Analysis of the stability of a family of singular-limit linear periodic systems in R4. Applications

In this paper we consider a 4D periodic linear system depending on a small parameter δ>0. We assume that the limit system has a singularity at t=0 of the form , with c1,c2>0 and c1→0 as δ→0. Using a blow up technique we develop an asymptotic formula for the stability parameters as δ goes to zero. As an example we consider the homographic solutions of the planar three body problem for an homogeneous potential of degree α∈(0,2). Newtonian three-body problem is obtained for α=1. The parameter δ can be taken as 1−e2 being e the eccentricity (or a generalised eccentricity if α≠1). The behaviour of the stability parameters predicted by the formula is checked against numerical computations and some results of a global numerical exploration are displayed.