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A heterodimensional cycle consists of a pair of heteroclinic connectionsbetween two saddle periodic orbits with unstable manifolds of differentdimensions. Recent theoretical work on chaotic dynamics beyond the uniformlyhyperbolic setting has shown that heterodimensional cycles may occur robustlyin diffeomorphisms of dimension at least three. We study a concrete example ofa heterodimensional cycle in the continuous-time setting, specifically in afour-dimensional vector field model of intracellular calcium dynamics. Byemploying advanced numerical techniques, Zhang, Krauskopf and Kirk [Discr.Contin. Dynam. Syst. A 32(8) 2825--2851 (2012)] found that a heterodimensionalcycle exists in this model.  We investigate the geometric structure of the associated stable and unstablemanifolds in the neighbourhood of this heterodimensional cycle, consisting of asingle connecting orbit of codimension one and an entire cylinder ofstructurally stable connecting orbits between two saddle periodic orbits. Weemploy a boundary-value problem set-up to compute their stable and unstablemanifolds, which we visualize in different projections of phase space and asintersection sets with a suitable three-dimensional Poincar\'e section. We showthat, locally near the intersection set of the heterodimensional cycle, themanifolds interact as described by the theory for three-dimensionaldiffeomorphisms. 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 sectionthat is transverse to the flow everywhere. Our results show that the abstractconcept of a heterodimensional cycle arises and can be studied incontinuous-time models from applications.

Global manifold structure of a continuous-time heterodimensional cycle