Spectra of Dynamical Systems

Abstract

We define the spectra of stretching numbers (i.e. short-time Lyapunov characteristic numbers) and of helicity angles (angles between successive infinitesimal vector deviations from a given orbit and a fixed direction). We give examples of the spectra of two different maps, that are quite different, even though their usual Lyapunov characteristic numbers are equal. The spectra of orbits in the same chaotic domain are invariant. We explain the form of the spectrum of the helicity angles by calculating the asymptotic curves of one simple unstable periodic orbit. The deviations of other orbits in the same chaotic domain are in general closely parallel to these asymptotic curves. The difference between ordered and chaotic spectra allows a fast separation of the chaotic and ordered domains.