Analysis of the stability of a family of singular-limit linear periodic systems in R4. Applications

Regina Martínez, Anna Samà, Carles Simó

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16 Citations (Scopus)

Abstract

In this paper we consider a 4D periodic linear system depending on a small parameter δ > 0. We assume that the limit system has a singularity at t = 0 of the form frac(1, c1 + c2 t2 + ⋯), with c1, c2 > 0 and c1 → 0 as δ → 0. Using a blow up technique we develop an asymptotic formula for the stability parameters as δ goes to zero. As an example we consider the homographic solutions of the planar three body problem for an homogeneous potential of degree α ∈ ( 0, 2 ). Newtonian three-body problem is obtained for α = 1. The parameter δ can be taken as 1 - e2 being e the eccentricity (or a generalised eccentricity if α ≠ 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. © 2005 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)652-686
JournalJournal of Differential Equations
Volume226
DOIs
Publication statusPublished - 15 Jul 2006

Keywords

  • Blow up
  • Homographic solutions
  • Near-singular periodic systems
  • Stability

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