Open Access

Seongchan Kim

• Since Poincaré, periodic orbits have been one of the most important objects in dynamical systems. However, searching them is in general quite difficult. A common way to find them is to construct families of periodic orbits which start at obvious periodic orbits. On the other hand, given two periodic orbits one might ask if they are connected by an orbit cylinder, i.e., by a one-parameter family of periodic orbits. In this thesis we study this question for the planar circular restricted three-body problem. More precisely, we first consider periodic orbits gamma^{RKP} and alpha^{Euler} in the rotating Kepler problem resp. in the Euler problem: The rotating Kepler problem is obtained by letting the mass ratio in the restricted three-body problem go to zero. One gets the Euler problem from the restricted three-body problem by setting the rotating term equal to zero. We assume that gamma^{RKP} and alpha^{Euler} are connected with periodic orbits gamma^{3BP} and alpha^{3BP} of theSince Poincaré, periodic orbits have been one of the most important objects in dynamical systems. However, searching them is in general quite difficult. A common way to find them is to construct families of periodic orbits which start at obvious periodic orbits. On the other hand, given two periodic orbits one might ask if they are connected by an orbit cylinder, i.e., by a one-parameter family of periodic orbits. In this thesis we study this question for the planar circular restricted three-body problem. More precisely, we first consider periodic orbits gamma^{RKP} and alpha^{Euler} in the rotating Kepler problem resp. in the Euler problem: The rotating Kepler problem is obtained by letting the mass ratio in the restricted three-body problem go to zero. One gets the Euler problem from the restricted three-body problem by setting the rotating term equal to zero. We assume that gamma^{RKP} and alpha^{Euler} are connected with periodic orbits gamma^{3BP} and alpha^{3BP} of the PCR3BP through Stark-Zeeman homotopies, respectively. We then ask for obstructions to find orbit cylinders in PCR3BP from gamma^{3BP} and alpha^{3BP}. Our strategy is to compare their Cieliebak-Frauenfelder-van Koert invariants which are obstructions to the existence of an orbit cylinder. We will prove that if gamma^{RKP} and alpha^{Euler} are contractible, then the invariants of gamma^{3BP} and alpha^{3BP} do not coincide with each other. Consequently, there exist no orbit cylinders connecting these periodic orbits in the PCR3BP.…