Analysis of the stability of a family of singular-limit linear periodic systems in R-4. Applications

摘要

In this paper we consider a 4D periodic linear system depending on a small parameter delta > 0. We assume that the limit system has a singularity at t = 0 of the form c(1)+c(2)t(2)+...1 with c(1), c(2) > 0 and c(1) -> 0 as delta -> 0. Using a blow up technique we develop an asymptotic formula for the stability parameters as 6 goes to zero. As an example we consider the homographic solutions of the planar three body problem for an homogeneous potential of degree alpha is an element of (0, 2). Newtonian three-body problem is obtained for alpha = 1. The parameter delta can be taken as 1 - e(2) being e the eccentricity (or a generalised eccentricity if alpha not equal 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. (c) 2005 Elsevier Inc. All rights reserved.