耦合向量场的可容许图的实现
The realization of admissible graphs for coupled vector fields
AI总结：
本研究解决耦合向量场的逆问题，提出构造所有互不等价可容许图的流程并给出图数上界，可用于研究不变子空间与同步态，为振子网络嵌合体态研究提供系统方法。
摘要：
在耦合网络中，细胞单元可通过多种方式相互作用。过去二十年间，大量文献采用图论形式化方法研究这类相互作用动力学，即赋予顶点集上输入等价关系的图，以此刻画支配网络动力学的可容许向量场。本研究致力于解决一个逆问题：当n≥2时，ℝⁿ上的任意映射都可被实现为某个图的可容许向量场，该图的顶点数依赖于n（但不一定等于n）。针对给定映射，我们提出一种构造所有互不等价的可容许图的流程，等价性依据相应的等价关系判定，同时给出了这类图数量的上界。由此，向量场下的不变子空间可被研究为可容许图所支撑的同步态轨迹，也就是说，可以选择合适的图来实现具有更多（或更少）同步性的耦合，这一同步性程度不同于同一向量场的另一可容许图。该方法尤其为van der Pol恒等振子网络中嵌合体态的出现提供了系统性研究途径。
英文摘要：
In a coupled network cells can interact in several ways. There is a vast literature from the last twenty years that investigates this interacting dynamics under a graph theory formalism, namely as a graph endowed with an input-equivalence relation on the set of vertices that enables a characterization of the admissible vector fields that rules the network dynamics. The present work goes in the direction of answering an inverse problem: for $n \geq 2$, any mapping on $\mathbb{R}^n$ can be realized as an admissible vector field for some graph with the number of vertices depending on (but not necessarily equal to) $n$. Given a mapping, we present a procedure to construct all non-equivalent admissible graphs, up to the appropriate equivalence relation. We also give an upper bound for the number of such graphs. As a consequence, invariant subspaces under the vector field can be investigated as the locus of synchrony states supported by an admissible graph, in the sense that a suitable graph can be chosen to realize couplings with more (or less) synchrony than another graph admissible to the same vector field. The approach provides in particular a systematic investigation of occurrence of chimera states in a network of van der Pol identical oscillators.
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