有界噪声激励下单势阱碰撞振动系统的混沌运动
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摘要
碰撞振动系统的混沌运动是非光滑系统动力学研究的热点问题之一.本文研究了谐和与有界噪声激励联合作用下带平方非线性项的单边碰撞振动系统的同宿轨与混沌运动.通过计算系统的Melnikov函数,推导出系统产生Smale马蹄混沌的必要条件,结合数值仿真验证了该条件的正确性.研究表明,在一定参数条件下有界噪声既可以诱导混沌运动,也可以抑制混沌运动.该研究结果为实现混沌控制提供理论指导.
The chaotic motion of the vibro-impact system is one of the hot topic in the research of non-smooth system dynamics.The homoclinic orbit and chaos for a single potential well vibro-impact system under harmonic and bounded noise excitation are investigated.By calculating the Melnikov function,the necessary conditions for occurrence of Smale horseshoe chaos are derived.Numerical simulations are carried out to verify the correctness of certain conditions.The results show that bounded noise can either induce or inhibit chaotic motion in certain parameter conditions,which provides the theoretical guidance for the realization of chaos control.
The chaotic motion of the vibro-impact system is one of the hot topic in the research of non-smooth system dynamics.The homoclinic orbit and chaos for a single potential well vibro-impact system under harmonic and bounded noise excitation are investigated.By calculating the Melnikov function,the necessary conditions for occurrence of Smale horseshoe chaos are derived.Numerical simulations are carried out to verify the correctness of certain conditions.The results show that bounded noise can either induce or inhibit chaotic motion in certain parameter conditions,which provides the theoretical guidance for the realization of chaos control.
引文
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[11]冯进钤,金宇寰,刘亚妮.谐和与白噪声激励下碰撞振动系统的混沌运动[J].纺织高校基础科学学报,2018,31(2):FENG J Q,JIN Y H,LIU Y N.Chaos of vibro-impact system under harmonic and white noise excitations[J].Basic Sciences Journal of Textile Universities,2018,31(2):(in Chinese)
[12]HOLMES P,MARSDEN J E.A partial differential equation with infinitely many periodic orbits:Chaotic oscillations of a forced beam[J].Rch Ration Mech Anal,1981,76(2):135-166.
[13]XU W,FENG J Q,RONG H W.Melnikov′s method for a general nonlinear vibro-impact oscillator[J].Nonlinear Analysis,2009,71(1):418-426.
[14]冯进钤.谐和激励下单边碰撞系统的同宿分岔[J].西安工程大学学报,2011,25(4):97-600.FENG J Q.Homoclinic bifurcation in vibro-impact system under harmonic excitation[J].Journal of Xi′an Polytechnic University,2011,25(4):97-600.(in Chinese)
[15]SHI L S,ZOU Y K,KUPPER T.Melnikov method and detection of chaos for non-smooth systems[J].Acta Mathematicae Applicatae Sinica,English Series,2013,29(4):881-896.
[16]赵茉莉,杨慧.Melnikov横截异宿定理的证明及应用[J].纺织高校基础科学学报,2005,18(2):151-154.ZHAO M L,YANG H.Proof and application of Melnikov transversal heteroclinic theorem[J].Basic Sciences Journal of Textile Universities,2005,18(2):151-154.(in Chinese)
[2]BERNARDO M D,BUDDD C J,CHAMPNEYS A R,et al.Piecewise-smooth dynamical systems theory and applications[M].London:Springer Verlag,2008:1-200.
[3]HOLMES P J.The dynamics of repeated impacts with a sinusoidally vibrating table[J].Journal of Sound and Vibration,1982,84(2):173-189.
[4]罗冠炜,谢建华.碰撞振动系统的周期运动和分岔[M].北京:科学出版社,2004:1-235.LUO G W,XIE J H.The periodic motion and bifurcation of the vibro-impact system[M].Beijing:Science Press,2004:1-235.(in Chinese)
[5]戎海武,王向东,徐伟,等.谐和与噪声联合作用下的Duffing振子的安全盆分叉与混沌[J].物理学报,2007,56(4):2005-2011.RONG H W,WANG X D,XU W,et al.Bifurcations of safe basins and chaos in softening Duffing oscillator under harmonic and bounded noise excitation[J].Acta Physics Sinica,2007,56(4):2005-2011.(in Chinese)
[6]甘春标,王振林.一类非线性振子中有界噪声诱发的混沌运动[J].振动工程学报,2004,17(3):321-325.GAN C B,WANG Z L.Chaotic motions induced by bounded noise in a nonlinear oscillator[J].Journal of Vibration Engineering,2004,17(3):321-325.(in Chinese)
[7]GREGORZ L,MAREK B,ARKADIUSZ S,et al.Transition to chaos in the self excited system with a cubic double well potential and parametric forcing[J].Science Direct,2007,40(5):1-15.
[8]LI X C,XU W,LI R H.Chaotic motion of the dynamical system under both additive and multiplicative[J].Chinese Physics B,2008,17(2):556-568.
[9]陈亚文,邹学文.二维非稳态对流扩散方程反问题的混沌粒子群算法[J].西安工业大学学报,2011,31(5):470-473.CHEN Y W,ZOU X W.The chaotic particle Swarm algorithm of two-dimensional non-steady-state convection-diffusion equation inverse[J].Journal of Xi′an Technological University,2011,31(5):470-473.(in Chinese)
[10]FENG J Q,LIU J L.Chaotic dynamics of vibro-impact system under bounded noise perturbation[J].Chaos,Solitons and Fractals,2015,73:10-16.
[11]冯进钤,金宇寰,刘亚妮.谐和与白噪声激励下碰撞振动系统的混沌运动[J].纺织高校基础科学学报,2018,31(2):FENG J Q,JIN Y H,LIU Y N.Chaos of vibro-impact system under harmonic and white noise excitations[J].Basic Sciences Journal of Textile Universities,2018,31(2):(in Chinese)
[12]HOLMES P,MARSDEN J E.A partial differential equation with infinitely many periodic orbits:Chaotic oscillations of a forced beam[J].Rch Ration Mech Anal,1981,76(2):135-166.
[13]XU W,FENG J Q,RONG H W.Melnikov′s method for a general nonlinear vibro-impact oscillator[J].Nonlinear Analysis,2009,71(1):418-426.
[14]冯进钤.谐和激励下单边碰撞系统的同宿分岔[J].西安工程大学学报,2011,25(4):97-600.FENG J Q.Homoclinic bifurcation in vibro-impact system under harmonic excitation[J].Journal of Xi′an Polytechnic University,2011,25(4):97-600.(in Chinese)
[15]SHI L S,ZOU Y K,KUPPER T.Melnikov method and detection of chaos for non-smooth systems[J].Acta Mathematicae Applicatae Sinica,English Series,2013,29(4):881-896.
[16]赵茉莉,杨慧.Melnikov横截异宿定理的证明及应用[J].纺织高校基础科学学报,2005,18(2):151-154.ZHAO M L,YANG H.Proof and application of Melnikov transversal heteroclinic theorem[J].Basic Sciences Journal of Textile Universities,2005,18(2):151-154.(in Chinese)