The power spectrum indicator: A new, efficient method for the early detection of chaos

To determine the regular or chaotic nature of the orbits in dynamical systems can be quite an issue. In this article, following Vozikis et al. (2000), we propose a new tool, namely, the Power Spectrum Indicator (PSI), $\psi^2$, that enables us to determine, as early as posible, whether an orbit of a two-dimensional map is chaotic or not. This new method is based on the frequency analysis of a data series constucted by recording the logarithm of the amplification factor of the deviation vector of nearby orbits. Accordingly, two datasets are recorded and the $\chi^2$-likelyhood of their power spectra is computed. Ordered orbits have always the same power spectrum, so their $\chi^2 \equiv \psi^2$ acquires a zero value. On the contrary, a chaotic orbit has a power spectrum that varies with time, hence, chaotic orbits always exhibit a non-zero $\psi^2$ value. Even as regards "sticky" orbits, the PSI method is very effective in the early detection of chaos, while the global behavior of the $\psi^2$ indicator can provide information (also) on the intense of the chaotic behavior, i.e., on how "strong" or "weak" the associated chaos may be.