# American Institute of Mathematical Sciences

March  2014, 34(3): 959-975. doi: 10.3934/dcds.2014.34.959

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

 1 Institut de Matemàtiques i Aplicacions de Castelló (IMAC) and Departament de Matemàtiques, Universitat Jaume I, E-12071 Castellón, Spain 2 Institut de Matemàtiques i Aplicacions de Castelló (IMAC), and Departament de Matemàtiques, Universitat Jaume I, E-12071 Castellón, Spain

Received  November 2012 Revised  March 2013 Published  August 2013

A perturbative procedure based on the Lie--Deprit algorithm of classical mechanics is proposed to compute analytic approximations to the fundamental matrix of linear differential equations with periodic coefficients. 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 system and also to its characteristic exponents. The procedure is computationally well adapted and converges for sufficiently small values of the perturbation parameter. 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.
Citation: Fernando Casas, Cristina Chiralt. A Lie--Deprit perturbation algorithm for linear differential equations with periodic coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 959-975. doi: 10.3934/dcds.2014.34.959
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