非线性Mathieu方程的局部分岔和在余维2退化点的Hopf分岔  

LOCAL BIFURCATION AND HOPF BIFURCATION AT THE DEGENERATE POINT OF CODIMENSION 2 FOR NONLINEAR MATHIEU EQUATION

作　　者：张海燕[1] 唐友刚[1] 陈芳启[2] 

机构地区：[1]天津大学建工学院船舶与海洋工程系,天津300072 [2]南京航空航天大学理学院数学系,南京210016

出　　处：《机械强度》2007年第5期717-721,共5页Journal of Mechanical Strength

基　　金：国家自然科学基金(50279026);教育部博士点基金资助项目(20050056052)~~

摘　　要：针对海洋工程中的潜水艇拖缆问题的参数激励方程,研究其稳定性和复杂动力学特性。方程中包括阻尼项、x与x.立方项等。利用多尺度法求解弱的非线性Mathieu方程的1/2亚谐共振解,得到局部分岔特性,并研究在余维2退化点的Hopf分岔和极限环的稳定性问题。用中心流形方法研究零解的稳定性,用Hopf分岔定理研究Hopf分岔产生的极限环的稳定性。The stability and complicated dynamic behaviors of a cable towed by a submarine are investigated. The system can be modeled as a parametrically excited equation with damping term and cubic terms of x and x, etc, The method of multiple scales is applied to solve the equation which is valid in the neighborhood of 2 ： 1 sub-harmonic resonance under the assumptions of small excitation and small nonlinearity. The characters of local bifurcation are obtained and the Hopf bifurcations together with the stability of limit circle are examined at the degenerate point with co-dimension 2. The stability of zero solution at the degenerate points is studied with center manifold method and the stability of limit cycle in the Hopf bifurcation is analyzed by means of Hopf bifurcation theorem.

关 键 词：参数激励 亚谐共振 HOPF分岔 稳定性 中心流形方法 

分 类 号：U661[交通运输工程—船舶及航道工程] O322[交通运输工程—船舶与海洋工程]