We classify simple heteroclinic networks for a $Γ$-equivariant system in ${\mathbb R}^4$ with finite $Γ\subset {\rm O}(4)$, proceeding as follows: we define a graph associated with a given $Γ\subset {\rm O}(n)$ and identify all so-called simple graphs associated with subgroups of ${\rm O}(4)$. Then, knowing the graph associated with a given $Γ$, we determine the types of heteroclinic networks that the group admits. Our study is restricted to networks that are maximal in the sense that they have the highest possible number of connections -- any non-maximal network can then be derived by deleting one or more connections. Finally, for networks of type A, i.e., admitted by $Γ\subset {\rm SO}(4)$, we give necessary and sufficient conditions for fragmentary and essential asymptotic stability. (For other simple heteroclinic networks the conditions for stability are known.) The results are illustrated by a numerical example of a simple heteroclinic network that involves two subcycles that can be essentially asymptotically stable simultaneously.
On the heteroclinic connection problem for multi-well gradient systems
我们重新审视的存在性问题的异宿连接$\mathbb{R}^N$与哈密顿系统涉及潜在的$W:\mathbb{R}^N\to \mathbb{R}$有几个全球最小值。在对$W$非常温和的假设下，我们提出了一种简单的变分方法，首先找到最小化连接任何两个势阱的曲线的长度的测地线，其中长度是相对于具有保形因子$\sqrt{W}.$的退化度量计算的。然后，我们表明，当这样的最小化测地线避免在中间时间通过其他势阱时，它在两个阱之间产生异宿连接。这项工作改进了P.Sternberg在$\texttt{Vector-valued local minimizers of nonconvex}$ $\texttt{variational problems}$中的方法，并代表了一种更几何的替代方法，用于寻找这种连接，例如，由N.D. Alikakos和G.Fusco在$\texttt{On the connection problem for potentials with}$ $\texttt{several global minima}$中，S.V. Bolotin在$\texttt{Libration motions of natural dynamical systems}$中，J. Byeon、P. Montecchiari和P. Rabinowitz在$\texttt{A double wel…
