Seminumerical Algorithms for Computing Invariant Manifolds of Vector Fields at Fixed Points

This chapter discusses computational aspects of invariant manifolds of vector fields at fixed points. It is focused on algorithms and implementations, since the theory is well established in many classical textbooks and in the foundational papers of the parameterization method. Special emphasis is given to the computation of semi-local expansions of invariant manifolds, for which algorithms are provided, based on the algebraic manipulation of power series and novel Automatic Differentiation techniques. The chapter illustrates the methodology with three detailed examples, which are: the 2D stable manifold of the origin of the Lorenz system, the 4D center manifold of a collinear point of the Restricted Three-Body Problem, and a 6D partial normal form in the same problem that allows the generation of Conley’s transit and non-transit trajectories associated to any object of the center manifold.

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