Dimensionality reduction using secant-based projection methods: the induced dynamics in projected systems (Q2499470)

An interesting class of dynamical systems which are defined on high-dimensional spaces are such which have a low-dimensional attractor. For such a system a reduced-order description can be found. How this can be done is a subject of intensive research, and the current paper gives a significant contribution. The present paper is a continuation of previous papers of the authors where they have developed an approach to the data reduction problem which is based on a well-known constructive proof of Whitney's embedding theorem [\textit{D.S. Broomhead} and \textit{M. Kirby}, SIAM J. Appl. Math. 60, No.~6, 2114--2142 (2000; Zbl 1038.65013); Neural Comput. 13, No.~11, 2595--2616 (2001; Zbl 1003.68112)]. This approach involves picking projections of the high-dimensional system which are optimised in the sense that they are easy to invert. This is done by considering the effect of projections on the set of unit secants constructed from the data. In the present paper the implication of this idea is discussed in the case when high-dimensional data is generated by a dynamical system. The question is asked whether the existence of an easily invertible projection leads to practical methods for the construction of an equivalent low-dimensional dynamical system. The paper consists of a review of the secant-based projection method and simple methods for finding good representations of the (nonlinear) inverse of the projections. Two variants of a way to find the dynamical system induced by a projection are discussed which leads to quite distinct numerical approximations. One of these is developed further describing various ways in which knowledge of the full dynamical system can be incorporated into the approximate projected system. The ideas of the paper are illustrated in some examples, which range from a simple system of nonlinear ODEs which have an attracting limit cycle, to low-dimensional solutions of Kuramoto-Sivashinsky equation which need many Galerkin modes for their description.