Bifurcations and Chaos in Duffng Equation with Damping and External Excitations  

作　　者：Mei-xiang CAI Jian-ping YANG Jin DENG 

机构地区：[1]Institute of Mathematics and Physics,Central South University of Forestry and Technology [2]College of Science,China Agriculture University [3]Department of Mathematics and Physics,Hunan Institute of Engineering

出　　处：《Acta Mathematicae Applicatae Sinica》2014年第2期483-504,共22页应用数学学报（英文版）

基　　金：Supported by the National Natural Science Foundation of China(No.10801135,11101170);Hunan Provincial Natural Science Foundation of China(No.13JJ4088);talent introduction fund(No.104-0163)

摘　　要：Duffing equation with damping and external excitations is investigated. By using Melnikov method and bifurcation theory, the criterions of existence of chaos under periodic perturbations are obtained. By using second-order averaging method, the criterions of existence of chaos in averaged system under quasi-periodic perturbations for Ω = nω ＋ εσ, n = 2,4,6 （where σ is not rational to ω） are investigated. However, the criterions of existence of chaos for n = 1, 3, 5, 7 - 20 can not be given. The numerical simulations verify the theoretical analysis, show the occurrence of chaos in the averaged system and original system under quasi- periodic perturbation for n = 1, 2, 3, 5, and expose some new complex dynamical behaviors which can not be given by theoretical analysis. In particular, the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations, and the period-doubling bifurcations to chaos has not been found under quasi-periodic perturbations.Duffing equation with damping and external excitations is investigated. By using Melnikov method and bifurcation theory, the criterions of existence of chaos under periodic perturbations are obtained. By using second-order averaging method, the criterions of existence of chaos in averaged system under quasi-periodic perturbations for Ω = nω ＋ εσ, n = 2,4,6 （where σ is not rational to ω） are investigated. However, the criterions of existence of chaos for n = 1, 3, 5, 7 - 20 can not be given. The numerical simulations verify the theoretical analysis, show the occurrence of chaos in the averaged system and original system under quasi- periodic perturbation for n = 1, 2, 3, 5, and expose some new complex dynamical behaviors which can not be given by theoretical analysis. In particular, the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations, and the period-doubling bifurcations to chaos has not been found under quasi-periodic perturbations.

关 键 词：Duffing equation Melnikov method second-order averaging method CHAOS 

分 类 号：O175[理学—数学]