Consecutive quasi-collisions in the planar circular RTBP. (Q2782717)

In his seminal work [\textit{H. Poincaré}, Les Méthodes Nouvelles de la Mécanique Celeste, I-III, Paris: Gauthier-Villars (1899)] Poincaré sets out an important restricted three-body problem (RTBP) consisting of two bodies \(E\) and \(M\) with masses \(m_1=1-\mu\) and \(m_2 =\mu\) performing a circular Keplerian motion, and the third massless body \(Z\). This problem has led to a great deal of research over the past and present centuries. Shown to be nonintegrable (i.e. the motion of \(Z\) is chaotic in general, and the only known first integral is the Jacobi constant \(C_J\)), this problem opens nevertheless many interesting perspectives and challenges. One of them is to prescribe in a random way the possible types of motion of the third particle \(Z\).NEWLINENEWLINE The paper under review develops a constructive approach to this problem based on the computation of an approximate expression of return map defined on a region of phase space whose projection is a circle around the small primary \(M\). This is done in several steps. First, a circle \(\mathbf C\) and a disk \(\mathbf D\) of radius \(\mu^{\alpha}, \alpha\in (\frac{1}{3},\frac{1}{2})\) centered at \(M\) are defined together with the so-called \(p-q\) resonant orbits \(\gamma(t)\) crossing \(\mathbf C\) for \(t=t_1,t_2, t_1<t_2\). Next, the authors derive a set of initial conditions on \(\mathbf C\) corresponding to \(p-q\) orbits (resonant strips) and give a detailed study of approximation of in (out) maps describing \(p-q\) orbits entering-leaving (leaving-entering) the disk \(\mathbf D\). The careful combination of these two maps together with optimization of \(\alpha\) leads to the construction of a global return map defined on the union of the resonant strips and which turns to be a horseshoe map (see [\textit{J. Moser}, Stable and random motions in dynamical systems. With special emphasis on celestial mechanics. With a new foreword by Philip J. Holmes. 3rd pbk-ed. Princeton, NJ: Princeton University Press (2001; Zbl 0991.70002)]). Techniques from symbolic dynamics are then used to show that the number of revolutions of the small bodies \(Z\) and \(M\) around the larger one between successive encounters can be chosen as two arbitrary sequences of natural numbers, with constraints depending on Jacobi constant. Numerous illustrations of this result based on precise numerical integrations of the equations of RTBP are given.