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

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