Regular and Chaotic
- Kluwer, J. L. Dordrecht Lebowitz
- Physica A,
We review recent progress in understanding the role of chaos in influencing the structure and evolution of galaxies. The orbits of stars in galaxies are generically chaotic: the chaotic behavior arises in part from the intrinsically grainy nature of a potential that is composed of point masses. Even if the potential is assumed to be smooth, however, much of the phase space of non-axisymmetric galaxies is chaotic due to the presence of central density cusps or black holes. The chaotic nature of orbits implies that perturbations will grow exponentially and this in turn is expected to result in a diffusion in phase space. We show that the degree of orbital evolution is not well predicted by the growth rate of infinitesimal perturbations, i.e. by the Liapunov exponent. A more useful criterion is whether perturbations continue to grow exponentially until their scale is of order the size of the system. We illustrate these ideas in a potential consisting of N fixed point masses. Liapunov exponents are large for all values of N , but orbits become increasingly regular in their behavior as N increases; the reason is that the exponential divergence saturates at smaller and smaller distances as N is increased. The objects which lend phase space its structure and impede diffusion are the invariant tori. In the triaxial potentials we discuss, a large fraction of the tori correspond to resonant (thin) orbits and their associated families of regular orbits. These tori are destroyed by perturbations to the potential. When only a few stable resonances remain, we find that the phase space distribution of an ensemble of chaotic orbits evolves rapidly toward a nearly stationary state. This mixing process is shown to occur on timescales of a few crossing times in triaxial potentials containing massive central singularities, consistent with the rapid evolution observed in N -body simulations of galaxies with central black holes.