Fourier-Taylor approximation of unstable manifolds for compact maps: numerical implementation and computer-assisted error bounds. (English) Zbl 1383.37066

Summary: We develop and implement a semi-numerical method for computing high-order Taylor approximations of unstable manifolds for hyperbolic fixed points of compact infinite-dimensional maps. The method can follow folds in the embedding and describes precisely the dynamics on the manifold. In order to ensure the accuracy of our computations in spite of the many truncation and round-off errors, we develop a posteriori error bounds for the approximations. Deliberate control of round-off errors (using interval arithmetic) in conjunction with explicit analytical estimates leads to mathematically rigorous computer-assisted theorems describing precisely the truncation errors for our approximation of the invariant manifold. The method is applied to the Kot-Schaffer model of population dynamics with spatial dispersion.


37L65 Special approximation methods (nonlinear Galerkin, etc.) for infinite-dimensional dissipative dynamical systems
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
34C45 Invariant manifolds for ordinary differential equations
37D10 Invariant manifold theory for dynamical systems
65P40 Numerical nonlinear stabilities in dynamical systems
Full Text: DOI


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