Transient Dynamics in Heteroclinic Networks
- Historically, the study of dynamical systems was primarily focussed on attractors, describing the dynamics in the large-time limit. However, many dynamical processes are intrinsically transient. Dynamics comprising long-lasting transients are relevant to applications in ecological systems or cognitive processes, for example. There are different ways in which transient dynamics are realized. One of them is via heteroclinic cycles or heteroclinic networks, which consist of saddles connected by heteroclinic orbits. In these systems, the dynamics continuously switches between different metastable states. When realized in the framework of evolutionary game theory, metastable states correspond to states with one dominating population, which is the winner only temporarily. Accordingly, these games are also termed winnerless competition. This thesis focusses on two aspects of heteroclinic dynamics. Firstly, we construct and analyse a hierarchically structured heteroclinic network. It constitutes a hierarchical attractor composed of multiple levels of winnerless competition. We demonstrate that this hierarchy in phase space translates to a hierarchy in timescales, whereby the heteroclinic switching on the slower timescale modulates the dynamics on the faster timescale. Our system thus presents a realization of dynamics that are nested, and in addition also self-similar by construction. We derive analytically the dependence of the ratio of timescales on model parameters and initial conditions via a Poincaré map approach. Moreover, we show how the hierarchy transfers to spatial patterns when multiple units of the system are assigned to a spatial lattice and coupled by diffusion. In this spatial system, we observe a large dimensional reduction when a multitude of hierarchical heteroclinic networks synchronizes to the dynamics of a single such network. In the broader context of evolutionary game theory, our construction method allows selecting in advance which [...]
