Chaotic transmission of waves and “cooling” of signals

Abstract

Ray dynamics in waveguide media exhibits chaotic motion. For a finite length of propagation, the large distance asymptotics is not uniform and represents a complicated combination of bunches of rays with different intermediate asymptotics. The origin of the phenomena that we call "chaotic transmission," lies in the nonuniformity of the phase space with sticky domains near the boundary of islands. We demonstrate different fractal properties of ray propagation using underwater acoustics as an example. The phenomenon of the kind of Levy flights can occur and it can be used as a mechanism of cooling of signals when the width of spatial spectra dispersion is significantly reduced. (c) 1997 American Institute of Physics.

Fractional Ginzburg–Landau equation for fractal media

“…The first term on the right-hand side is related to the wave propagation in a media with fractal properties. The fractional derivative may also appear as a result of ray chaos [16,17] or due to superdiffusive wave propagation (see also the discussion in Refs. [1,16] and corresponding references therein).…”

Some applications of fractional equations

“…For t and x < ∼ 1, so that both Gaussian and normal transport may occur we may compare (15), (16) When |x| ∼ 1, P and Q are substantially different; however for t ∼ 1, |x| > 1, one may use (16), and then the leading order terms in (16) (20) and (23) for P (x, t), which again matches Q(x, t) in (30), but with corrections. Finally if |x| and t are both large, as is |x| α /t, then we may conclude that (14) and (16) apply for P (x, t) which approximates Q(x, t) as given by (31).…”

Ray chaos and ray clustering in an ocean waveguide

“…2) with a periodic perturbation [23,26,30] c(z, r) = c 0 1 + µ (η − 1 + e −η ) + ε z B e −2z/B cos 2πr λ , Fig. 9 but with the parameters of the periodic perturbation, λ = 10 km and ε = 0.005.…”

“…It can be used as a stochasticity criterion for nonlinear systems under a noisy-like perturbation in a close analogy with the Chirikov's criterion for deterministic dynamical systems since the rate of decreasing of Fourier amplitudes is defined mainly by the dependence of the frequency of spatial oscillations ω on the action I. It is a linear function with Model 1 (26). In Model 2 the respective dependence ω(I) has a local maximum at I r ≃ I(H r ).…”