  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.

Chaotic Dynamics