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This book demonstrates how a 'triangulation method' can be used to model and control dynamical systems defined only from experimental data. For example, they are used to reconstruct the Hénon map. This is a simple two dimensional model found by a French astronomer Michel Hénon, which has a chaotic behaviour. A trajectory is shown in Figure 1.1. For details, see Section 4.3.1 on page 61 of this book. For comparison, we will apply the same control method to the true Hénon map. The idea of the 'triangulation method' is to construct a "net" of geometrical objects spanning the experimental data points. More precisely, the plane is filled with triangles, three dimensional space is filled with tetrahedra, and, in general, m-dimensional space is filled with simplices, which are simple m-dimensional objects, determined by the minimum number of points: 3 for the plane and m + 1 for an m-dimensional space. We require that all the vertices are the data points and no data point lies within any simplex. Now, we use this natural grid to approximate an unknown map by linear interpolation for every simplex separately. |
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