Lyapunov characteristic numbers and the structure of phasespace
Abstract
The Liapunov characteristics numbers (LCNs) have been used extensively as a criterion of stochasticity of dynamical systems. LCNs in many two and threedimensional systems suggest that the orbits must be calculated for long enough times, otherwise the values found for the LCNs are unreliable. In twodimensional systems, regions of phase space separated by closed invariant surfaces give different LCNs, but the LCNs are approximately constant in the same stochastic region. In threedimensional systems, the invariant surfaces do not separate the different stochastic regions of phasespace and Arnold diffusion may take place. However, for small perturbations Arnold diffusion is quite inefficient and the LCNs of various stochastic regions are different over extremely long times. As the perturbation increases the different stochastic regions communicate and their LCNs tend to become equal. The study of the structure of phasespace is performed, finding an approximate boundary between the ordered regions (LCN = zero) and the stochastic regions. LCNs vary as certain parameters of the system vary.
 Publication:

Astronomy and Astrophysics
 Pub Date:
 September 1989
 Bibcode:
 1989A&A...222..329C
 Keywords:

 Liapunov Functions;
 Orbit Calculation;
 Stellar Motions;
 Stochastic Processes;
 Degrees Of Freedom;
 Hamiltonian Functions;
 Numerical Analysis;
 Astrophysics