  We consider ray propagation in a waveguide with a designed sound-speed
profile perturbed by a range-dependent perturbation caused by internal waves in
deep ocean environments. The Hamiltonian formalism in terms of the action and
angle variables is applied to study nonlinear ray dynamics with two
sound-channel models and three perturbation models: a single-mode perturbation,
a random-like sound-speed fluctuations, and a mixed perturbation. In the
integrable limit without any perturbation, we derive analytical expressions for
ray arrival times and timefronts at a given range, the main measurable
characteristics in field experiments in the ocean. In the presence of a
single-mode perturbation, ray chaos is shown to arise as a result of
overlapping nonlinear ray-medium resonances. Poincar\'e maps, plots of
variations of the action per a ray cycle length, and plots with rays escaping
the channel reveal inhomogeneous structure of the underlying phase space with
remarkable zones of stability where stable coherent ray clusters may be formed.
We demonstrate the possibility of determining the wavelength of the
perturbation mode from the arrival time distribution under conditions of ray
chaos. It is surprising that coherent ray clusters, consisting of fans of rays
which propagate over long ranges with close dynamical characteristics, can
survive under a random-like multiplicative perturbation modelling sound-speed
fluctuations caused by a wide spectrum of internal waves.
