We study a nonlinear dynamical system on networks inspired by the pitchfork bifurcation normal form. The system has several interesting interpretations: as an interconnection of several pitchfork systems, a gradient dynamical system and the dominating behaviour of a general class of nonlinear dynamical systems. The equilibrium behaviour of the system exhibits a global bifurcation with respect to the system parameter, with a transition from a single constant stationary state to a large range of possible stationary states. Our main result classifies the stability of (a subset of) these stationary states in terms of the effective resistances of the underlying graph; this classification clearly discerns the influence of the specific topology in which the local pitchfork systems are interconnected. We further describe exact solutions for graphs with external equitable partitions and characterize the basins of attraction on tree graphs. Our technical analysis is supplemented by a study of the system on a number of prototypical networks: tree graphs, complete graphs and barbell graphs. We describe a number of qualitative properties of the dynamics on these networks, with promising modelling consequences.

我们研究了基于干草叉分叉法线形式的网络上的非线性动力学系统。该系统有几种有趣的解释：作为多个干草叉系统，梯度动力学系统和通用非线性动力学系统的主导行为的互连。系统的平衡行为相对于系统参数表现出全局分叉，具有从单个恒定稳态到大范围可能的稳态的过渡。我们的主要结果是根据有效电阻将这些稳态（一部分）的稳定性分类基础图；该分类清楚地识别了本地干草叉系统互连的特定拓扑的影响。我们进一步描述了具有外部公平分区的图的精确解，并描述了树图上的吸引盆地。通过对许多原型网络上的系统进行研究，可以补充我们的技术分析：树形图，完整图和杠铃图。我们描述了这些网络动力学的许多定性性质，并带来了有希望的建模结果。