Attractor reconstruction for nonlinear systems: A methodological note (Q5942613)

Lyapunov spectra and prediction algorithms provide important information about a dynamical system and can be used, for instance, for selection of an appropriate scale for study. However, extracting this information from an attractor reconstruction requires great care. The authors present methods for attractor reconstruction and illustrate the consequences for estimation of prediction error and the Lyapunov spectrum. NEWLINENEWLINENEWLINEThe main idea of attractor reconstruction is to use a single system output to create a set of new coordinates that preserve the invariant properties of the system. This may be accomplished by exploiting the method introduced by \textit{F. Takens} [Detecting strange attractors in turbulence, in: Dynamical systems and turbulence, Proc. Symp., Coventry 1980, Lect. Notes Math. 898, 366-381 (1981; Zbl 0513.58032)] One uses delay coordinates to construct a set of state vectors NEWLINE\[NEWLINE x(n)=(x(n),x(n+T),x(n+2T),\ldots,x(n+(d_e-1)T)), NEWLINE\]NEWLINE where \(x(n)\) is the observable state variable at discrete time \(n, T\) is delay, and \(d_e\) is called the embedding dimension. Results highlight the need of care to be taken in choosing \(d_e\) such that underlying dynamical attractor is faithfully reconstructed, and the number of vectors chosen to represent the system must capture all of the essential dynamics.