论文标题
马鞍不变的物体及其全球流形在案例B的同层翻转分叉附近
Saddle Invariant Objects and their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of Case B
论文作者
论文摘要
当三维矢量场中的真实马鞍平衡发生同型分叉时，平衡的相关二维不变歧管以可定向或不可方向的方式在自身上关闭。我们对鞍座的全局不变流形和鞍形周期轨道之间的相互作用感兴趣，该矢量场接近codimension-two two two two two two syoclinic flip分叉，即具有定向或不可方向的两维表面之间的过渡点。在这里，我们专注于案例$ \ textbf {b} $的同层面翻转分叉，其特点是，Codimension-Two点会引起附加的同型分叉，即是两型型斜角轨道。为了解释全球流形如何组织相空间，我们考虑了Sandstede的三维矢量场模型，该模型具有倾斜度和轨道翻转分叉。我们通过延续合适的两点边界问题来计算全局不变的歧管及其相交集，以理解它们作为吸引周期性轨道的盆地分离的作用。我们在相空间和球体中显示了代表性的图像，因此我们可以识别参数空间不同区域和涉及的同层分叉中歧管的拓扑特性。我们发现马鞍周期轨道和平衡之间的杂斜轨道产生了无限的许多杂斜轨道区域。 Sandstede的模型中存在其他平衡，我们压缩了相位空间，以捕获平衡如何从或逃脱到无限。我们介绍了这些分叉图的图像，其中概述了靠近case $ \ textbf {b} $的同层面翻转分叉的平衡的不同配置;此外，我们表征了Sandstede在Infinity上的动力学。
When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or non-orientable way. We are interested in the interaction between global invariant manifolds of saddle equilibria and saddle periodic orbits for a vector field close to a codimension-two homoclinic flip bifurcation, that is, the point of transition between having an orientable or non-orientable two-dimensional surface. Here, we focus on homoclinic flip bifurcations of case $\textbf{B}$, which is characterized by the fact that the codimension-two point gives rise to an additional homoclinic bifurcation, namely, a two-homoclinic orbit. To explain how the global manifolds organize phase space, we consider Sandstede's three-dimensional vector field model, which features inclination and orbit flip bifurcations. We compute global invariant manifolds and their intersection sets with a suitable sphere, by means of continuation of suitable two-point boundary problems, to understand their role as separatrices of basins of attracting periodic orbits. We show representative images in phase space and on the sphere, such that we can identify topological properties of the manifolds in the different regions of parameter space and at the homoclinic bifurcations involved. We find heteroclinic orbits between saddle periodic orbits and equilibria, which give rise to regions of infinitely many heteroclinic orbits. Additional equilibria exist in Sandstede's model and we compactify phase space to capture how equilibria may emerge from or escape to infinity. We present images of these bifurcation diagrams, where we outline different configurations of equilibria close to homoclinic flip bifurcations of case $\textbf{B}$; furthermore, we characterize the dynamics of Sandstede's model at infinity.
