A perturbative algorithm for quasi-periodic linear systems close to constant coefficients

A perturbative procedure is proposed to formally construct analytic solutions for a linear differential equation with quasi-periodic but close to constant coefficients. The scheme constructs the necessary linear transformations involved in the reduction process up to an arbitrary order in the perturbation parameter. It is recursive, can be implemented in any symbolic algebra package and leads to accurate analytic approximations sharing with the exact solution important qualitative properties. This algorithm can be used, in particular, to carry out systematic stability analyses in the parameter space of a given system by considering variational equations.