Bifurcation at complex instability

Abstract

In an autonomous Hamiltonian system with three or more degrees of freedom, a family of periodic orbits may become unstable when two pairs of characteristic multipliers coallesce on the unit circle at points not equal to ±1, and then move off the unit circle. This paper develops normal forms suitable for the neighbourhood of such an, instability, and, at this approximation, demonstrates the bifurcation from the periodic orbit of a family of invariant two-dimensional tori. The theory is illustrated with numerical computations of orbits of the planar general three-body problem.