A Numerical, Literal, and Converged Perturbation Algorithm
The KAM theorem and von Ziepel’s method are applied to a perturbed harmonic oscillator, and it is noted that the KAM methodology does not allow for necessary frequency or angle corrections, while von Ziepel does. The KAM methodology can be carried out with purely numerical methods, since its generating function does not contain momentum dependence. The KAM iteration is extended to allow for frequency and angle changes, and in the process apparently can be successfully applied to degenerate systems normally ruled out by the classical KAM theorem. Convergence is observed to be geometric, not exponential, but it does proceed smoothly to machine precision. The algorithm produces a converged perturbation solution by numerical methods, while still retaining literal variable dependence, at least in the vicinity of a given trajectory.
Journal of the Astronautical Sciences
Wiesel, W.E. A Numerical, Literal, and Converged Perturbation Algorithm. J of Astronaut Sci 64, 251–266 (2017). https://doi.org/10.1007/s40295-016-0112-2