Bifurcation of Periodic Orbits and Chaos in Duffing Equation  

作　　者：Mei-xiang Cai Jian-ping Yang 

机构地区：[1]Department of Mathcmatics, Hunan Normal University, Changsha 410081, China [2]Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, Chian, and Graduate School of the Chinese Academy of Sciences, Beijing 100039

出　　处：《Acta Mathematicae Applicatae Sinica》2006年第3期495-508,共14页应用数学学报（英文版）

基　　金：Supported by the National Natural Science Foundation of China （No.10371037）, and by Chinese Academy Scicnces （KZCX2-SW-118）

摘　　要：Duffing equation with fifth nonlinear-restoring force, one external forcing and a phase shift is investigated, The conditions of existences for primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using second-averaging method, Melnikov methods and bifurcation theory. Numerical simulations including bifurcation diagrams, bifurcation surfaces, phase portraits, not only show the consistence with the theoretical analysis, but also exhibit the new dynamical behaviors. We show the onset of chaos, chaos suddenly disappearing to period orbit, one-band and double-band chaos, period-doubling bifurcations from period 1, 2, and 3 orbits, period-windows （period-2, 3 and 5） in chaotic regions.Duffing equation with fifth nonlinear-restoring force, one external forcing and a phase shift is investigated, The conditions of existences for primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using second-averaging method, Melnikov methods and bifurcation theory. Numerical simulations including bifurcation diagrams, bifurcation surfaces, phase portraits, not only show the consistence with the theoretical analysis, but also exhibit the new dynamical behaviors. We show the onset of chaos, chaos suddenly disappearing to period orbit, one-band and double-band chaos, period-doubling bifurcations from period 1, 2, and 3 orbits, period-windows （period-2, 3 and 5） in chaotic regions.

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

分 类 号：O17[理学—数学]