3,686 research outputs found
Computation of maximal local (un)stable manifold patches by the parameterization method
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. In the present work we treat
explicitly the case where the eigenvalues satisfy a certain non-resonance
condition.Comment: Revised version, typos corrected, references adde
Localization theorems for nonlinear eigenvalue problems
Let T : \Omega \rightarrow \bbC^{n \times n} be a matrix-valued function
that is analytic on some simply-connected domain \Omega \subset \bbC. A point
is an eigenvalue if the matrix is singular.
In this paper, we describe new localization results for nonlinear eigenvalue
problems that generalize Gershgorin's theorem, pseudospectral inclusion
theorems, and the Bauer-Fike theorem. We use our results to analyze three
nonlinear eigenvalue problems: an example from delay differential equations, a
problem due to Hadeler, and a quantum resonance computation.Comment: Submitted to SIMAX. 22 pages, 11 figure
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