Shadowing and prediction of dynamical systems
The author deals with the dynamical system (1) \(y_{t+1}= f(y_t)\), \(t=0,1,\dots\), where \(y\in X\) (\(X\) some bounded set of \(\mathbb{R}^n\)), \(f:X\to X\) is at least \(C^2\) and the initial value \(y_0\) is specified. The author considers instead of (1) following system (2) \(x_{t+1}= \widehat{f}(x_t)\), \(t=0,1,\dots\), where \(\widehat{f}\) is an application of \(f\) and \(x\in X\). The main goal of the author is to generate sequences of vectors \(x_t\) using the system in (2) that corresponds to shadows of the vectors \(y_t\) observed from the real system (1). Under suitable assumptions on \(f\), the author shows that there exists a sequence of data points \(\{y_t\}_{t=0}^N\) such that when the mapping \(\widehat{f}\) in (2) is applied to the sequence, one can obtain (3) \(\|y_{t+1}- \widehat{f}(y_t)\|\leq \delta\), \(t=0,1,\dots, N-1\), thus modifying the technique of \textit{S. N. Chow} and \textit{K. J. Palmer} [J. Complexity 8, No.~4, 398-423 (1992; Zbl 0769.58040)].
scientific article; zbMATH DE number 1755396