TITLE: Stability diagram for 4D linear periodic systems with applications to homographic solutions AUTHORS: Regina Mart\'{\i}nez^(1), Anna Sam\`a^(1) and Carles Sim\'o^(2) (1) Dept. de Matem\`atiques Universitat Aut\`onoma de Barcelona, Bellaterra 08193, Barcelona, Spain. (2) Dept. de Matem\`atica Aplicada i An\`alisi, Univ. de Barcelona, Gran Via 585, 08007 Barcelona, Spain E-mail: reginamb@mat.uab.es, sama@mat.uab.es, carles@maia.ub.es ABSTRACT We consider a family of 4-dimensional Hamiltonian time-periodic linear systems depending on three parameters, $\lambda_1,\lambda_2$ and $\varepsilon$ such that for $\varepsilon=0$ the system becomes autonomous. Using Normal Form techniques we study stability and bifurcations for $\varepsilon > 0 $ small enough. We pay special attention to the d'Alembert 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 $-\alpha,0<\alpha<2$ including the Newtonian case.