__Ray
chaos: or why Hamiltonian dynamics might be relevant to__

__probing
pattern formation in dissipative systems__

__ __

** Randall Tagg** and Masoud Asadi-Zeydabadi

University of Colorado at Denver

Most patterns are observed experimentally by direct visualization,
but in some circumstances it may be necessary to probe patterns indirectly by
passing beams of light, sound, or particles through them. In particular, moving
fluids will affect the transmission of sound while density gradients can alter
both light and sound transmission. A seemingly simple approach to understanding
these effects is to study the "geometric optics" approximation to
signal propagation, where the problem is reduced to one of ray tracing.
Interestingly, this reduction produces a Hamiltonian system of equations for
the ray dynamics in which the axial distance of propagation "z" plays
the role of time. Moreover, all the interesting features of complex Hamiltonian
dynamics come into play: resonance, homoclinic tangles, and chaos. We have
studied the effect of imposing a particularly simple pattern on a waveguide which
could occur either in an optical fiber or in the ocean. Appropriate scaling
renders a wide class of physical situations into a two-parameter problem, where
the parameters are the amplitude and the wavelength of the pattern. The effect
on the optical/acoustical ray dynamics is viewed in terms of Poincare sections,
both in natural variables and "action-angle" variables. A beautiful
structure emerges that might even by exploited in real signal propagation
applications. This depends on where the structure survives transforming the
problem back to finite wavelength of the

probing signal, an issue related to questions in quantum chaos.