Set-valued Markov chains and negative semitrajectories of discretized dynamical systems
Any computer simulation of dynamical systems involves a finite set phase space caused by the nature of finiteness of used machine arithmetic and a rounding to this grid. This paper deals with a mathematical justification of how certain asymptotic distributions arise in computer implementation on models of fixed-point computer arithmetic. It is supposed that the underlying system is a differential equation or an iterated function \(T\) defined on a continuum \(X \subset \mathbb{R}^n\), resulting to its computer implementation by an iterative algorithm, represented by a function \(T_\varepsilon\) defined on the \(\varepsilon\)-grid \(X_\varepsilon = X \cap (\varepsilon \mathbb{Z}^n)\) of \(X\). As we all know, if \(T\) is a smooth invertible function on a bounded open set \(X \subset \mathbb{R}^n\), then its discretization induces a dynamical system on \(X_\varepsilon\) and the discretized function \(T_\varepsilon : X_\varepsilon \to X_\varepsilon\) is neither injective nor surjective. On the one hand, positive semitrajectories can collaps to a common orbit. On the other hand, there might be unreachable points in \(X_\varepsilon\), apart from being initial values themselves. Asymptotic probabilistic properties of iterations of \(T_\varepsilon\) as \(\varepsilon \to 0\) are investigated by considering negative semitrajectories of the discretized system. In contrast to the original invertible \(T\), for further asymptotic studies, one considers associated set-valued Markov chains. In fact, the main theorem of the authors relates the asymptotic behavior of negative semitrajectories of the discretized system to an associated set valued Markov chain. The evolution of these set-valued semitrajectories is studied by the quantization errors of the compensating system generated by discretizing \(T^{-1}\). It is important to emphasize that the considered asymptotic probabilistic laws are not necessarily uniform distributions in general, compared to other texts. The results of this paper can be applied to give a justification for often rather heuristic models for computational collapse in chaotic systems.
scientific article; zbMATH DE number 1931216