具有轨道翻转的三点异维环“∞”型奇异轨分支

Bifurcation of Three-Point Heterodimensional Cycles with Orbit Flip in Shape of “∞”

摘要: 本文主要研究了一类具有轨道翻转的三点异维环分支问题。在未扰动异维环Γ的小管状邻域内，通过构建局部活动坐标架，建立Poincaré映射得到后继函数。再通过对分支方程的分析，得到了三点异维环Γ的小邻域内“∞”型双异维环、双同宿环的存在性和双异维环与1-周期轨、2重周期轨的共存性。另外，我们还得到了轨道的存在区域和分支曲面的表达式。

Abstract: In this paper, bifurcation of three-point heterodimensional cycles with orbit flip is studied in a three-dimensional vector field. By establishing local moving frame systems in a small tubular neighborhood of unperturbed heterodimensional cycles, we build a Poincaré return map and obtain succeed functions. Based on analysis of the bifurcation equations, the existence of “∞” type double heterodimensional cycles, homoclinic loops, and the coexistence of hetrodimensional cycle with 1-periodic orbit or 2-fold periodic orbit near Γ are received. Moreover, we give the existence regions of the above orbits and the expression of bifurcation surfaces.