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

• Main Authors: Vozikis, Christos, Kleidis, Konstantinos, Papaioannou, Stavros
• Format: Journal Article
• Language: English
• Published: 01-08-2018
• Subjects: Mathematics - Dynamical Systems; Physics - Chaotic Dynamics
• DOI: 10.48550/arxiv.1808.00223

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.