In network dynamics, synchrony between nodes defines an equivalence relation, usually represented as a colouring. If the colouring is balanced, meaning that nodes of the same colour have colour-isomorphic inputs, it determines a subspace that is flow-invariant for any ODE compatible with the network structure. Therefore any state lying in such a subspace has the synchrony pattern determined by that balanced colouring. In 2005 Golubitsky and coworkers proved a strong converse for synchronous equilibria: every rigid synchrony colouring for a hyperbolic equilibrium is balanced, where rigidity means that the pattern persists under small admissible perturbations. We give a different proof of this theorem, based on overdetermined constraint equations, Sard’s Theorem, bump functions, and groupoid symmetrisation.

中文翻译：

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网络平衡的超定约束和刚性同步模式

在网络动力学中，节点之间的同步定义了等价关系，通常表示为着色。如果着色是平衡的，则意味着相同颜色的节点具有同色输入，它将确定一个子空间，该子空间对于与网络结构兼容的任何ODE都是流量不变的。因此，处于这种子空间中的任何状态都具有由该平衡着色确定的同步模式。在2005年，Golubitsky及其同事证明了同步平衡的强烈反面：双曲线平衡的每个刚性同步着色都是平衡的，其中刚性意味着该模式在较小的容许扰动下仍然存在。基于超定约束方程，Sard定理，凸点函数和群对称，我们给出了该定理的另一种证明。