On the C 8/3-regularisation of simultaneous binary collisions in the planar four-body problem

The dynamics of the four-body problem allows for two binary collisions to occur simultaneously. It is known that in the collinear four-body problem this simultaneous binary collision (SBC) can be block-regularised, but that the resulting block map is only C 8/3 differentiable. In this paper, it is proved that the C 8/3 differentiability persists for the SBC in the planar four-body problem. The proof uses several geometric tools, namely, blow-up, normal forms, dynamics near normally hyperbolic manifolds of equilibrium points, and Dulac maps.

[1]  Nathan Duignan.  Normal Forms for Manifolds of Normally Hyperbolic Singularities and Asymptotic Properties of Nearby Transitions , 2019, Qualitative Theory of Dynamical Systems.

[2]  H. Dullin,et al.  On the C8/3-regularisation of simultaneous binary collisions in the collinear 4-body problem , 2019, 1908.05576.

[3]  Robert H. Roussarie.  Bifurcations of planar vector fields and Hilbert''''s 16th Problem , 1995 .

[4]  Richard McGehee,et al.  Triple collision in the collinear three-body problem , 1974 .

[5]  T. Ouyang,et al.  Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem , 2010, 1002.2161.

[6]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[7]  A. Glutsyuk.  Lectures on analytic differential equations (Graduate Studies in Mathematics 86) , 2010 .

[8]  G. Belitskii.  C∞-Normal Forms of Local Vector Fields , 2002 .

[9]  Carles Simó,et al.  Simultaneous binary collisions in the planar four-body problem , 1999 .

[10]  Carles Simó,et al.  Some questions looking for answers in dynamical systems , 2018 .

[11]  F. Takens,et al.  Topological equivalence of normally hyperbolic dynamical systems , 1977 .

[12]  Realizing all reduced syzygy sequences in the planar three-body problem , 2015 .

[13]  W. Sweatman,et al.  A global regularisation for integrating the Caledonian symmetric four-body problem , 2010 .

[14]  The flow of the N-body problem near a simultaneous-binary-collision singularity and integrals of motion on the collision manifold , 1996 .

[15]  W. Kyner.  Invariant Manifolds , 1961 .

[16]  J. Barrow-Green.  Poincare and the Three Body Problem , 1996 .

[17]  M. Elbialy.  Collision singularities in celestial mechanics , 1990 .

[18]  F. Dumortier,et al.  Smooth Normal Linearization of Vector Fields Near Lines of Singularities , 2010 .

[19]  L. Stolovitch.  Progress in normal form theory , 2009 .

[20]  T. Levi-Civita,et al.  Sur la régularisation du problème des trois corps , 1920 .

[21]  Martha Alvarez-Ramírez,et al.  Global Regularization of a Restricted Four-Body Problem , 2014, Int. J. Bifurc. Chaos.

[22]  A. Chenciner.  Poincaré and the Three-Body Problem , 2015 .

[23]  E. Belbruno.  On simultaneous double collision in the collinear four-body problem , 1984 .

[24]  H. Dullin,et al.  Regularisation for Planar vector fields , 2019, Nonlinearity.

[25]  Eric Lombardi,et al.  Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation , 2010 .

[26]  A. Mourtada,et al.  Cyclicite finie des polycycles hyperboliques de champs de vecteurs du plan mise sous forme normale , 1990 .

[27]  A. Roy,et al.  The Caledonian Symmetrical Double Binary Four-Body Problem I: Surfaces of Zero-Velocity Using the Energy Integral , 2000 .

[28]  R. Easton.  Regularization of Vector Fields by Surgery , 1971 .

[29]  G. Belitskii.  Invariant normal forms of formal series , 1979 .

[30]  C. Conley,et al.  Isolated invariant sets and isolating blocks , 1971 .

[31]  R. Roussarie,et al.  Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem , 1998 .