On the C 8/3-regularisation of simultaneous binary collisions in the planar four-body problem

The dynamics of the 4-body problem allows for two binary collisions to occur simultaneously. It is known that in collinear this simultaneous collision (SBC) can be block-regularised, but resulting block map only $C^{8/3}$ differentiable. In paper, it proved differentiability persists SBC planar problem. proof uses several geometric tools, namely, blow-up, normal forms, near normally hyperbolic manifolds equilibrium points, and Dulac maps.