Invariant spectra of orbits in dynamical systems

Abstract

We show that in deterministic dynamical systems any orbit is associated with an invariant spectrum of stretching numbers, i.e. numbers expressing the logarithmic divergences of neighbouring orbits within one period. The first moment of this invariant spectrum is the maximal Lyapunov characteristic number (LCN). In the case of a chaotic domain, a single invariant spectrum characterizes the whole domain. The invariance of this spectrum allows the estimation of the LCN by calculating, for short times, many orbits with initial conditions in the same chaotic region instead of calculating one orbit for extremely long times. However, if part of the initial conditions are in an ordered region, the average of the short-time calculations may deviate considerably from the LCN. Invariant spectra appear not only for conservative but also for dissipative systems. A few examples are given.