In a recent paper Dias and Stewart studied the existence, branching geometry, and stability of secondary branches of equilibria in all-to-all coupled systems of differential equations, that is, equations that are equivariant under the permutation action of the symmetric group S{sub}N. They consider the most general cubic order system of this type. Primary branches in such systems correspond to partitions of N into two parts p, q with p + q =N. Secondary branches correspond to partitions of N into three parts a, b, c with a + b + c =N. They prove that except in the case a =b = c secondary branches exist and are (generically) globally unstable in the cubic order system. In this work they realized that the cubic order system is too degenerate to provide secondary branches if a = b = c. In this paper we consider a general system of ordinary differential equations commuting with the permutation action of the symmetric group S{sub}(3n) on R{sup}(3n). Using singularity theory results, we find sufficient conditions on the coefficients of the fifth order truncation of the general smooth S{sub}(3n)-equivariant vector field for the existence of a secondary branch of equilibria near the origin with Sn × Sn × Sn symmetry of such system. Moreover, we prove that under such conditions the solutions are (generically) globally unstable except in the cases where two tertiary bifurcations occur along the secondary branch. In these cases, the instability result holds only for the equilibria near the secondary bifurcation points. We show an example where stability between tertiary bifurcation points on the secondary branch occurs.
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机译：迪亚斯和斯图尔特（Dias and Stewart）在最近的一篇论文中研究了微分方程的所有耦合方程组（即，在对称群S { sub} N。他们考虑了这种类型的最一般的三次系统。这样的系统中的主要分支对应于将N划分为两个部分p，q，其中p + q = N。次要分支对应于N分为三个部分a，b，c，其中a + b + c = N。他们证明，除非存在a = b = c的情况，否则次分支存在并且在三次系统中（通常）是全局不稳定的。在这项工作中，他们意识到，如果a = b = c，则三次阶系统过于简并无法提供次级分支。在本文中，我们考虑了一个常微分方程组的通用系统，该系统通过对称群S {sub}（3n）对R {sup}（3n）的置换作用进行换向。利用奇异性理论的结果，我们找到了一般光滑S {sub}（3n）-等变矢量场的五阶截断系数的充分条件，以便在原点附近具有Sn×Sn×Sn的平衡点存在次要分支这种系统的对称性。此外，我们证明，在这种情况下，解决方案（通常）总体上是不稳定的，除非在二级分支上发生两个三级分支。在这些情况下，不稳定性结果仅对次级分叉点附近的平衡成立。我们显示一个示例，其中二级分支上的三级分支点之间发生稳定性。
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