下产生混沌的准则,但数值模拟显示了原系统出现混沌,用数值模拟我们给出十个系统参数变化时系统动态的变化。
The complex dynamical behaviours as the system's parametrics varing in Duffing-Van der Pol equation with nonlinear restoring and external excitations are investigated by using bifurcation theory,second-order averaging methods,Melnikov methods and chaotic theory.Wo prove that conditions of existence of harmoniz and its bifurcations,and The threshold values of existence of chaotic motion under the periodic perturbation.By applying the second-order averaging method and Melnikov method,we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω_2 = nω_1 +∈ν,n=1,2,3,4,5,7,and cannot prove the criterion of existence of chaos in second-order averaged system under quasiperiodic perturbation forω_2 = nω_1 +∈ν,n = 6,8,9 - 15,whereω_1 is rational toν, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams,Lyapunov exponent,phase portraits and Poincare(?) map,not only show the consistence with the theoretical analysis but also exhibit some new complex dynamics,we found that periodic doubling bifurcation and the reversed periodic from period-one-three doubling bifurcation leading to chaos,and the interleaving occurrences of chaotic behaviors and periodic windows,the onset of chaos and chaos suddenly conventing to periodic-two ubits,the large chaotic regions with complex period-windows and without period-windows,the chaotic regious with complex invariant torus,the region of invariant torus without period-windows,and the interior crisis.
The paper consists of three chapters,as the following,Chapter 1 is the preparation knowledge.A brief review of second-order averaging methods and Melnikov methods,chaos and some routes to chaos for continual dynamical system is presented.
In chapter 2,we briefly introduce the backgrounds and histories of Duffing Duffing-Van der Pol equations.
In chapter 3,we study Duffing-Van der Pol equation with nonlinear restoring and external excitations.we prove that the conditions of harmoniz and its bifurcation, and the threshold values of existence of chaotic motion under the periodic perturbation,and the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω= nω_1 +∈ν,n = 1,2,3,4,5,7,and cannot prove the criterion of existence of chaos under quasi-periodic perturbation forω_2= nω_1 +∈ν,n = 6,8,9 - 15,but can show the occurrence of chaos in original system by numerical simulation.we find many new complex dynamics as the ten-system's parametries varing by numerical simulation.
全文共分三章。第一章是关于连续动力系统的分支理论、二阶平均方法、Melnikov方法与混沌理论的预备知识。
第二章简单介绍了Duffing-Van der Pol方程的一些历史背景知识。
第三章应用二阶平均方法和Melnikov理论研究具有非线性恢复力和外力激励的Duffing-Van der Pol方程,给出了谐波介和其分支存在条件以及周期扰动下系统产生混沌的准则,也给出了在ω=nω_1+εv,n=1,2,3,4,5,7的拟周期扰动下平均系统产生混沌的准则,而不能给出在ω=nω_1+εv,n=6,8,9-15的拟周期扰动下产生混沌的准则,但数值模拟显示了原系统出现混沌,用数值模拟我们给出十个系统参数变化时系统动态的变化。
The complex dynamical behaviours as the system's parametrics varing in Duffing-Van der Pol equation with nonlinear restoring and external excitations are investigated by using bifurcation theory,second-order averaging methods,Melnikov methods and chaotic theory.Wo prove that conditions of existence of harmoniz and its bifurcations,and The threshold values of existence of chaotic motion under the periodic perturbation.By applying the second-order averaging method and Melnikov method,we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω_2 = nω_1 +∈ν,n=1,2,3,4,5,7,and cannot prove the criterion of existence of chaos in second-order averaged system under quasiperiodic perturbation forω_2 = nω_1 +∈ν,n = 6,8,9 - 15,whereω_1 is rational toν, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams,Lyapunov exponent,phase portraits and Poincare(?) map,not only show the consistence with the theoretical analysis but also exhibit some new complex dynamics,we found that periodic doubling bifurcation and the reversed periodic from period-one-three doubling bifurcation leading to chaos,and the interleaving occurrences of chaotic behaviors and periodic windows,the onset of chaos and chaos suddenly conventing to periodic-two ubits,the large chaotic regions with complex period-windows and without period-windows,the chaotic regious with complex invariant torus,the region of invariant torus without period-windows,and the interior crisis.
The paper consists of three chapters,as the following,Chapter 1 is the preparation knowledge.A brief review of second-order averaging methods and Melnikov methods,chaos and some routes to chaos for continual dynamical system is presented.
In chapter 2,we briefly introduce the backgrounds and histories of Duffing Duffing-Van der Pol equations.
In chapter 3,we study Duffing-Van der Pol equation with nonlinear restoring and external excitations.we prove that the conditions of harmoniz and its bifurcation, and the threshold values of existence of chaotic motion under the periodic perturbation,and the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω= nω_1 +∈ν,n = 1,2,3,4,5,7,and cannot prove the criterion of existence of chaos under quasi-periodic perturbation forω_2= nω_1 +∈ν,n = 6,8,9 - 15,but can show the occurrence of chaos in original system by numerical simulation.we find many new complex dynamics as the ten-system's parametries varing by numerical simulation.
引文
[1]Cai,M.X.＆ Yang,J.P.Bifurcation of Periodic Orbits and Chaos in Duffing Equation.Acta Mathematicae Applicatae Sinica.2006,22(3):495-508.
[2]Cartwright,M.L.＆ Littlewood,J.E.On nonlinear differential equations of the second order.Ⅰ.The equation(y|¨) + k(1 - y~2)(?) + y = bλk cos(λt + a),k large.J.Lond.Math.Soc,1945,20:180-189.
[3]Devaney,R.L.An Introduction to Chaotic Dynamical Systems.Aoldison-Wesley,Reduood City,Calif.1989.
[4]Duffing,G.Erzwungene Schwingungen bei Ver(a|¨)nderlicher Eigenfrequenz.F.Vieweg u.Sohn:Braunshweig.
[5]FitzHugh,R.Impulses,physiological states in theoretical models of nerve membrane.Biophys.J,1961,1:445-466.
[6]Guckenheimer,J.& Hoffman,K& Weckesser,W.The forced Van der Pol equation.I.The slow flow and its bifurcations.SIAM J Appl Dyn Syst,2003,2(1):1-35.
[7]Guckenheimer,J& Holmes,P.Nonlinear Oscillations,Dynamical Systems and Bifurcations of Vector Fields.New York:Springer-Verlag;1983.
[8]Guckenheimer,J.& Holmes,P.Dynamical systems and bifurcations of vector fields.New York:Springer-Verlag;1997.
[9]Hirano,N.& Rybicki,S.Existence of limit cycles for coupled van der Pol equations.J.Differential Equations,2003,195:194-209.
[10]Hodgkin,A.L.& Huxley,A.F.A quantitative description of membrane current and its application to conduction and excitation in nerve.J.Physiol,1952,117:500-544.
[11]Hale,J.K.Ordinary Differential Equation.New York:John Wiley and Sons;1969.
[12]Holden,A.V.Chaos.Princeton,NJ,Princeton University Press;1986.
[13]Holmes,C.＆ Holmes,P.Second order averaging and bifurcations to subharmonics in DuJfing's equation.Journal of Sound and Vibration,1981,78(2):161-174.
[14]Holmes,P.& Whitley,D.On the attracting set for Duffing's equation.Physica D 1983,111-23.
[15]井竹君.混沌简介.数学的实践与认识,1991,1:81-94.
[16]Jing,Z.J.Chaotic behavior in the Josephson equations with periodic force.Siam.J.Appl.Math,1989,4.
[17]Jing,Z.J,Huang JC,Deng J.Complex dynamics in three-well Duffing system with two external forcings.Chaos,Solitons and Fractals.2007,33:795-812.
[18]Jing,Z.J.& Wang,P.G.Analysis of Duffing's equation with periodic force and without damping harmonic solutions and subharmonic solutions and chaotic solutions-A study of biomathematical model of aneurysm of Wills.Ann.of Diff.Eqs,1989,5.
[19]Jing,Z.J.& Wang,R.Q.Complex dynamics in Duffing system with two external forcings.Chaos,Solitons and Fractals,2005,23:399-411.
[20]Jing,Z.J.& Xu,P.C.Chaos behaviors for Duffing equation with two external forcing terms.Progr Natural Sci.1999,9(3):171-179.
[21]Kenfark,A.Bifurcation structure of the compel periodically driven doublewell duffing oscillations.Chaos,Solitons and Fractals.2003,15:205-218.
[22]Kakmeni,F.M.M.& Bowvng,S.& Tchawoua,C.& Kaptouom,E.Strange attractors an chaos control in a Duffing-Van der oscillator with two external periodic forces.J.Sound Viber,2004,227:783-799.
[23]Kao,Y.H.& Wang,C.S.Analog study of bifurcation structures in a Van der Pol oscillator with a nonlinear restoring force.Phys.Rev.E,1993,48:2514-2520.
[24]Kapitaniak,T.Analytical condition for chaotic behavior of the Duffing oscillation.Phys Lett A,1990,144(6,7):322-324.
[25]Kapitaniak,T.& Steeb,W.H.Transition to chaos in a generalized Van der Pol's equation.Journal of Sound and Vibration,1990,143(1):167-170.
[26]Kapitaniak,T.Chaotic oscillations in mechanical systems.Manchester University Press,UK;1991.
[27]Kenfark,A.Bifurcation structure of two coupled periodically driven doublewell Duffing oscillators.Chaos,Solitons and Fractals,2003,15:205-218.
[28]Lakshmanan,M.& Murali,M.Chaos in nonlinear oscillations.World Scientific;1996.
[29]Leung,A.Y.T.& Zhang,Q.L.Complex normal form for strongly non-linear vibration systems exemplified by Duffing-Van der Pol equation.J.Sound Viber,1998,213(5):907-914.
[30]Levinson,N.A second-order differential equation with singular solutions.Ann.Math,1949,50:127-153.
[31]Li GX,Moon FC.Criteria for chaos of a three-well potential oscillator with homoclinic and heteroclinic orbits.Journal of Sound and Vibration,1996,136(1):17-34.
[32]Li,T.Y.& Yorke,J.A.Period Three Implies Chaos.Amer.Math.Monthly,1975,82:985-992.
[33]Liang,Y.& Namachchuraya,N.S.P-bifurcation in the noise Duffing-Van der Pol equation.Springer,New York:Stochastic Dynamics,1999,pp.49-70.
[34]Marinca,V.＆ Herisanu,N.Periodic solutions of Duffing equation with strong non-linearity.Chaos,Solitons and Fractals.2008,37:144-149.
[35]Moon,F.C.Chaotic and fractal dynamics.New York:Wiley;1992.
[36]Murakami,W.&Murakami,C.&Hirose,Kei-ichi.＆ Ichikawa,Y.H.Intcgrable duff-ing's maps and solutions of the duffing equation.Chaos,Solitons and Fractals.2003,15:425-443.
[37]Marotto,F.R.Snap-back Repellers Imply Chaos in R~n.J.Mathematical Analysis and Applications,1978,63:199-223.
[38]Medio,A.& Lines,M.Nonlinear Dynamics.Cambridge:Cambridge University Press;2001.
[39]Melnikov,V.K.On the Stability of the Center for Time Periodic Perturbations.Translation of the Moscow Mathematical Society,1963,12:1-56..
[40]Moon,F.C.& Holmes,P.J.A maganetoelastic strange attractor.J.Sound Vib,1979,65(2):285-296.
[41]Moon,F.C.& Holmes,P.J.Addendum:a magnetoelastic strange attractor J.Sound Vib,1980,69(2):339.
[42]Nagumo,J.＆ Arimoto,S.＆ Yoshizawa,S.An active pulse transmission line simulating nerve axon.Proc.IRE,1962,50:493-498.
[43]Nawachchivaga,N.S.＆ Sowers,R.B.& Vedula,L.Non-standard reduction of noise Duffing- Van der Pol equation.Dyn.Syst,2001,16(3):223-245.
[44]Nitecki,Z.Differentiable Dynamics.M.IT Press;1971.
[45]Ott,E.Chaos in Dynamical Systems.Cambridge University Press;1993.
[46]Panagotoumakos,D.E.& Theotokoglou,E.E.& Markakis,M.P.Exact analytic solutions for the dumped Duffing nonlinear oscillator.C.R.Mecanique.2006,334:311-316.
[47]Parlitz,V.& Lauterborn,W.Supersturcture in the bifurcation set of Du ± ng equation.Phys Lett A.1985,107:351-355.
[48]Parlitz,U.& Lauterborn W.Period-doubling cascades and Peril's staircases of the Driven Van der Pol oscillator.Phys.Rev.A,1987,36:1482-1434.
[49]Rajasikar,S.& Parthasarathy,S.& Lakshmanan,M.Prediction of horseshoe chaos in BVP and DVP oscillators.Chaos,Solitons and Fractals,1992,2:271-280.
[50]Ramos,J.I.On linstedt Poincar(?) techniques for the quintiz Duffing equation.Applied Mathematics and Computation.2007,193:303-310.
[51]Rio,E.D.& Velarde,M.G.& Lozanno,A.R.Long time date series and di ± culties with characterization of chaotic attractors:a case with intermittency Ⅲ.Chaos,Solitons and Fractals 1994,4(12):2169-79.
[52]Richard,H.R.& Armbruster,D.Perturbation Methods,Bifurcation Theory and Computer Algebra.Springer-Verlag;1987.
[53]Sanders,J.A.& Verhulst,F.Averaging Methods in Nonlinear Dynamical Systems.New York:Springer-Verlag;1985.
[54]Sparrow,C.The Lorenz Equations.Berlin:Springer Verlag;1982.
[55]Touhey,P.Yet Another Definition of Chaos.Amer.Math.Monthly,1997,104:411-414.
[56]Ueda,Y.Strange attractors and the origin of chaos.Nonlinear Science Today 1992;2(2):1-16.中译文见:数学译林,1995,2:117-136
[57]Van der Pol,B.Forced oscillations in a circuit with nonlinear resistance(receptance with reactive triode).London,Edinburgh and Dublin Phil.Mag,1927,3:65-80.
[58]Van der Pol,B.& Van der Mark,J.Frequency demultiplication.Nature,1927,120:363-364.
[59]文兰.动力系统简介.数学进展,2002;31(4).
[60]Wakako,M.& Chieko,M.& Koi-ichi,H.& Yoshi,H.I.Integrable Duffing's maps and solutions of the Duffing equation.Chaos,Solitons and Fractals,2003,15(3):425-443.
[61]Wiggins,S.Global Bifurcations and Chaos,Analytical Methods.Berlin:Springer-Verlag;1988.
[62]Wiggins,S.Introduction to applied nonlinear dynamical System and chaos.New York:Springer-Verlag;1990.
[63]Yagasaki,K.Second-order averaging and chaos in quasi-periodically forced weakly nonlinear oscillators.Physica D,1990,44:445-458.
[64]Yagasaki,K.Homoclinic tangles,phase locking,and chaos in a two frequency perturbation of Duffing's equation.J.Nonlinear Sci,1999,9:131-148.
[65]Yagasaki,K.Detection of bifurcation structures by higher-order averaging for Duffing's equation.Nonlinear Dynamics,1999,18:121-158.
[66]Yagasaki,K.Homoclinic motions and chaos in the quasiperiodically forced van der Pol-Duffing oscillator with single well potential.Proc.R.Soc.Lond.A,1994,445:59-617.
[67]Yagasaki,K.Second-order averaging and Melnikov analysis for forced nonlinear oscillators.J.Sound and Vibration,1996,190(4):587-609.
[68]Yagasaki,K.Higher-order averaging and ultra subharmonics in forced oscillators.Joural of Sound and Vibration,1998,210(4):529-553.
[69]Yagasaki,K.& Zchikawa,T.Higher-order averaging for periodically forced,weakly nonlinear systems.International J.of Bifur.and Chaos,1999,9(3):519-531,.
[70]Zhang,M.&Yang,J.P.Bifuractions and chaos in Duffing equation.Acta Mathematicae Applicatae Sinica.2007,23:655-684.
[2]Cartwright,M.L.＆ Littlewood,J.E.On nonlinear differential equations of the second order.Ⅰ.The equation(y|¨) + k(1 - y~2)(?) + y = bλk cos(λt + a),k large.J.Lond.Math.Soc,1945,20:180-189.
[3]Devaney,R.L.An Introduction to Chaotic Dynamical Systems.Aoldison-Wesley,Reduood City,Calif.1989.
[4]Duffing,G.Erzwungene Schwingungen bei Ver(a|¨)nderlicher Eigenfrequenz.F.Vieweg u.Sohn:Braunshweig.
[5]FitzHugh,R.Impulses,physiological states in theoretical models of nerve membrane.Biophys.J,1961,1:445-466.
[6]Guckenheimer,J.& Hoffman,K& Weckesser,W.The forced Van der Pol equation.I.The slow flow and its bifurcations.SIAM J Appl Dyn Syst,2003,2(1):1-35.
[7]Guckenheimer,J& Holmes,P.Nonlinear Oscillations,Dynamical Systems and Bifurcations of Vector Fields.New York:Springer-Verlag;1983.
[8]Guckenheimer,J.& Holmes,P.Dynamical systems and bifurcations of vector fields.New York:Springer-Verlag;1997.
[9]Hirano,N.& Rybicki,S.Existence of limit cycles for coupled van der Pol equations.J.Differential Equations,2003,195:194-209.
[10]Hodgkin,A.L.& Huxley,A.F.A quantitative description of membrane current and its application to conduction and excitation in nerve.J.Physiol,1952,117:500-544.
[11]Hale,J.K.Ordinary Differential Equation.New York:John Wiley and Sons;1969.
[12]Holden,A.V.Chaos.Princeton,NJ,Princeton University Press;1986.
[13]Holmes,C.＆ Holmes,P.Second order averaging and bifurcations to subharmonics in DuJfing's equation.Journal of Sound and Vibration,1981,78(2):161-174.
[14]Holmes,P.& Whitley,D.On the attracting set for Duffing's equation.Physica D 1983,111-23.
[15]井竹君.混沌简介.数学的实践与认识,1991,1:81-94.
[16]Jing,Z.J.Chaotic behavior in the Josephson equations with periodic force.Siam.J.Appl.Math,1989,4.
[17]Jing,Z.J,Huang JC,Deng J.Complex dynamics in three-well Duffing system with two external forcings.Chaos,Solitons and Fractals.2007,33:795-812.
[18]Jing,Z.J.& Wang,P.G.Analysis of Duffing's equation with periodic force and without damping harmonic solutions and subharmonic solutions and chaotic solutions-A study of biomathematical model of aneurysm of Wills.Ann.of Diff.Eqs,1989,5.
[19]Jing,Z.J.& Wang,R.Q.Complex dynamics in Duffing system with two external forcings.Chaos,Solitons and Fractals,2005,23:399-411.
[20]Jing,Z.J.& Xu,P.C.Chaos behaviors for Duffing equation with two external forcing terms.Progr Natural Sci.1999,9(3):171-179.
[21]Kenfark,A.Bifurcation structure of the compel periodically driven doublewell duffing oscillations.Chaos,Solitons and Fractals.2003,15:205-218.
[22]Kakmeni,F.M.M.& Bowvng,S.& Tchawoua,C.& Kaptouom,E.Strange attractors an chaos control in a Duffing-Van der oscillator with two external periodic forces.J.Sound Viber,2004,227:783-799.
[23]Kao,Y.H.& Wang,C.S.Analog study of bifurcation structures in a Van der Pol oscillator with a nonlinear restoring force.Phys.Rev.E,1993,48:2514-2520.
[24]Kapitaniak,T.Analytical condition for chaotic behavior of the Duffing oscillation.Phys Lett A,1990,144(6,7):322-324.
[25]Kapitaniak,T.& Steeb,W.H.Transition to chaos in a generalized Van der Pol's equation.Journal of Sound and Vibration,1990,143(1):167-170.
[26]Kapitaniak,T.Chaotic oscillations in mechanical systems.Manchester University Press,UK;1991.
[27]Kenfark,A.Bifurcation structure of two coupled periodically driven doublewell Duffing oscillators.Chaos,Solitons and Fractals,2003,15:205-218.
[28]Lakshmanan,M.& Murali,M.Chaos in nonlinear oscillations.World Scientific;1996.
[29]Leung,A.Y.T.& Zhang,Q.L.Complex normal form for strongly non-linear vibration systems exemplified by Duffing-Van der Pol equation.J.Sound Viber,1998,213(5):907-914.
[30]Levinson,N.A second-order differential equation with singular solutions.Ann.Math,1949,50:127-153.
[31]Li GX,Moon FC.Criteria for chaos of a three-well potential oscillator with homoclinic and heteroclinic orbits.Journal of Sound and Vibration,1996,136(1):17-34.
[32]Li,T.Y.& Yorke,J.A.Period Three Implies Chaos.Amer.Math.Monthly,1975,82:985-992.
[33]Liang,Y.& Namachchuraya,N.S.P-bifurcation in the noise Duffing-Van der Pol equation.Springer,New York:Stochastic Dynamics,1999,pp.49-70.
[34]Marinca,V.＆ Herisanu,N.Periodic solutions of Duffing equation with strong non-linearity.Chaos,Solitons and Fractals.2008,37:144-149.
[35]Moon,F.C.Chaotic and fractal dynamics.New York:Wiley;1992.
[36]Murakami,W.&Murakami,C.&Hirose,Kei-ichi.＆ Ichikawa,Y.H.Intcgrable duff-ing's maps and solutions of the duffing equation.Chaos,Solitons and Fractals.2003,15:425-443.
[37]Marotto,F.R.Snap-back Repellers Imply Chaos in R~n.J.Mathematical Analysis and Applications,1978,63:199-223.
[38]Medio,A.& Lines,M.Nonlinear Dynamics.Cambridge:Cambridge University Press;2001.
[39]Melnikov,V.K.On the Stability of the Center for Time Periodic Perturbations.Translation of the Moscow Mathematical Society,1963,12:1-56..
[40]Moon,F.C.& Holmes,P.J.A maganetoelastic strange attractor.J.Sound Vib,1979,65(2):285-296.
[41]Moon,F.C.& Holmes,P.J.Addendum:a magnetoelastic strange attractor J.Sound Vib,1980,69(2):339.
[42]Nagumo,J.＆ Arimoto,S.＆ Yoshizawa,S.An active pulse transmission line simulating nerve axon.Proc.IRE,1962,50:493-498.
[43]Nawachchivaga,N.S.＆ Sowers,R.B.& Vedula,L.Non-standard reduction of noise Duffing- Van der Pol equation.Dyn.Syst,2001,16(3):223-245.
[44]Nitecki,Z.Differentiable Dynamics.M.IT Press;1971.
[45]Ott,E.Chaos in Dynamical Systems.Cambridge University Press;1993.
[46]Panagotoumakos,D.E.& Theotokoglou,E.E.& Markakis,M.P.Exact analytic solutions for the dumped Duffing nonlinear oscillator.C.R.Mecanique.2006,334:311-316.
[47]Parlitz,V.& Lauterborn,W.Supersturcture in the bifurcation set of Du ± ng equation.Phys Lett A.1985,107:351-355.
[48]Parlitz,U.& Lauterborn W.Period-doubling cascades and Peril's staircases of the Driven Van der Pol oscillator.Phys.Rev.A,1987,36:1482-1434.
[49]Rajasikar,S.& Parthasarathy,S.& Lakshmanan,M.Prediction of horseshoe chaos in BVP and DVP oscillators.Chaos,Solitons and Fractals,1992,2:271-280.
[50]Ramos,J.I.On linstedt Poincar(?) techniques for the quintiz Duffing equation.Applied Mathematics and Computation.2007,193:303-310.
[51]Rio,E.D.& Velarde,M.G.& Lozanno,A.R.Long time date series and di ± culties with characterization of chaotic attractors:a case with intermittency Ⅲ.Chaos,Solitons and Fractals 1994,4(12):2169-79.
[52]Richard,H.R.& Armbruster,D.Perturbation Methods,Bifurcation Theory and Computer Algebra.Springer-Verlag;1987.
[53]Sanders,J.A.& Verhulst,F.Averaging Methods in Nonlinear Dynamical Systems.New York:Springer-Verlag;1985.
[54]Sparrow,C.The Lorenz Equations.Berlin:Springer Verlag;1982.
[55]Touhey,P.Yet Another Definition of Chaos.Amer.Math.Monthly,1997,104:411-414.
[56]Ueda,Y.Strange attractors and the origin of chaos.Nonlinear Science Today 1992;2(2):1-16.中译文见:数学译林,1995,2:117-136
[57]Van der Pol,B.Forced oscillations in a circuit with nonlinear resistance(receptance with reactive triode).London,Edinburgh and Dublin Phil.Mag,1927,3:65-80.
[58]Van der Pol,B.& Van der Mark,J.Frequency demultiplication.Nature,1927,120:363-364.
[59]文兰.动力系统简介.数学进展,2002;31(4).
[60]Wakako,M.& Chieko,M.& Koi-ichi,H.& Yoshi,H.I.Integrable Duffing's maps and solutions of the Duffing equation.Chaos,Solitons and Fractals,2003,15(3):425-443.
[61]Wiggins,S.Global Bifurcations and Chaos,Analytical Methods.Berlin:Springer-Verlag;1988.
[62]Wiggins,S.Introduction to applied nonlinear dynamical System and chaos.New York:Springer-Verlag;1990.
[63]Yagasaki,K.Second-order averaging and chaos in quasi-periodically forced weakly nonlinear oscillators.Physica D,1990,44:445-458.
[64]Yagasaki,K.Homoclinic tangles,phase locking,and chaos in a two frequency perturbation of Duffing's equation.J.Nonlinear Sci,1999,9:131-148.
[65]Yagasaki,K.Detection of bifurcation structures by higher-order averaging for Duffing's equation.Nonlinear Dynamics,1999,18:121-158.
[66]Yagasaki,K.Homoclinic motions and chaos in the quasiperiodically forced van der Pol-Duffing oscillator with single well potential.Proc.R.Soc.Lond.A,1994,445:59-617.
[67]Yagasaki,K.Second-order averaging and Melnikov analysis for forced nonlinear oscillators.J.Sound and Vibration,1996,190(4):587-609.
[68]Yagasaki,K.Higher-order averaging and ultra subharmonics in forced oscillators.Joural of Sound and Vibration,1998,210(4):529-553.
[69]Yagasaki,K.& Zchikawa,T.Higher-order averaging for periodically forced,weakly nonlinear systems.International J.of Bifur.and Chaos,1999,9(3):519-531,.
[70]Zhang,M.&Yang,J.P.Bifuractions and chaos in Duffing equation.Acta Mathematicae Applicatae Sinica.2007,23:655-684.
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