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| In this talk, we will describe an algorithm for the computation of high-order power series expansions of invariant manifolds of fixed points of vector fields that is inspired in the parametrization method. It is very general, in the sense that works in any dimension (examples will be provided for 2, 4, 6) and provides a unified approach to the computation of different kinds of manifolds (like stable/unstable, center, or slow manifold inside a stable/unstable one) with different kind of expansions (as a graph, as a general parametrization with the reduced vector field in normal form, or a combination of both strategies). The actual implementation is especially efficient thanks to the use of automatic differentiation ideas and a recursive representation of polynomials in several variables. Three applications will be presented, that consist on the computation of expansions for: the 2D Lorenz manifold, the 4D center manifold of a collinear fixed point of the spatial, circular Restricted Three--Body Problem, and a 6D reparametrization of the full neighborhood of the same point of the same problem that non-linearly separates the central dynamics (periodic orbits, 2D tori, and chaotic dynamics in between) from the normal hyperbolic part (stable/unstable manifolds of all the previous objects). |
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