Periodic motion in perturbed elliptic oscillators
N. D. Caranicolas, K. A. Innanen · The Astronomical Journal · 1992
Perturbed elliptic oscillators in galactic-type Hamiltonian systems are analyzed to evaluate the positions and periods of their resonant periodic orbits and compare the results to those of integrated equations of motion. The resonant periodic orbits are studied in the contexts of different perturbation functions by solving the systems of differential equations to yield frequencies. The required empirical formulas are derived with a linear-least-squares fitting and by slightly modifying the numerical coefficients. The present method yields positions of the elliptic periodic orbits that are as accurate as those derived with the second integral of motion. A formula is presented which describes the frequency of the elliptic periodic orbits as a function of all the parameters of the Hamiltonian. The present method also confirms the finding given by the second formal integral of motion that elliptic periodic orbits going around the origin are ellipses intersecting both axes perpendicularly.