ISSN:
1432-0606
Keywords:
Key words. Galerkin bases, Eigenvalues, Average curvature. AMS Classification. 34C35, 34A26.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. The local adaptive Galerkin bases for large-dimensional dynamical systems, whose long-time behavior is confined to a finite-dimensional manifold, are optimal bases chosen by a local version of a singular decomposition analysis. These bases are picked out by choosing directions of maximum bending of the manifold restricted to a ball of radius ɛ . We show their geometrical meaning by analyzing the eigenvalues of a certain self-adjoint operator. The eigenvalues scale according to the information they carry, the ones that scale as ɛ 2 have a common factor that depends only on the dimension of the manifold, the ones that scale as ɛ 4 give the different curvatures of the manifold, the ones that scale as ɛ 6 give the third invariants, as the torsion for curves, and so on. In this way we obtain a decomposition of phase space into orthogonal spaces E m , where E m is spanned by the eigenvectors whose corresponding eigenvalues scale as ɛ m . This decomposition is analogous to the Frenet frames for curves. We also discover a practical way to compute the dimension and local structure of the invariant manifold.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s0024599110160
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