Dynamics of (4+1)-dihedrally symmetric nearly parallel vortex filaments

We give a detailed analytical and numerical description of the global dynamics of 4+1 points interacting through the singular logarithmic potential and subject to the following symmetry constraint: at each instant 4 of them form an orbit of the Klein group D ₂ of order 4. The main device in order to achieve our results is to use a McGehee-like transformation introduced in (Paparella and Portaluri in Global dynamics of the dihedral singular logarithmic potential and nearly parallel vortex filaments, 2011) for a problem analogous to the present one. After performing this transformation in order to regularize the total collision, we study the rest-points of the flow, the invariant (stable and unstable) manifolds and we derive some interesting information about the global dynamics.