Time-delay coordinates and polynomial mappings. (Q1400953)

A simple model of a dynamical system is a diffeomorphism \(f: M \to M\) of a compact finite-dimensional manifold \(M\) as its state space. Consider the sequence of scalers \[ g(x), g(f(x)), g(f^2(x)),\dots, g(f^n(x)), \dots \] where \(g : M \to \mathbb R^1 \in C^\infty\), \(x \in M\) initial state and \(f^n\) represents the \(n\)-fold composition of \(f\) with itself. Then the method of time-delay coordinates involves reconstructing \(M\) by an embedding \(\varphi : M \to \mathbb R^N\) defined by \[ \varphi (x)=(f(x),g(f(x)),\dots,g(f^{N-1}(x))) \in \mathbb R^N \] with an appropriate \(N\) under suitable hypotheses on \(g\). Takens, Sauer and others gave a rigorous basis for this method in the case of generic systems (i.e., embedding theorems and methods which are commonly used for the reconstruction of attractors from data series). After some preliminary considerations such as Whitney's and Nash's theorems, the author presents an analogue of the results of \textit{T. Sauer, J. A. Yorke} and \textit{M. Casdagli} [J. Stat. Phys. 65, 579--616 (1991; Zbl 0943.37506)] in the corresponding setting of polynomial mappings. The use of time-delay coordinates is illustrated, proofs and some examples are provided as well.