Resonant Periodic Orbits

Resonant periodic orbits, making n oscillations along the x-axis and m oscillations along the y-axis, appear in a two-dimensional potential V = ~(Ax2 + By2) + higher-order terms whenever the ratio A1/2/B'12 is near n/rn. Such orbits can be found by means of a "third" integral in which small divisor terms have been eliminated. Orbits whose initial conditions are near those of a stable resonant periodic orbit are tube orbits. By finding the main periodic orbits, one can derive the main characteristics of the totality of orbits. An application in the case A'12/B1/2 ~ is given. These considerations can be applied to find the main types of orbits in various galactic models