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Computation
of maximal local (un)stable manifold patches by the
parameterization method
Maxime Breden,
Jean-Philippe Lessard and Jason D. Mireles James
In this work we develop some automatic
procedures for computing high order polynomial expansions
of local (un)stable manifolds for equilibria of
differential equations. Our method incorporates validated
truncation error bounds, and maximizes the size of the
image of the polynomial approximation relative to some
specified constraints. More precisely we use that the
manifold computations depend heavily on the scalings of
the eigenvectors: indeed we study the precise effects of
these scalings on the estimates which determine the
validated error bounds. This relationship between the
eigenvector scalings and the error estimates plays a
central role in our automatic procedures. In order to
illustrate the utility of these methods we present several
applications, including visualization of invariant
manifolds in the Lorenz and FitzHugh-Nagumo systems and an
automatic continuation scheme for (un)stable manifolds in
a suspension bridge problem.
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