Relative equilibria, stability and bifurcations in Hamiltonian galactic tidal models

We study special solutions of Galactic Tidal models and the existence, stability and bifurcations of relative equilibria. Precisely, we found four equilibria and rectilinear, as well as circular solutions, previous to any manipulation of the original system. Moreover, by averaging of the Keplerian energy and making use of Reeb's theorem, we find four periodic orbits. The stability of these relative equilibria is analysed and determined in several cases that involve a relation between the Keplerian energy and the gravitational parameter. Further averaging on the angle giving the orbital plane node provides the Matese-Whitman model. This Hamiltonian is expressed as a function of the invariants of the twice-reduced space, which is given by three connected components. Forthe three connected components become attached to each other by two singular points. Although only the bounded component is endowed with physical meaning, we provide a full analysis of the stability and bifurcations for all the equilibria. By doing this, we get an insight into the bifurcation process. Precisely, we observe that for the singular leaf of the reduced space, some of the equilibria in the bounded and unbounded components switch their locations.