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A unified approach to compute foliations, inertial manifolds, and tracking solutions


Authors: Y.-M. Chung and M. S. Jolly
Journal: Math. Comp. 84 (2015), 1729-1751
MSC (2010): Primary 34C40, 34C45, 37L25
DOI: https://doi.org/10.1090/S0025-5718-2014-02904-9
Published electronically: December 9, 2014
MathSciNet review: 3335889
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Abstract: Several algorithms are presented for the accurate computation of the leaves in the foliation of an ODE near a hyperbolic fixed point. They are variations of a contraction mapping method used by Ricardo Rosa in 1995 to compute inertial manifolds, which represents a particular leaf in the unstable foliation. Such a mapping is combined with one for the leaf in the stable foliation to compute tracking solutions. The algorithms are demonstrated on the Kuramoto-Sivashinsky equation.


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Additional Information

Y.-M. Chung
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email: ychung@wm.edu

M. S. Jolly
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: msjolly@indiana.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02904-9
Keywords: Foliations, inertial manifolds, tracking solution, spectral gap condition
Received by editor(s): September 25, 2012
Received by editor(s) in revised form: September 12, 2013, and October 27, 2013
Published electronically: December 9, 2014
Additional Notes: This work was supported in part by NSF grant numbers DMS-1008661 and DMS-1109638. The authors thank Ricardo Rosa for several stimulating discussions, and the referees for their helpful comments.
Article copyright: © Copyright 2014 American Mathematical Society