Perturbation theory in the vicinity of a periodic orbit by repeated linear transformations

Abstract

The usual description of motion near a periodic orbit as a solution to a linear time-periodic system (a Floquet problem) can be formally extended to higher orders of approximation. Each subsequent order problem is a linear, time-multiply periodic system. In formulating the second order problem the order of the Hamiltonian can be locally reduced by one degree of freedom for every exact integral of the motion present in the original problem. As in the Floquet problem, the system's state transition matrix at each order can be formally decomposed into the product of a bounded multiply-periodic matrix and the exponential of a constant matrix times the time. This yields stability information for the second and higher order approximations, analogous to Poincaré exponents in the Floquet problem. However, second and higher order stability exponents are functions of the displacement from the original periodic orbit, so this method may be able to predict the size of the region of stable motion surrounding a periodic orbit.