Stability diagram for 4D linear periodic systems with applications to homographic solutions

We consider a family of 4-dimensional Hamiltonian time-periodic linear systems depending on three parameters, λ1, λ2 and ε such that for ε=0 the system becomes autonomous. Using normal form techniques we study stability and bifurcations for ε>0 small enough. We pay special attention to the dʹAlembert case. The results are applied to the study of the linear stability of homographic solutions of the planar three-body problem, for some homogeneous potential of degree −α, 0<α<2, including the Newtonian case.