Analytic continuation in the case of non-regular dependency on a small parameter with an application to celestial mechanics

Abstract

We consider a non-autonomous system of ordinary differential equations. Assume that the time dependence is periodic with a very high frequency 1/ε, where ε is a small parameter and differentiability with respect to the parameter is lost when ε equals zero. We derive from Arenstorf's implicit function theorem a set of conditions to show the existence of periodic solutions. These conditions look formally like the standard analytic continuation method, namely, checking that a certain minor does not vanish. We apply this result to show the existence of a new class of periodic orbits of very large radii in the three-dimensional elliptic restricted three-body problem for arbitrary values of the masses of the primaries.

Families of first kind periodic solutions of the spatial and planar elliptic restricted three-body problem with collision

“…The rest of the proof is as in Brandão Dias and Vidal (2008) orCors et al (2005). Now, we are in position to prove the following result.…”

“…Using the same arguments as in Cors et al (2001Cors et al ( , 2005 and Brandão Dias and Vidal (2008), we will show how solutions of the spatial ERTBP with collision in which the infinitesimal body keeps far away from the primaries can be approximated by means of successive corrections of the Keplerian motion. The following two lemmas show that the solutions of the system (33) can be written as solutions of the unperturbed system Z = F 0 (Z) plus terms which are, in our case, of order 4 , and the same is true for its partial derivatives with respect to the initial conditions.…”

“…is differentiable with respect to X ∈ U , for all ∈ V and satisfies the hypotheses of the Arenstorf Theorem (see details for example in Brandão Dias and Vidal 2008; Cors et al 2005 or Meyer and Howison 2000), therefore there is a function X( ) ∈ U in a neighborhood of the origin = 0, V ⊂ V such that φ(X( ), ) = 0 for ∈ V . Then, we can conclude that there are families of initial conditions (indexed by m,m, j, µ) in one parameter ( ), X m,m, j,µ ( ) such that φ(X m,m, j,µ ( ), ) = 0.…”

“…The proofs of these lemmas follow from Taylor's expansion and the repeated use of Gronwall's inequality. See more details in Cors et al (2005) and Meyer and Howison (2000).…”

“…Periodic orbits of a collinear restricted three-body problem in Llibre and Corbera (2003) was considered. The Arenstorf method has been considered in Cors et al (2001Cors et al ( , 2005 in order to prove the existence of symmetric periodic solutions in the study of a new class of periodic orbits in the three-dimensional elliptic restricted three-body problem in the case of equal masses of the primaries. Also in Llibre and Roberto (2009) Arenstorf's Theorem has been considered in order to get symmetric periodic solutions in the (N + 1)-body problem.…”