Bifurcations in the Mass Ratio of the Planar Isosceles Three-Body Problem

Abstract

We study a Poincaré map in the planar isosceles three-body problem, but we emphasize the mapping of areas in phase space over few iterations rather than single points over many iterations. This map, with a complementary symbol ic dynamics, l ea ds to global information. We identify a s table fixed poi nt of the mapping wi t h associat ed qu asi- perio dic motionion for sm alle r massss ratios. The invariant KAM region around this fixed point vanishes at an inverse period doubling bifurcation at m ₃/m ₁ ≅ 2.581. We also find a set on which a horseshoe map completely describes the motion. This simply chaotic set is destroyed at mass ratio m ₃/m ₁ ≅ 2.662 leading to an interesting global bifurcation. Ranges of the mass ratio are identified on which the dynamics is qualitatively similar in a global sense. We also study the motion at the limiting values of the mass ratio.