Invariant Spectra of Dynamical Systems

Details Invariant Spectra of Dynamical Systems Invariant Spectra of Dynamical Systems

Jan 1995

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995nyasa.773..145c&link_type=abstract

Waves in Astrophysics, vol. Volume 773, p. 145-168

Mathematics

Dynamical Systems

1

Liapunov Functions, Hubble Constant, Galaxies, Hamiltonian Functions, Orbits, Dynamical Systems, Eigenvalues, Invariance, Strange Attractors, Chaos, Poincare Problem

Scientific paper

In recent years the problem of finite-time Lyapunov numbers has attracted much interest. In astronomical problems an appropriate time is Hubble time which, in the case of stars in galaxies, is about 100 periods. It is of interest to find what are the effective Lyapunov numbers over one Hubble time. It is also of interest to find the behavior of the Lyapunov numbers over much shorter times. The usual (maximal) Lyapunov characteristic number (LCN) is equal to the limit (as t approaches infinity) of (chi(t) = (1/t)ln(d(t)/d(0))), where d(t) is the deviation from a given orbit at time t and the initial deviation is d(0). The details of the spectrum of values of chi(t) are washed out as time t increases. We have adopted the extreme case of t equal to one period, namely one iteration in the case of a map. In the case of continuous time we have considered one intersection of a Poincare surface of section, but even more details can be found if we consider very small time intervals. In this paper we consider two-dimensional maps, both conservative and dissipative; two-dimensional Hamiltonian systems; and four-dimensional maps, which correspond to Hamiltonian systems of 3 degrees of freedom.

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