Topological analysis for designing a suspension of the Hénon map

•A suspension of a map is the flow for which the Poincaré section is that map.•Under orientation preserving conditions, the suspension for the Hénon map is 3D.•Under orientation reversing conditions, the suspension for the Hénon map is 4D.•Topological analysis of suspensions for the Hénon map are performed for both cases.

A suspension of a map consists of the flow for which the Poincaré section is that map. Designing a suspension of a given map remains a non-trivial task in general. The case of suspending the Hénon map is here considered. Depending on the parameter values, the Hénon map is orientation preserving or reversing; it is here shown that while a tridimensional suspension can be obtained in the former case, a four-dimensional flow is required to suspend the latter. A topological characterization of the three-dimensional suspension proposed by Starrett and Nicholas for the orientation preserving area is performed. A template is proposed for the four-dimensional case, for which the governing equations remain to be obtained.