Search for Low-Dimensional Nonlinear Behavior in Irregular Variable Stars. the Global Flow Reconstruction Method.

We describe a powerful, recently developed method for nonlinear time-series analysis. The method allows one to check whether the given signal has a dominant component that been generated by a low-dimensional nonlinear dynamics. Furthermore it allows one to extract properties of this dynamics from the mere knowledge of the given scalar (single) observed quantity. In the context of variable stars this is normally the luminosity of the star, or possibly its radial velocity. The method is tailored to irregular signals and thus complements the classical techniques of analysis which apply only to multi-periodic signals. The ultimate purpose is to develop a better physical understanding of the pulsations and to derive novel astrophysical constraints from irregular light curves.Before applying the global flow reconstruction to signals of unknown properties it is imperative to test it on a well known system. For that purpose we have applied it to the well studied Rossler oscillator which, we note, has a behavior that is similar to the one encountered in the numerical model pulsations of W Virginis stars. For the analysis we allow ourselves only a short section of the temporal behavior of only one of the 3 Rossler variables to infer properties of the whole attractor. In order to make the test more realistic, Gaussian noise has also been added to the Rossler oscillator data. The method is shown to perform very well, producing synthetic signals that, when chaotic, are very close to the original one, with similar Fourier spectra. 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 exponents and fractal dimensions of the synthetic signals. These quantities appear to be fairly robust.The minimum embedding dimension for the reconstruction with the first Rossler variable is found to be 3. Importantly, also, even though only one Rossler variable is assumed to be known the method recovers the 'physical' dimension 3 of the RGssler band. 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 pulsations of a W Virginis model. Applications to observational data of R Set and of AC Her are presented in companion papers.