Exact periodic orbits and chaos in polynomial potentials

Using the theory of the inverse problem, we construct galactic type potentials based on perturbed harmonic oscillators, producing exact analytic trajectories represented by second-order curves. The various families of orbits are examined in these potentials. It is found, that there are cases where our galactic models produce exact analytic orbits and chaotic orbits as well. The coexistence of analytic and chaotic orbits, happens when the energy of the test particle is close to the energy of escape. Comparison with previous work is also made.