Analytic investigation of chaos in a class of parabolic ray systems

Abstract

It has been shown that acoustic ray paths in range-dependent ocean models exhibit chaotic behavior. Most of the investigations into the ray chaos phenomenon have been primarily numerical in nature. Analytical derivation of sufficient conditions for chaos in acoustic systems has been restricted to inherently discrete problems. This article reports a theoretical study of the existence of ray chaos in a class of continuous parabolic ray systems. This class of ray systems is indexed by a family of analytically prescribed double-channel sound-speed profiles perturbed by periodic range-dependent disturbances. The perturbed Hamiltonian ray systems are studied analytically via the Melnikov method. It is shown that, under certain conditions, ray trajectories of the systems are equivalent to trajectories of a classic chaotic system known as the horseshoe map when the perturbation is periodic and small. These conditions are sufficient for ray chaos and easily satisfied, thus explaining why double-channel propagation is very likely to exhibit chaotic behavior. © 1997 Acoustical Society of America. ͓S0001-4966͑97͒00503-1͔ PACS numbers: 43.30.Cq, 43.25.Rq, 43.25.Ts ͓MBP͔ INTRODUCTIONWhen sound speed varies as a function of depth and range, the acoustic wave equation can be studied by a variety of methods. When the index of refraction changes slowly with respect to the wavelength of the field, the Helmholtz equation may be reduced to a ray acoustic model. 1,2 Such a model provides a computationally efficient and intuitive means of analysis.Typical high-frequency approximations to the Helmholtz equation give the standard ͑two-way͒ elliptic ray equations. The assumption that rays travel only in the direction of increasing range coordinate yields the one-way elliptic ray equations. Requiring ray angles with respect to the range axis to be small yields the parabolic ray equations. All three sets of ray equations have been shown to be integrable when the sound speed is independent of range. 2 The integrability of ray systems guarantees regular ray motion, i.e., neighboring ray trajectories diverge and converge at a polynomial rate. It should be clear that when we speak of two rays converging we mean that all components ͑i.e., position and angle͒ of the ray vector become arbitrarily close. In the generic range-dependent problem, however, the ray equations are not integrable and some systems may be chaotic. A chaotic system possesses at least some trajectories which are extremely sensitive to initial conditions. Such chaotic trajectories converge and diverge in an apparently random fashion. Locally, these orbits may diverge/converge at an exponential rate.The study of ray chaos in underwater acoustics involves the application of results from the study of nonintegrable Hamiltonian systems. This topic has been reviewed and discussed by many authors. Palmer et al. 3 showed numerically that acoustic ray paths in a weakly range-dependent deterministic ocean model exhibit chaotic behavior. Tappert et al. 4 and Brown et al. 5 investigated so...