Coupled heteroclinic networks

受我们以前关于瞬时无赢家竞争博弈的研究结果的启发，我们希望把我们的研究重点放在异宿轨道、异宿圈和异宿网络上。此外，我们将考虑的情况下，这些网络在相空间耦合的空间网格。异宿网络由对应于鞍点的节点（特别是鞍点不动点）和形成异宿连接的链路组成。我们要追求的问题是由神经元动力学和竞争物种的生态系统的认知过程提出的。在我们以前的项目的扩展中，我们希望探索如何沿着预先选择的路径引导轨迹沿着存储二进制字符串序列的信息，动态粗粒度网络结构朝向更大的循环，可以重新组合，并分析切换和灭绝事件的噪声的微妙作用。当这些网络耦合在一个空间网格上的吸引力或排斥性的耦合，扩散，或延迟，丰富多样的同步模式，降维以及混沌行为的预期，这取决于参数的选择和网络拓扑结构。在这里，我们将分析动力学和拓扑结构之间的相互作用，以区分哪些组合保留耦合系统中的个体动力学特征，哪些组合会导致新的涌现特征。

Inspired by results from our previous work on transient winnerless games of competition, we want to focus our studies on heteroclinic orbits, heteroclinic cycles and heteroclinic networks. Moreover, we will consider the case, where these networks in phase space are coupled on a spatial grid. Heteroclinic networks consist of nodes that correspond to saddles, in particular saddle fixed points, and of links that form heteroclinic connections. The questions we want to pursue are suggested by cognitive processes of neuronal dynamics and ecological systems with competing species. In extension of our previous project, we want to explore how to direct trajectories along pre-selected paths to store information of sequences of binary strings, to dynamically coarse-grain the network structure towards larger cycles, which can be recombined, and to analyze the subtle role of noise for switching and extinction events. When these networks are coupled on a spatial grid with attractive or repulsive couplings, with diffusion, or with delay, a rich variety of synchronization patterns, dimensional reduction as well as chaotic behavior are expected, depending on the choice of parameters and the network topology. Here we will analyze the interplay between dynamics and topology to distinguish which combinations preserve features of the individual dynamics in the coupled system and which lead to new emergent features as a result of the coupling.