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Integration of Perturbed Initial Value Problems through Reduction Theory

We propose a method to integrate an initial value problem formed by the differential equations of a perturbed dynamical system plus the initial condition. The approach consists of several steps. First of all, if the original system does not enjoy a continuous symmetry, it is transformed up to a certain order of approximation M into an equivalent but equivariant one after truncation at order M. Second, this new symmetry allows us to define a reduction map which produces in its whole phase space a splitting of the transformed equations. Specifically, the latter system is decoupled into two subsystems, also initial value problems, one of them defined in the orbit space associated with the reduction and the other linear in its coordinates. As a third step the reduced subsystems are analytically resolved or numerically integrated using standard methods. Finally, the solution of the original system is recovered by inverting back the reduction map and thereafter the analytic transformations. We apply our procedure to three examples.