SEARCH FOR LOW-DIMENSIONAL NONLINEAR BEHAVIOR IN IRREGULAR VARIABLE-STARS - THE GLOBAL HOW RECONSTRUCTION METHOD

We describe a powerful, recently developed method for nonlinear time-s eries analysis. The method allows one to check whether the given signa l has a dominant component that been generated by a low-dimensional no nlinear dynamics. Furthermore it allows one to extract properties of t his dynamics from the mere knowledge of the given scalar (single) obse rved quantity. In the context of variable stars this is normally the l uminosity of the star, or possibly its radial velocity. The method is tailored to irregular signals and thus complements the classical techn iques of analysis which apply only to multi-periodic signals. The ulti mate purpose is to develop a better physical understanding of the puls ations and to derive novel astrophysical constraints from irregular li ght curves. Before applying the global flow reconstruction to signals of unknown properties it is imperative to test it on a well known syst em. For that purpose we have applied it to the well studied Rossler os cillator which, we note, has a behavior that is similar to the one enc ountered in the numerical model pulsations of W Virginis stars. For th e analysis we allow ourselves only a short section of the temporal beh avior of only one of the 3 Rossler variables to infer properties of th e whole attractor. In order to make the test more realistic, Gaussian noise has also been added to the Rossler oscillator data. The method i s shown to perform very well, producing synthetic signals that, when c haotic, are very close to the original one, with similar Fourier spect ra. The map that is obtained from the data allows one then to quantify the complexity of a chaotic signal, e.g. with the help of Lyapunov ex ponents and fractal dimensions of the synthetic signals. These quantit ies appear to be fairly robust. The minimum embedding dimension for th e reconstruction with the first Rossler variable is found to be 3. Imp ortantly, also, even though only one Rossler variable is assumed to be known the method recovers the 'physical' dimension 3 of the RGssler b and. The method works well even when large amounts of noise are added to the signal prior to the analysis. In a twin paper we analyze the pu lsations of a W Virginis model. Applications to observational data of R Set and of AC Her are presented in companion papers.