RUELLE'S INEQUALITY FOR ISOTROPIC ORNSTEIN–UHLENBECK FLOWS

Ruelle's inequality asserts that the entropy of a dynamical system is bounded from above by the Lyapunov characteristic numbers counted with their multiplicities. We show that this inequality holds true in the case of a random dynamical system deduced from an isotropic Ornstein–Uhlenbeck-flow (IOUF).

The Role of Chaos in Barred Galaxies

Ordered orbits in barred galaxies appear along the bar and between the −4/1 and −2/1 resonances of the outer spiral. Chaotic orbits appear mainly near corotation. Such orbits support the bar and the spiral for long times and they are important for self-consistency. There are three main mechanisms for transition from order to chaos: (a) infinite bifurcations, (b) infinite gaps, and (c) infinite spirals. The Lyapunov characteristic number is zero for ordered orbits and positive for chaotic orbits. But much more information is provided by the distribution of the stretching numbers (one-period Lyapunov characteristic numbers). The spectrum of stretching numbers is invariant with respect to initial conditions in a connected chaotic domain. We provide examples of such spectra for 2-D maps, plane galactic orbits, 2-D dissipative systems, 3-D systems (represented by 4-D maps), and systems depending periodically on the time.

Pluto's Lyapunov numbers in different dynamical models

We present the results of numerical integrations of Pluto and some fictitious Plutos in three different models (the circular and the elliptic restricted three body problem and the outer solar system). We determined the “extension” of the stable region in these models by means of the Lyapunov Characteristic Numbers and by an analysis of the orbital elements.