Asymptotic behaviour of the stability parameter for a family of singular-limit Hill's equation

Under some nondegeneracy conditions we give asymptotic formulae for the stability parameter of a family of singular-limit Hill's equation which depends on three parameters. We use the blow-up techniques introduced in [R. Martínez, A. Samà, C. Simó, Analysis of the stability of a family of singular-limit linear periodic systems in R4. Applications, J. Differential Equations 226 (2006) 652-686]. The main contribution of this paper concerns the study of the nondegeneracy conditions. We give a geometrical interpretation of them, in terms of heteroclinic orbits for some related systems. In this way one can determine values of the parameters such that the nondegeneracy conditions are satisfied. As a motivation and application we consider the vertical stability of homographic solutions in the three-body problem. © 2008 Elsevier Inc. All rights reserved.