Asymptotic behaviour of the stability parameter for a family of singular-limit Hill's equation
The authors consider the following Hill's equation \[ \ddot{x} - (\lambda_1 + \lambda_2 g^{\alpha-2})x = 0 \] where \(\alpha \in (0, 2)\) and \(g=g(t; \delta)\) is a periodic function in \(t\), that depends on a small parameter \(\delta\), \(g(t; \delta) > 0\) for all \(t\) and \(g(0; t) \rightarrow 0\) as \(\delta \rightarrow 0\). This problem occurs in the study of the linear stability of homographic solutions in the planar three-body problem. The authors derive nondegeneracy conditions for the asymptotic behaviour of the stability parameter, which is the trace of the fundamental solution matrix. Using a blow-up of the singularity, they demonstrate that the degeneracies occur, if heteroclinic orbits exist in some related systems. The stability and instability regions are also computed numerically and show a very complicated pattern.
scientific article; zbMATH DE number 5382303