A Lie-Deprit perturbation algorithm for linear differential equations with periodic coefficients

Abstract

A perturbative procedure based on the Lie-Deprit algorithm of classical mechanics is proposed to compute analytic approximations to the fundamental matrix of linear di erential equations with periodic coe cients. These approximations reproduce the structure assured by the Floquet theorem. Alternatively, the algorithm provides explicit approximations to the Lyapunov transformation reducing the original periodic problem to an autonomous sys- tem and also to its characteristic exponents. The procedure is computationally well adapted and converges for su ciently small values of the perturbation pa- rameter. Moreover, when the system evolves in a Lie group, the approximations also belong to the same Lie group, thus preserving qualitative properties of the exact solution.