论文标题
计算将轨道连接到与同型翻转分叉相关的无穷大的
Computing connecting orbits to infinity associated with a homoclinic flip bifurcation
论文作者
论文摘要
我们在$ \ mathbb {r}^3 $中的合适参数平面中考虑了分叉图，该图具有最复杂类型的同层面翻转分叉。这种编码两个分叉的特征是相关的二维流形的可方向性变化，并产生了次要分叉的无限家族。我们表明，在涉及无穷大的杂斜分叉的曲线上积累了$ n $ homoclinic分叉的曲线。我们提出了称为Lin方法的技术的改编，使我们能够计算此类连接轨道到无穷大。我们首先执行$ \ mathbb {r}^3 $的加权定向压实，随后对无穷大的非纤维性马鞍进行了爆炸。然后，我们为一个共同的二维部分设置了两个轨道段的边界值问题：第一个是在常规坐标中的有限鞍座，第二个是从吹牛图中无穷大的鞍座附近的。然后，通过延续将所谓的LIN间隙沿截面中的固定一维方向带到零。一旦以这种方式找到连接轨道，就可以将其位点作为参数平面中的曲线追溯到。
We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in $\mathbb{R}^3$ that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary $n$-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity. We present an adaptation of the technique known as Lin's method that enables us to compute such connecting orbits to infinity. We first perform a weighted directional compactification of $\mathbb{R}^3$ with a subsequent blow-up of a non-hyperbolic saddle at infinity. We then set up boundary-value problems for two orbit segments from and to a common two-dimensional section: the first is to a finite saddle in the regular coordinates, and the second is from the vicinity of the saddle at infinity in the blown-up chart. The so-called Lin gap along a fixed one-dimensional direction in the section is then brought to zero by continuation. Once a connecting orbit has been found in this way, its locus can be traced out as a curve in a parameter plane.
