具有共振特征值的轨道翻转双同宿环分支  

Bifurcations of Double Homoclinic Flip Orbits With Resonant Eigenvalues

作　　者：张天四[1] 朱德明[2] 

机构地区：[1]上海理工大学理学院,上海200093 [2]华东师范大学数学系,上海200062

出　　处：《应用数学和力学》2007年第11期1353-1362,共10页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目(10371040)

摘　　要：该文研究了具有轨道翻转的双同宿环四维系统,在主特征值共振和沿轨道奇点处切方向共振下的两种分支.我们分别在系统奇点小邻域内利用规范型的解构造一个奇异映射,再在双同宿环的管状邻域内引起局部活动坐标架,利用系统线性变分方程的解定义了一个正则映射,通过复合两个映射而得到分支研究中一类重要的Poincaré映射,经过简单的计算最终得到后继函数的精确表达式.对分支方程细致地研究,我们给出了原双同宿环的保存性条件,并证明了"大"1-同宿环分支曲面,2-重"大"1-周期轨分支曲面,"大"2-同宿环分支曲面的存在性、存在区域和近似表达式,及其分支出的"大"周期轨和"大"同宿轨的存在性区域和数量.Concerns double homoclinic loops with orbit flips and two resonant eigenvalues in a fourdimensional system. We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium,and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops, then compose them to get the import,ant Poincaré map.A simple calculation gives explicitly an expression of the associated successor function. By a delicate analysis of the bifurcation equation, we obtain the condition that the original double homoclinic loops are kept, and prove the existence and the existence regions of the large 1-homoclinic orbit bifurcation surface,2-fold large 1-periodic orbit bifurcation surface, large 2-homoclinic orbit bifurcation surface and their approximate expressions. We also locate the large periodic orbits and large homoclinic orbits and their number.

关 键 词：双同宿轨 轨道翻转 周期轨 共振 

分 类 号：O175.12[理学—数学]