一类余维3的鞍-焦点异宿环分支  

A Codimension 3 Bifurcation of Heteroclinic Contour Involving a Hyperbolic and a Non-hyperbolic Saddle-foci

作　　者：邓桂丰[1] 朱德明[1] 

机构地区：[1]华东师范大学数学系,上海200062

出　　处：《数学年刊（A辑）》2007年第5期667-678,共12页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No.10371040)资助的项目

摘　　要：对余维3系统Xμ（x）具有包含一个双曲鞍-焦点O1和一个非双曲鞍-焦点O2的异宿环f进行了研究.证明了在f的邻域内有可数无穷条周期轨线和异宿轨线,当非粗糙异宿轨线Γ^0破裂时Xμ（x）会产生同宿轨分支,并给出了相应的分支曲线和两种同宿环共存的参数值.在3参数扰动下Γ^0破裂和O2点产生Hopf分支的情况下,在f的邻域内有一条含O1点同宿环,可数无效多条的轨线同宿于O2点分支出的闭轨H0,一条或无穷多条（可数或连续统的）异宿轨线等.The codimension 3 bifurcations associated with a heteroclinic contour formed with two saddle-loci （among which, one is a weak saddle-focus） and two of their heteroclinic orbits are studied. In certain neighborhood of the contour, the existence of countably infinite 1-periodic orbits and 11/2-heteroclinic orbits is given for the unperturbed system. Meanwhile, the complicated bifurcation patterns under the generic 3-parameter perturbation are also established, such as the Hopf bifurcation, 1-homoclinic bifurcation, 1-heteroclinic bifurcation, 11/2-heteroclinic bifurcation and the coexistence of different kinds of bifurcated orbits, etc.. Furthermore, the number of the produced homoclinic or herteroclinic orbits can be finite, denumerable infinite or uncountable infinite, etc..

关 键 词：异宿环 同宿环 极限环 分支曲面 HOPF分支 

分 类 号：O175[理学—数学]