Coupled heteroclinic networks

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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.