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Global Heteroclinic Rebel Dynamics Among Large 2-Clusters in Permutation Equivariant Systems
SIAM Journal on Applied Dynamical Systems ( IF 2.1 ) Pub Date : 2021-07-22 , DOI: 10.1137/20m1361493
Bernold Fiedler , Sindre W. Haugland , Felix P. Kemeth , Katharina Krischer

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 3, Page 1277-1319, January 2021.
We explore equivariant dynamics under the symmetric group $S_N$ of all permutations of $N$ elements. Specifically we study one-parameter vector fields, up to cubic order, which commute with the standard real $(N-1)$-dimensional irreducible representation of $S_N$. The parameter is the linearization at the trivial 1-cluster equilibrium of total synchrony. All equilibria are cluster solutions involving up to three clusters. The resulting global dynamics is of gradient type: all bounded solutions are cluster equilibria and heteroclinic orbits between them. In the limit of large $N$, we present a detailed analysis of the web of heteroclinic orbits among the plethora of 2-cluster equilibria. Our focus is on the global dynamics of 3-cluster solutions with one rebel cluster of small size. These solutions describe slow relative growth and decay of 2-cluster states. For $N\rightarrow\infty$, the limiting heteroclinic web defines an integrable rebel flow in the space of 2-cluster equilibrium configurations. We identify and study the seven qualitatively distinct global rebel flows which arise in this setting. Applications include oscillators with all-to-all coupling, and electrochemistry. For illustration we consider synchronization clusters among $N$ complex Stuart--Landau oscillators with complex linear global coupling.


中文翻译:

置换等变系统中大型 2 簇之间的全局异斜反叛动力学

SIAM Journal on Applied Dynamical Systems,第 20 卷,第 3 期,第 1277-1319 页,2021 年 1 月。
我们在 $N$ 元素的所有排列的对称群 $S_N$ 下探索等变动力学。具体来说,我们研究单参数向量场,最高可达三次,它与 $S_N$ 的标准实数 $(N-1)$ 维不可约表示相通。该参数是总同步的平凡 1 簇平衡处的线性化。所有均衡都是最多包含三个集群的集群解决方案。由此产生的全局动力学是梯度类型的:所有有界解都是簇平衡和它们之间的异宿轨道。在大 $N$ 的限制下,我们详细分析了大量 2 簇平衡之间的异宿轨道网络。我们的重点是具有一个小型反叛集群的 3 集群解决方案的全球动态。这些解决方案描述了 2 簇状态的相对缓慢的增长和衰减。对于 $N\rightarrow\infty$,限制异斜网定义了 2 簇平衡配置空间中的可积反叛流。我们确定并研究了在这种情况下出现的七种性质不同的全球反叛流动。应用包括具有全对全耦合的振荡器和电化学。为了说明,我们考虑具有复杂线性全局耦合的 $N$ 复杂 Stuart--Landau 振荡器之间的同步集群。和电化学。为了说明,我们考虑具有复杂线性全局耦合的 $N$ 复杂 Stuart--Landau 振荡器之间的同步集群。和电化学。为了说明,我们考虑具有复杂线性全局耦合的 $N$ 复杂 Stuart--Landau 振荡器之间的同步集群。
更新日期:2021-07-23
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