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.