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Invariant manifolds of an autonomous ordinary differential equation from its generalized normal forms.

A method to approximate some invariant sets of dynamical systems defined through an autonomous m-dimensional ordinary differential equation is presented. Our technique is based on the calculation of formal symmetries and generalized normal forms associated with the system of equations, making use of Lie transformations for smooth vector fields. Once a symmetry is determined up to a certain order, a reduction map allows us to pass from the equation in normal form to a related equation in a certain reduced space, the so-called reduced system of dimension s<m. Now, under certain regularity conditions, a nondegenerate p-dimensional invariant set of the reduced system is formally transformed into a (p+m-s)-dimensional invariant set of the original equation. Moreover, the existence of some actual (p+m-s)-dimensional invariant manifolds of the initial equations related to the ones determined through our analysis can be proven under certain hypotheses that we make explicit. The procedure is illustrated by characterizing the set of all periodic orbits sufficiently close to the origin of the Hamiltonian vector field defined by the Henon and Heiles family.