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We consider transport properties of the chaotic (strange) attractor along unfolded trajectories of the dissipative standard map. It is shown that the diffusion process is normal except for the cases when a control parameter is close to some special values that correspond to the ballistic mode dynamics. Diffusion near the related crises is anomalous and nonuniform in time; there are large time intervals during which the transport is normal or ballistic, or even superballistic. The anomalous superdiffusion seems to be caused by stickiness of trajectories to a nonchaotic and nowhere dense invariant Cantor set that plays a similar role as cantori in Hamiltonian chaos. We provide a numerical example of such a sticky set. Distribution function on the sticky set almost coincides with the distribution function (SRB measure) of the chaotic attractor., ((c) 2008 American Institute of Physics.)

We consider a chain of nonlinear oscillators with long-range interaction of the type 1l(1+alpha), where l is a distance between oscillators and 0

Mixing and transport of passive particles are studied in a simple kinematic model of a meandering jet flow motivated by the problem of lateral mixing and transport in the Gulf Stream. We briefly discuss a model stream function, Hamiltonian advection equations, stationary points, and bifurcations. The phase portrait of the chosen model flow in the moving reference frame consists of a central eastward jet, chains of northern and southern circulations, and peripheral westward currents. Under a periodic perturbation of the meander's amplitude, the topology of the phase space is complicated by the presence of chaotic layers and chains of oscillatory and ballistic islands with sticky boundaries immersed into a stochastic sea. Typical chaotic trajectories of advected particles are shown to demonstrate a complicated behavior with long flights in both the directions of motion intermittent with trapping in the circulation cells being stuck to the boundaries of vortex cores and resonant islands. Transport is asymmetric in the sense that mixing between the circulations and the peripheral currents is, in general, different from mixing between the circulations and the jet. The transport properties are characterized by probability distribution functions (PDFs) of durations and lengths of flights. Both the PDFs exhibit at their tails power-law decay with different values of exponents.

We consider a one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction. The corresponding term in dynamical equations is proportional to 1//n-m/alpha+1. It is shown that the equation of motion in the infrared limit can be transformed into the medium equation with the Riesz fractional derivative of order alpha, when 0

We consider wave propagation in a model of a deep ocean acoustic wave guide with a periodic range dependence. It is assumed that the wave field is governed by the parabolic equation. Formally the mathematical model of the wave guide coincides with that of a quantum system with time-dependent Hamiltonian. From the analysis of Floquet modes of the wave guide it is shown that there exists a "scarring" effect similar to that observed in quantum systems. It turns out that the segments of an unstable periodic ray trajectory may be distinguished in the spatial distribution of the wave field intensity at a finite wavelength. Besides the scarring effect, it is found that the so-called "stable islands" in the phase space of ray dynamics reveal themselves in the coarse-grained Wigner functions of the Floquet modes.

This paper provides some historical comments on the study of the Fermi, Pasta, and Ulam (FPU) paper and its influence on the development of the theory of chaos. We also discuss some problems raised in the FPU paper and the links of these problems to such contemporary notions in chaos theory as ergodicity, mixing, recurrences, pseudochaos, kinetics, intermittency, etc.