Duffing系统在双参数平面上的动力学特性分析
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摘要
将单参数最大Lyapunov指数的计算推广到双参数平面上,数值计算Duffing系统在双参数平面上的最大Lyapunov指数,得到系统在参数平面上周期运动、混沌运动、各种分岔曲线的参数区域;结合系统单参数分岔图、相图、庞加莱截面图讨论了系统在参数平面上的分岔混沌过程以及阻尼系数对系统双参数特性的影响。结果表明:在双参数平面上系统出现了周期跳跃、周期倍化分岔、叉式分岔等复杂的分岔曲线,而且这些分岔曲线随阻尼系数的增加不断发生着复杂变化;得到系统在以往单参数分岔过程中很少出现的分岔曲线相交、嵌套、演变等特殊现象;阻尼系数对系统双参数耦合动力学特性有重要的影响。本文对工程中其它多参数系统的参数耦合特性的研究具有一定的参考价值。
The calculation of single parameter Top Lyapunov exponent is extended to the two-parameter plane,and the parameter regions of periodic motion,chaotic motion and bifurcation curves are obtained by calculating the Top Lyapunov exponent of Duffing system in two-parameter plane.Combining with the single-parameter bifurcation diagrams,phase diagrams and Poincarésection maps,the dynamic characters of system in two-parameter and the impact of damping coefficient a on two-parameter characters of system are discussed.Results obtained indicate that the complicated bifurcation curves of periodic jump,periodic-doubling bifurcation and pitchfork bifurcation of system take place in two-parameter plane,and the bifurcation curves continue to change with the increase of damping coefficient.Some special phenomenon,such as bifurcation curves intersect,nesting and evolution,that the system has not met in the past single-parameter bifurcation,are obtained.It can be concluded that the damping coefficient a has significant influence on dynamic character of system in two-parameter plane.Current research has some reference values for the study of the parameter coupling characteristics of other multi-parameter systems in engineering.
The calculation of single parameter Top Lyapunov exponent is extended to the two-parameter plane,and the parameter regions of periodic motion,chaotic motion and bifurcation curves are obtained by calculating the Top Lyapunov exponent of Duffing system in two-parameter plane.Combining with the single-parameter bifurcation diagrams,phase diagrams and Poincarésection maps,the dynamic characters of system in two-parameter and the impact of damping coefficient a on two-parameter characters of system are discussed.Results obtained indicate that the complicated bifurcation curves of periodic jump,periodic-doubling bifurcation and pitchfork bifurcation of system take place in two-parameter plane,and the bifurcation curves continue to change with the increase of damping coefficient.Some special phenomenon,such as bifurcation curves intersect,nesting and evolution,that the system has not met in the past single-parameter bifurcation,are obtained.It can be concluded that the damping coefficient a has significant influence on dynamic character of system in two-parameter plane.Current research has some reference values for the study of the parameter coupling characteristics of other multi-parameter systems in engineering.
引文
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[15]DAI Guowei.Two global several-parameter bifurcation theorems and their applications[J].Journal of Mathematical Analysis and Applications,2016,433(2):749-761.
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[17]李瑞红,徐伟,李爽.含三次耦合项的两自由度Duffing系统的共振及混沌行为[J].应用力学学报,2007,24(2):200-203.(LI Ruihong,XU Wei,LI Shuang.Resonance and chaos behavior in a two-degree-of-freedom Duffing system with cubic coupled Terms[J].Chinese Journal of Applied Mechanics,2007,24(2):200-203(in Chinese)).
[2]SOUZA S L T D,CALDAS I L.Calculation of Lyapunov exponents in systems with impacts[J].Chaos Solitons&Fractals,2004,19(3):569-579.
[3]魏艳辉,徐洁琼,黄龙生.两自由度碰撞振动系统的Lyapunov指数谱分析[J].振动与冲击,2009,28(1):60-65.(WEI Yanhui,XU Jieqiong,HUANG Longsheng.Spectral analysis of Lyapunov exponents for a two-degree-of-freedom vibro-impact system[J].Journal of Vibration and Shock,2009,28(1):60-65(in Chinese)).
[4]GOU Xiangfeng,ZHU Lingyun,CHEN Dailin.Bifurcation and chaos analysis of spur gear pair in two-parameter plane[J].Nonlinear Dynamics,2014,79(3):2225-2235.
[5]吴淑花,孙毅,郝建红,等.耦合发电机系统的分岔和双参数特性[J].物理学报,2011,60(1):139-157.(WU Shuhua,SUN Yi,HAO Jianhong,et al.Bifurcation and dual-parameter characteristic of the coupled dynamos system[J].Acta Physica Sinica,2011,60(1):139-157(in Chinese)).
[6]ZHANG Chun,BI Qinsheng.On two-parameter bifurcation analysis of the periodic parameter-switching Lorenz oscillator[J].Nonlinear Dynamics,2015,81(1/2):1-7.
[7]QIN Zhang,CHEN Yushu.Singular analysis of bifurcation systems with two parameters[J].Acta Mechanica Sinica,2010,26(3):501-507.
[8]ZHANG Chun,BI Qinsheng.On two-parameter bifurcation analysis of switched system composed of Duffing and van der Pol oscillators[J].Communications in Nonlinear Science and Numerical Simulation,2014,19(3):750-757.
[9]LUO G W,LV X H,SHI Y Q.Vibro-impact dynamics of a two-degree-of freedom periodically-forced system with a clearance:diversity and parameter matching of periodic-impact motions[J].International Journal of Non-Linear Mechanics,2014,65(4):173-195.
[10]秦朝红.非线性动力学双参量奇异性方法及其工程应用[D].哈尔滨:哈尔滨工业大学,2010.(QIN Chaohong.Singularity method for nonlinear dynamical analysis of systems with two parameters and its application in engineering[D].Harbin:Harbin Institute of Technology,2010(in Chinese)).
[11]TOANNA J F,PIIROINEN P T.Interactions between global and grazing bifurcations in an impacting system[J].Chaos,2011,21(1):1121-1122.
[12]THOTA P,KRAUSKOPF B,LOWENBERG M.Multi-parameter bifurcation study of shimmy oscillations in a dual-wheel aircraft nose landing gear[J].Nonlinear Dynamics,2012,70(2):1675-1688.
[13]LI Xiaofeng,CHU Yandong,ZHANG Hui.Fractal structures in a generalized square map with exponential terms[J].Chinese Physics B,2012(3):15-21.
[14]杨娟,卞保明,彭刚,等.随机信号双参数脉冲模型的分形特征[J].物理学报,2011,60(1):139-157.(YANG Juan,BIAN Baoming,PENG Gang,et al.The fractal character of two-parameter pulse model for random signal[J].Acta Physica Sinica,2011,60(1):139-157(in Chinese)).
[15]DAI Guowei.Two global several-parameter bifurcation theorems and their applications[J].Journal of Mathematical Analysis and Applications,2016,433(2):749-761.
[16]LOI N Y.On two-parameter global bifurcation of periodic solutions to a class of differential variational inequalities[J].Nonlinear Analysis,2015,122:83-99.
[17]李瑞红,徐伟,李爽.含三次耦合项的两自由度Duffing系统的共振及混沌行为[J].应用力学学报,2007,24(2):200-203.(LI Ruihong,XU Wei,LI Shuang.Resonance and chaos behavior in a two-degree-of-freedom Duffing system with cubic coupled Terms[J].Chinese Journal of Applied Mechanics,2007,24(2):200-203(in Chinese)).