Astronomy and Astrophysics – Astrophysics
Scientific paper
Jan 1995
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995nyasa.773..189s&link_type=abstract
Waves in Astrophysics, vol. Volume 773, p. 189-204
Astronomy and Astrophysics
Astrophysics
1
Time Dependence, Time Functions, Liapunov Functions, Orbits, Orbital Mechanics, Galaxies, Orbit Perturbation, Chaos, Angular Momentum, Nonlinear Systems, Many Body Problem
Scientific paper
There is experimental evidence that there may be long-lived global oscillations of galaxies. Such oscillations have been seen in the kinetic and potential energies as well as in the density profiles of computed N-body systems. The presence of such oscillations is consistent with the virial equations, and at least one detailed self-consistent model has been constructed having such oscillations. Our particular interest is in the characteristics of orbits in such systems, in which the potential is time-dependent, and especially in manifestations of chaos in some of those orbits. To identify the character of the orbits we first reduce the problem from three dimensions to one using the vector angular momentum. Then, to allow for the time dependence of the gravitational potential we employ the usual extended phase space, with additional canonical phase-space coordinates. Because the perturbation is strictly periodic we can fold the time dimension into itself at intervals of the oscillation period. Integration of the orbits was done with a fourth-order explicit symplectic integrator. The spectra of finite-time Lyapunov exponents are useful. The most detailed information is found if we take very small time intervals, while for longer intervals many of the details are lost. If we take a sufficiently large number of iterations of the map, the spectrum of the one-period Lyapunov numbers is invariant (1) with respect to the initial conditions along an orbit; (2) with respect to different initial directions of the perturbation to an orbit; and (3) with respect to the initial conditions of orbits in a given connected chaotic zone. In this paper we find short-time Lyapunov spectra for a system dependent periodically on the time, using various sampling time intervals. We find that that we can derive all the information about such spectra if we calculate the spectrum for a very small interval, namely the integration timestep itself.
Contopoulos George
Smith Haywood Jr.
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