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Local adaptive Galerkin bases for large-dimensional dynamical systems. (English) Zbl 0729.58034

This paper is concerned with the problem of describing the dynamics of the (finite dimensional) attractor of an infinite dimensional dynamical system, typically a partial differential equation. An example of such an attractor would be an inertial manifold but the authors do not restrict themselves to cases where such exists. Because a global description of the attractor will still involve too many dimensions to be practical, they seek a method which will focus on the local, significant dynamics which will occur in a phase space of more manageable dimension. The problem is thus to find local coordinate systems for the attractor related by patching functions (matrices) which encode the transitional dynamics. These coordinate systems will have time dependent bases as is already seen in the case of action-angle variables for integrable, Hamiltonian evolution equations, such as the Korteweg-de Vries (KdV) equation on the circle. The authors employ a local version of singular valued decomposition of the data set by breaking the data set up into points lying within \(\epsilon\)-balls and applying the usual decomposition within each ball. Those directions whose singular values scale as \(\epsilon\) span the \(\epsilon\)-tangent space, the dynamics on which governs that in the other directions. A first approximation can be gleaned from crude data and then used in refining the data. In this way a dynamical system on the \(\epsilon\)-tangent spaces is obtained.
Throughout the authors employ the KdV on the circle as an example. In this example the numerical procedure can be compared with the explicit calculation of the \(\epsilon\)-tangent space.

MSC:

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
35G20 Nonlinear higher-order PDEs
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