Synchronization in complex networks is an evergreen subject with many practical applications across the natural and social sciences. The stability of synchronization is thereby crucial for determining whether the dynamical behavior is stable or not. The master stability function is commonly used to that effect. In this paper, we study whether there is a relation between the stability of synchronization and the proximity to certain bifurcation types. We consider four different nonlinear dynamical systems, and we determine their master stability functions in dependence on key bifurcation parameters. We also calculate the corresponding bifurcation diagrams. By means of systematic comparisons, we show that, although there are some variations in the master stability functions in dependence on bifurcation proximity and type, there is in fact no general relation between synchronization stability and bifurcation type. This has important implication for the restrained generalizability of findings concerning synchronization in complex networks for one type of node dynamics to others.

中文翻译：

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同步稳定性和分岔类型之间有关系吗？

复杂网络中的同步是一门常青的学科，在自然科学和社会科学中有许多实际应用。因此，同步的稳定性对于确定动态行为是否稳定至关重要。主稳定性功能通常用于该效果。在本文中，我们研究了同步的稳定性与某些分岔类型的接近度之间是否存在关系。我们考虑四种不同的非线性动力系统，并根据关键分岔参数确定它们的主稳定性函数。我们还计算了相应的分岔图。通过系统比较，我们表明，虽然主稳定性函数存在一些变化，但取决于分岔的接近程度和类型，事实上，同步稳定性和分岔类型之间没有普遍的关系。这对于限制将复杂网络中的一种节点动态同步到其他节点的发现的普遍性具有重要意义。