Coupled heteroclinic networks

受我们以前关于瞬时无赢者竞争对策的研究结果的启发，我们想要集中研究异宿轨道、异宿圈和异宿网络。此外，我们还将考虑相空间中的这些网络耦合在空间网格上的情况。异宿网络包括对应于鞍点的节点，特别是鞍点不动点，以及形成异宿连接的链路。我们想要追寻的问题是由神经元动力学和生态系统与竞争物种的认知过程所暗示的。在我们上一个项目的扩展中，我们想要探索如何沿着预先选择的路径引导轨迹来存储二进制字符串序列的信息，动态地将网络结构粗粒度化为可以重组的更大的循环，并分析噪声对开关和灭绝事件的微妙作用。当这些网络以吸引耦合或排斥耦合、扩散耦合或时滞耦合在空间网格上时，根据参数和网络拓扑的选择，期望有丰富多样的同步模式、降维以及混沌行为。在这里，我们将分析动力学和拓扑学之间的相互作用，以区分哪些组合保留了耦合系统中单个动力学的特征，哪些组合导致了作为耦合结果的新的涌现特征。

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.