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.