Sensitivity of ray paths to initial condition

TL;DR

This paper analyzes how initial conditions affect ray travel times in a waveguide with nonlinear oscillator dynamics, revealing step-like behaviors, chaos, and clustering effects through analytical and model-based approaches.

Contribution

It introduces an analytical framework for understanding the dependence of ray travel time on initial conditions in a nonlinear waveguide model, explaining step-like behavior and chaos.

Findings

Travel time exhibits step-like dependence on initial conditions.

Periodic perturbations induce wave and ray chaos.

Chaotic ray travel times are clustered around maxima related to ray semi-cycles.

Abstract

Using a parabolic equation, we consider ray propagation in a waveguide with the sound speed profile that corresponds to the dynamics of a nonlinear oscillator. An analytical consideration of the dependence of the travel time on the initial conditions is presented. Using an exactly solvable model and the path integral representation of the travel time, we explain the step-like behavior of the travel time (T) as a function of the starting momentum (p_0) (related to the starting ray grazing angle (\chi_0) by (p_0=\tan\chi_0)). A periodic perturbation of the waveguide along the range leads to wave and ray chaos. We explain an inhomogeneity of distribution of the chaotic ray travel times, which has obvious maxima. These maxima lead to the clustering of rays and each maximum relates to a ray identifier, {\em i.e.} to the number of ray semi--cycles along the ray path.