论文标题
非线性网络动力学具有共识 - 分叉分叉
Non-linear network dynamics with consensus-dissensus bifurcation
论文作者
论文摘要
我们研究了受干草叉分叉正常形式启发的网络上的非线性动力系统。该系统有几种有趣的解释：作为几个干草叉系统的互连，梯度动力学系统以及一类非线性动力学系统的主导行为。系统的平衡行为相对于系统参数表现出全局分叉，并从单个恒定固定态转变为可能的固定状态。我们的主要结果将这些固定状态的（子集的一个子集的稳定性都根据基础图的有效抗性为角度分类；该分类清楚地辨别了当地干草叉系统相互联系的特定拓扑的影响。我们进一步描述了具有外部公平分区的图形的精确解决方案，并表征了树图上的吸引力盆地。我们的技术分析补充了对许多原型网络上的系统的研究：树图，完整的图和杠铃图。我们描述了这些网络上动力学的许多定性属性，并具有令人鼓舞的建模后果。
We study a non-linear 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 non-linear 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 modeling consequences.
