Determining the global manifold structure of a continuous-time heterodimensional cycle

<p style='text-indent:20px;'>A heterodimensional cycle consists of two saddle periodic orbits with unstable manifolds different dimensions and a pair connecting between them. Recent theoretical work on chaotic dynamics beyond the uniformly hyperbolic setting has shown that cycles may occur robustly in diffeomorphisms dimension at least three. We consider first explicit example continuous-time setting, which been identified by Zhang, Krauskopf Kirk [<i>Discr. Contin. Dynam. Syst. A</i> <b>32</b>(8) 2825-2851 (2012)] four-dimensional vector-field model intracellular calcium dynamics.</p><p style='text-indent:20px;'>We show here how boundary-value problem set-up can be employed to determine organization neighborhood phase space this cycle, single orbit codimension one an entire cylinder structurally stable orbits. More specifically, we compute relevant manifolds, visualize projections as intersection sets suitable three-dimensional Poincaré section. In way, that, locally near set interact described theory for diffeomorphisms. On other hand, their global structure is more intricate, due fact it not possible find section transverse flow everywhere. generally, our results advanced numerical continuation techniques enable investigate abstract concepts â€" such diffeomorphism arise manifest themselves systems from applications.</p>