Rigorously computed orbits of dynamical systems without the wrapping effect
Consider the discrete dynamic system \[ \Omega_n=f_n(\Omega_{n-1}), \quad\Omega_0\in R^d,\tag{1} \] \noindent where \(f_n\) are the maps, \(f_n:R^d\rightarrow R^d\) for stages \(n=1,2,\dots\). The goal is to construct supersets (enclosures) for the orbit of (1) such that the overestimation is kept small. (1) is not immediately suitable for the iteration on a computer. Therefore, the overestimation of the range \(f_n(\Omega_{n-1})\) at every stage \(n\) by wrapping it into a feasible superset is necessary to get a result suitable for the iteration on a computer (marching method). A collection of such sets and maps \(F_n\) operating on this collection are selected such that \(f_n(\Theta)\subseteq F_n(\Theta)\) for \(\Omega_0\subseteq\Theta_0\). Such maps are called extensions of \(f_n\) over the particular collection. The iterates are enclosures for (1). The paper presents a marching method (called cascade reduction) which beats the wrapping effect effectively and efficiently by constructing high order zonotope extensions of the orbit. A zonotope is the Minkowski sum of straight line segments. The number of line segments is always a multiple \(m\) of \(d\) here. The integer \(m\) is a freely selected measure for geometric complexity and is considered as a performance parameter for the method. The cascade reduction for finite precision, parallel computing and sparse maps are analyzed. Several examples are supplied.
scientific article; zbMATH DE number 1199878
scientific article