Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem

Abstract

By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j–k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ– C characteristic curves of these families extend to the left of the μ– C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* ( μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun–Jupiter and Earth–Moon systems. We found new μ– C and x– C characteristic curves. Along each μ– C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x– C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ– C or the x– C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L 4 and L 5 are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ– C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x– C curve, that spiral into the points L 4 and L 5, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem.