Chaotic Dynamics

We address scaling in inhomogeneous and anisotropic turbulent flows by decomposing structure functions into their irreducible representation of the SO(3) symmetry group which are designated by j,m indices. Employing simulations of channel flows with Re λ ≈70 we demonstrate that different components characterized by different j display different scaling exponents, but for a given j these remain the same at different distances from the wall. The j=0 exponent agrees extremely well with high Re measurements of the scaling exponents, demonstrating the vitality of the SO(3) decomposition.

We show that an integro-differential equation model for pulse propagation in optical transmission lines with dispersion management, is integrable at the {\it leading nonlinear order}. This equation can be transformed into the nonlinear Schroedinger equation by a near-identity canonical transformation for the case of weak dispersion. We also derive the next order (nonintegrable) correction.

We compute analytically the probability distribution function P(ϵ) of the dissipation field ϵ=(∇θ ) 2 of a passive scalar θ advected by a d -dimensional random flow, in the limit of large Peclet and Prandtl numbers (Batchelor-Kraichnan regime). The tail of the distribution is a stretched exponential: for ϵ→∞ , lnP(ϵ)∼−( d 2 ϵ ) 1/3 .

I investigate the propagator of the Wigner function for a dissipative chaotic quantum map. I show that a small amount of dissipation reduces the propagator of sufficiently smooth Wigner functions to its classical counterpart, the Frobenius-Perron operator, if ℏ→0 . Several consequences arise: The Wigner transform of the invariant density matrix is a smeared out version of the classical strange attractor; time dependent expectation values and correlation functions of observables can be evaluated via hybrid quantum-classical formulae in which the quantum character enters only via the initial Wigner function. If a classical phase-space distribution is chosen for the latter or if the map is iterated sufficiently many times the formulae become entirely classical, and powerful classical trace formulae apply.

The full nonlinear dissipative quasigeostrophic model is shown to have a unique temporally almost periodic solution when the wind forcing is temporally almost periodic under suitable constraints on the spatial square-integral of the wind forcing and the β parameter, Ekman number, viscosity and the domain size. The proof involves the pullback attractor for the associated nonautonomous dynamical system.

We consider the tunneling of a wave packet through a potential barrier which is coupled to a nonintegrable classical system and study the interplay of classical chaos and dissipation in the tunneling dynamics. We show that chaos-assisted tunneling is further enhanced by dissipation, while tunneling is suppressed by dissipation when the classical subsystem is regular.

The zeroes of the Husimi function provide a minimal description of individual quantum eigenstates and their distribution is of considerable interest. We provide here a numerical study for pseudo- integrable billiards which suggests that the zeroes tend to diffuse over phase space in a manner reminiscent of chaotic systems but nevertheless contain a subtle signature of pseudo-integrability. We also find that the zeroes depend sensitively on the position and momentum uncertainties with the classical correspondence best when the position and momentum uncertainties are equal. Finally, short range correlations seem to be well described by the Ginibre ensemble of complex matrices.

The quantitative contributions of a mixed phase-space to the mean characterizing the distribution of diagonal transition matrix elements and to the variance characterizing the distributions of non-diagonal transition matrix elements are studied. It is shown that the mean can be expressed as the sum of suitably weighted classical averages along an ergodic trajectory and along the stable periodic orbits. Similarly, it is shown that the values of the variance are well reproduced by the sum of the suitably weighted Fourier transforms of classical autocorrelation functions along an ergodic trajectory and along the stable periodic orbits. The illustrative numerical computations are done in the framework of the Hydrogen atom in a strong magnetic field, for three different values of the scaled energy.

We argue that Gaspard and coworkers do not give evidence for microscopic chaos in the sense in which they use the term. The effectively infinite number of molecules in a fluid can generate the same macroscopic disorder without any intrinsic instability. But we argue also that the notion of chaos in infinitely extended systems needs clarification: In a wider sense, even some systems without local instabilities can be considered chaotic.

We analyze characteristics of drifter trajectories from the Adriatic Sea with recently introduced nonlinear dynamics techniques. We discuss how in quasi-enclosed basins, relative dispersion as function of time, a standard analysis tool in this context, may give a distorted picture of the dynamics. We further show that useful information may be obtained by using two related non-asymptotic indicators, the Finite-Scale Lyapunov Exponent (FSLE) and the Lagrangian Structure Function (LSF), which both describe intrinsic physical properties at a given scale. We introduce a simple chaotic model for drifter motion in this system, and show by comparison with the model that Lagrangian dispersion is mainly driven by advection at sub-basin scales until saturation sets in.