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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 particles transport in the Sinai billiard with infinite horizon. The simulation shows that the transport is superdiffusive in both continuous and discrete time. Also, it is shown that the moments do not converge to the Gaussian moments even in the logarithmically renormalized time scale, at least for a fairly long computational time. These results are discussed with respect to the existent rigorous theorems. Similar results are obtained for the stadium billiard.

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

It is shown that two coupled oscillators perturbed by periodic kicks generate a thin stochastic web in the four-dimensional phase space, which differs from the Arnold web. Under some resonance-type condition the web possesses a quasicrystal-type symmetry. In three-dimensional coordinate space, the web's symmetry corresponds to the icosahedral one and, due to that, the original four-dimensional map can be considered as a dynamical generator of the quasicrystal-type tiling of three-dimensional space.

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 a nonlinear oscillator of the Duffing type with fractional derivative of the order 1

Hamiltonian chaotic dynamics is, in general, not ergodic and the boundaries of the ergodic or quasiergodic area (stochastic sea, stochastic layers, stochastic webs, etc.) are sticky, i.e., trajectories can spend an arbitrarily long time in the vicinity of the boundaries with a nonexponentially small probability. Segments of trajectories imposed by the stickiness are called flights. The flights have polynomial dispersion that can lead to non-Gaussian statistics of displacements and to anomalous transport in phase space. In particular, the presence of flights influences the distribution of Poincaré recurrences. We use the distribution function of (l,t;epsilon, epsilon0) -separation of trajectories that at time instant t and trajectory length l are separated for the first time by epsilon<<1, being initially at a distance epsilon0 <

Sound-wave propagation in a strongly idealized model of the deep-water acoustic waveguide with a periodic range dependence is considered. It is investigated how the phenomenon of ray and wave chaos affects the sound scattering at a strong mesoscale inhomogeneity of the refractive index caused by the synoptic eddy. Methods derived in the theory of dynamical and quantum chaos are applied. When studying the properties of wave chaos we decompose the wave field into a sum of Floquet modes analogous to quantum states with fixed quasi-energies. It is demonstrated numerically that the "stable islands" from the phase portrait of the ray system reveal themselves in the coarse-grained Wigner functions of individual Floquet modes. A perturbation theory has been derived which gives an insight into the role of the mode-medium resonance in the formation of Floquet modes. It is shown that the presence of a weak internal-wave-induced perturbation giving rise to ray and wave chaos strongly increases the sensitivity of the monochromatic wave field to an appearance of the eddy. To investigate the sensitivity of the transient wave field we have considered variations of the ray travel times--arrival times of sound pulses coming to the receiver through individual ray paths--caused by the eddy. It turns out that even under conditions of ray chaos these variations are relatively predictable. This result suggests that the influence of chaotic-ray motion may be partially suppressed by using pulse signals. However, the relative predictability of travel time variations caused by a large-scale inhomogeneity is not a general property of the ray chaos. This statement is illustrated numerically by considering an inhomogeneity in the form of a perfectly reflecting bar.

The transport properties of particles evolving in a system governed by the Charney-Hasegawa-Mima equation are investigated. Transport is found to be anomalous with a nonlinear evolution of the second moments with time. The origin of this anomaly is traced back to the presence of chaotic jets within the flow. All characteristic transport exponents have a similar value around mu = 1.75, which is also the one found for simple point vortex flows in the literature, indicating some kind of universality. Moreover, the law gamma = mu + 1 linking the trapping-time exponent within jets to the transport exponent is confirmed, and an accumulation toward zero of the spectrum of the finite-time Lyapunov exponent is observed. The localization of a jet is performed, and its structure is analyzed. It is clearly shown that despite a regular coarse-grained picture of the jet, the motion within the jet appears as chaotic, but that chaos is bounded on successive small scales.

The paper describes the complex topological structure of invariant surfaces that appears in a quasi-stationary regime of the tokamak plasma, and it considers in detail anomalous transport of particles along the invariant surfaces (isosurfaces) that have topological genus greater than 1. Such dynamics is pseudochaotic; i.e. it has a zero Lyapunov exponent. Simulations discover such surfaces in confined plasmas under a fairly low ratio of pressure to the magnetic field energy (beta). The isosurfaces correspond to quasi-coherent structures called "streamers" and the streamers are connected by filaments. We study distribution of time of particle separation, Poincaré; recurrences of trajectories, and first time arrival to the system's edge. A model of a multibar-in-square billiard, introduced by Carreras et al. [Chaos 13, 1175 (2003)] is studied with renormalization group method to obtain a distribution of the first time of particles arrival to the edge as a function of the number of bars, which appears to be power-like. The characteristic exponent of this distribution is discussed with respect to its dependence on the number of filaments that connect adjacent streamers.

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.

We study how the chaotic ray motion manifests itself at a finite wavelength at long-range sound propagation in the ocean. The problem is investigated using a model of an underwater acoustic waveguide with a periodic range dependence. It is assumed that the sound propagation is governed by the parabolic equation, similar to the Schrodinger equation. When investigating the sound energy distribution in the time-depth plane, it has been found that the coexistence of chaotic and regular rays can cause a "focusing" of acoustic energy within a small temporal interval. It has been shown that this effect is a manifestation of the so-called stickiness, that is, the presence of such parts of the chaotic trajectory where the latter exhibit an almost regular behavior. Another issue considered in this paper is the range variation of the modal structure of the wave field. In a numerical simulation, it has been shown that the energy distribution over normal modes exhibits surprising periodicity. This occurs even for a mode formed by contributions from predominantly chaotic rays. The phenomenon is interpreted from the viewpoint of mode-medium resonance. For some modes, the following effect has been observed. Although an initially excited mode due to scattering at the inhomogeneity breaks up into a group of modes its amplitude at some range points almost restores the starting value. At these ranges, almost all acoustic energy gathers again in the initial mode and the coarse-grained Wigner function concentrates within a comparatively small area of the phase plane., ((c) 2004 American Institute of Physics)

In dynamical systems with a zero Lyapunov exponent, weak mixing can be governed by a specific topological structure of some surfaces that are invariant with respect to particle dynamics. In particular, when the genus of the invariant surfaces is more than one, they may have weak mixing and the corresponding fractional kinetics. This possibility is demonstrated by using a typical example from plasma physics, a three-dimensional resistive pressure-gradient-driven turbulence model. In a toroidal geometry and with a low-pressure gradient, this model shows the emergence of quasicoherent structures. In this situation, the isosurfaces of the velocity stream function have a web structure with filamentary surfaces emerging from the outer region of the torus and covering the inner region. The filamentary surfaces can result in stochastic jets of particles that cause a "topological instability." In such a situation, particle transport along the surfaces is of the anomalous superdiffusion type., (Copyright 2003 American Institute of Physics.)

A ray-based wave-field description is employed in the interpretation of broadband basin-scale acoustic propagation measurements obtained during the Acoustic Thermometry of Ocean Climate program's 1994 Acoustic Engineering Test. Acoustic observables of interest are wavefront time spread, probability density function (PDF) of intensity, vertical extension of acoustic energy in the reception finale, and the transition region between temporally resolved and unresolved wavefronts. Ray-based numerical simulation results that include both mesoscale and internal-wave-induced sound-speed perturbations are shown to be consistent with measurements of all the aforementioned observables, even though the underlying ray trajectories are predominantly chaotic, that is, exponentially sensitive to initial and environmental conditions. Much of the analysis exploits results that relate to the subject of ray chaos; these results follow from the Hamiltonian structure of the ray equations. Further, it is shown that the collection of the many eigenrays that form one of the resolved arrivals is nonlocal, both spatially and as a function of launch angle, which places severe restrictions on theories that are based on a perturbation expansion about a background ray.

New notions of the complexity function C(epsilon;t,s) and entropy function S(epsilon;t,s) are introduced to describe systems with nonzero or zero Lyapunov exponents or systems that exhibit strong intermittent behavior with "flights," trappings, weak mixing, etc. The important part of the new notions is the first appearance of epsilon-separation of initially close trajectories. The complexity function is similar to the propagator p(t(0),x(0);t,x) with a replacement of x by the natural lengths s of trajectories, and its introduction does not assume of the space-time independence in the process of evolution of the system. A special stress is done on the choice of variables and the replacement t-->eta=ln t, s-->xi=ln s makes it possible to consider time-algebraic and space-algebraic complexity and some mixed cases. It is shown that for typical cases the entropy function S(epsilon;xi,eta) possesses invariants (alpha,beta) that describe the fractal dimensions of the space-time structures of trajectories. The invariants (alpha,beta) can be linked to the transport properties of the system, from one side, and to the Riemann invariants for simple waves, from the other side. This analog provides a new meaning for the transport exponent mu that can be considered as the speed of a Riemann wave in the log-phase space of the log-space-time variables. Some other applications of new notions are considered and numerical examples are presented., ((c) 2003 American Institute of Physics.)

Recent results relating to ray dynamics in ocean acoustics are reviewed. Attention is focused on long-range propagation in deep ocean environments. For this class of problems, the ray equations may be simplified by making use of a one-way formulation in which the range variable appears as the independent (timelike) variable. Topics discussed include integrable and nonintegrable ray systems, action-angle variables, nonlinear resonances and the KAM theorem, ray chaos, Lyapunov exponents, predictability, nondegeneracy violation, ray intensity statistics, semiclassical breakdown, wave chaos, and the connection between ray chaos and mode coupling. The Hamiltonian structure of the ray equations plays an important role in all of these topics.

Quantization of energy balance equations, which describe a separatrixlike motion is presented. The method is based on an exact canonical transformation of the energy-time pair to the action-angle canonical pair, (E,t)-->(I,theta). Quantum mechanical dynamics can be studied in the framework of the new Hamiltonian. This transformation also establishes a relation between a wide class of the energy balance equations and dynamical localization of classical diffusion by quantum interference, that was studied in the field of quantum chaos. An exact solution for a simple system is presented as well.

A model of a quantum dissipative system is considered in the regime when the classical limit corresponds to a chaotic attractor, and the breaking time tau(Planck) of the classical-quantum correspondence is obtained. The model describes a periodically kicked harmonic oscillator (or a particle in a constant magnetic field) with a dissipation. Another analog of this problem is the dissipative kicked Harper model. It is shown that in the limit of the so-called dying attractor, the breaking time tau(Planck) can be arbitrarily large.

The structure of the homoclinic tangle of 11 / 2 degrees of freedom Hamiltonian systems in the neighborhood of the saddle point is invariant under discrete rescaling of the system's parameters. The rescaling constant is derived from the separatrix map and the Melnikov formula. Invariant manifolds for the periodically modulated Duffing oscillator are computed numerically to confirm this property. The scaling is related to the recently found invariance of the separatrix map under a discrete renormalization group. A possibility to extend the scaling invariance to different systems is demonstrated. The equivalency conditions under which two systems have the similarity of their chaotic layer structure near the saddle are derived. A numerical example shows a Duffing oscillator and a pendulum (acted on by different periodic perturbations) with the same structure of the tangle.

We study dynamics of the atom-photon interaction in cavity quantum electrodynamics, considering a cold two-level atom in a single-mode high-finesse standing-wave cavity as a nonlinear Hamiltonian system with three coupled degrees of freedom: translational, internal atomic, and the field. The system proves to have different types of motion including Lévy flights and chaotic walkings of an atom in a cavity. The corresponding equations of motion for expectation values of the atom and field variables have two characteristic time scales: fast Rabi oscillations of the internal atomic and field quantities and slow translational oscillations of the center of the atom mass. It is shown that the translational motion, related to the atom recoils, is governed by an equation of a parametric nonlinear pendulum with a frequency modulated by the Rabi oscillations. This type of dynamics is chaotic with some width of the stochastic layer that is estimated analytically. The width is fairly small for realistic values of the control parameters, the normalized detuning delta and atomic recoil frequency alpha. We consider the Poincaré sections of the dynamics, compute the Lyapunov exponents, and find a range of the detuning, |delta| less, similar 3, where chaos is prominent. It is demonstrated how the atom-photon dynamics with a given value of alpha depends on the values of delta and initial conditions. Two types of Lévy flights, one corresponding to the ballistic motion of the atom and the other corresponding to small oscillations in a potential well, are found. These flights influence statistical properties of the atom-photon interaction such as distribution of Poincaré recurrences and moments of the atom position x. The simulation shows different regimes of motion, from slightly abnormal diffusion with approximately tau(1.13) at delta=1.2 to a superdiffusion with approximately tau(2.2) at delta=1.92 that corresponds to a superballistic motion of the atom with an acceleration. The obtained results can be used to find new ways to manipulate atoms, to cool and trap them by adjusting the detuning delta.

Ray in a waveguide can be considered as a trajectory of the corresponding Hamiltonian system, which appears to be chaotic in a nonuniform environment. From the experimental and practical viewpoints, the ray travel time is an important characteristic that, in some way, involves an information about the waveguide condition. It is shown that the ray travel time as a function of the initial momentum and propagation range in the unperturbed waveguide displays a scaling law. Some properties of the ray travel time predicted by this law still persist in periodically nonuniform waveguides with chaotic ray trajectories. As examples we consider few models with special attention to the underwater acoustic waveguide. It is demonstrated for a deep ocean propagation model that even under conditions of ray chaos the ray travel time is determined, to a considerable extent, by the coordinates of the ray endpoints and the number of turning points, i.e., by a topology of the ray path. We show how the closeness of travel times for rays with equal numbers of turning points reveals itself in ray travel time dependencies on the starting momentum and on the depth of the observation point. It has been shown that the same effect is associated with the appearance of the gap between travel times of chaotic and regular rays. The manifestation of the stickiness (the presence of such parts in a chaotic trajectory where the latter exhibits an almost regular behavior) in ray travel times is discussed. (c) 2002 American Institute of Physics.

Dynamical and statistical properties of the vortex and passive particle advection in chaotic flows generated by 4- and 16-point vortices are investigated. General transport properties of these flows are found to be anomalous and exhibit a superdiffusive behavior with typical second moment exponent mu approximately 1.75. The origin of this anomaly is traced to the presence of coherent structures within the flow, the vortex cores, and the region far from where vortices are located. In the vicinity of these regions stickiness is observed and the motion of tracers is quasiballistic. The chaotic nature of the underlying flow dictates the choice for thorough analysis of transport properties. Passive tracer motion is analyzed by measuring the mutual relative evolution of two nearby tracers. Some tracers travel in each other's vicinity for relatively long times. This is related to a hidden order for the tracers, which we call jets. Jets are localized and found in sticky regions. Their structure is analyzed and found to be formed of a nested set of jets within jets. The analysis of the jet trapping time statistics shows a quantitative agreement with the observed transport exponent.

Dynamical localization of classical superdiffusion for the quantum kicked rotor is studied in the semiclassical limit. Both classical and quantum dynamics of the system become more complicated under the conditions of mixed phase space with accelerator mode islands. Recently, long time quantum flights due to the accelerator mode islands have been found. By exploration of their dynamics, it is shown here that the classical-quantum duality of the flights leads to their localization. The classical mechanism of superdiffusion is due to accelerator mode dynamics, while quantum tunneling suppresses the superdiffusion and leads to localization of the wave function. Coupling of the regular type dynamics inside the accelerator mode island structures to dynamics in the chaotic sea proves increasing the localization length. A numerical procedure and an analytical method are developed to obtain an estimate of the localization length which, as it is shown, has exponentially large scaling with the dimensionless Planck's constant (tilde)h<<1 in the semiclassical limit. Conditions for the validity of the developed method are specified.

Chaotic ray dynamics in deep sea propagation models is considered using the approaches developed in the theory of dynamical chaos. It has been demonstrated that the mechanism of emergence of ray chaos due to overlapping of nonlinear ray-medium resonances should play an important role in long range sound propagation. Analytical estimations, supported by numerical simulations, show that for realistic values of spatial periods and sound speed fluctuation amplitudes associated with internal-wave-induced perturbations, the resonance overlapping causes stochastic instability of ray paths. The influence of the form of the smooth unperturbed sound speed profile on ray sensitivity to the perturbation is studied. Stability analysis has been conducted by constructing the Poincaré maps and examining depth differences of ray trajectories with close take-off angles. The properties of ray travel times, including fractal properties of the time front fine structures, under condition of ray chaos have been investigated. It has been shown that the coexistence of chaotic and regular rays, typical for dynamical chaos, leads to the appearance of gaps in ray travel time distributions, which are absent in unperturbed waveguides. This phenomenon has a prototype in theory of dynamical chaos called the stochastic particle acceleration. It has been shown that mesoscale inhomogeneities with greater spatial scales than that of internal waves, create irregular local waveguide channels in the vicinity of the axis (i.e., sound speed minimum) of the unperturbed waveguide. Near-axial rays propagating at small grazing angles, "jump" irregularly between these microchannels. This mechanism determines chaotic behavior of the near-axial rays.

Kinetic equations used to describe systems with dynamical chaos may contain fractional derivatives of an order alpha in space and beta in time in order to represent processes of stickiness, intermittency, and so on. We demonstrate for a simple example that the kinetics is anisotropic not only in the angular dependence of the diffusion constant, but also in the angular dependence of the exponents alpha and beta. A theory of such kinetic processes has been developed on the basis of integral representation and asymptotic solutions for different cases have been obtained. The results show the existence of self-similar solutions as well as possible logarithmic deviations. (c) 2001 American Institute of Physics.

We consider chaotic properties of a particle in a square billiard with a horizontal bar in the middle. Such a system can model field-line windings of the merged surfaces. The system has weak-mixing properties with zero Lyapunov exponent and entropy, and it can be also interesting as an example of a system with intermediate chaotic properties, between the integrability and strong mixing. We show that the transport is anomalous and that its properties can be linked to the ergodic properties of continued fractions. The distribution of Poincare recurrences, distribution of the displacements, and the moments of the truncated distribution of the displacements are obtained. Connections between different exponents are found. It is shown that the distribution function of displacements and its truncated moments as a function of time exhibit log-periodic oscillations (modulations) with a universal period T(log)=pi(2)/12 ln 2. We note that similar results are valid for a family of billiard, particularly for billiards with square-in-square geometry. (c) 2001 American Institute of Physics.

We show that the breaking time of quantum-classical correspondence depends on the type of kinetics and the dominant origin of stickiness. For sticky dynamics of quantum kicked rotor, when the hierarchical set of islands corresponds to the accelerator mode, we demonstrate by simulation that the breaking time scales as tau(Planck's over 2pi) approximately (1/Planck's over 2pi)(1/mu) with the transport exponent mu>1 that corresponds to superdiffusive dynamics [B. Sundaram and G. M. Zaslavsky, Phys. Rev. E 59, 7231 (1999)]. We discuss also other possibilities for the breaking time scaling and transition to the logarithmic one tau(Planck's over 2pi) approximately ln(1/Planck's over 2pi) with respect to Planck's over 2pi.

Dynamical and statistical properties of tracer advection are studied in a family of flows produced by three point-vortices of different signs. Tracer dynamics is analyzed by numerical construction of Poincaré sections, and is found to be strongly chaotic: advection pattern in the region around the center of vorticity is dominated by a well developed stochastic sea, which grows as the vortex system's initial conditions are set closer to those leading to the collapse of the vortices; at the same time, the islands of regular motion around vortices, known as vortex cores, shrink. An estimation of the core's radii from the minimum distance of vortex approach to each other is obtained. Tracer transport was found to be anomalous: for all of the three numerically investigated cases, the variance of the tracer distribution grows faster than a linear function of time, corresponding to a superdiffusive regime. The transport exponent varies with time decades, implying the presence of multi-fractal transport features. Yet, its value is never too far from 3/2, indicating some kind of universality. Statistics of Poincaré recurrences is non-Poissonian: distributions have long power-law tails. The anomalous properties of tracer statistics are the result of the complex structure of the advection phase space, in particular, of strong stickiness on the boundaries between the regions of chaotic and regular motion. The role of the different phase space structures involved in this phenomenon is analyzed. Based on this analysis, a kinetic description is constructed, which takes into account different time and space scalings by using a fractional equation.

We study transport of tracer particles in a two-dimensional incompressible inviscid flow produced by three point vortices of equal strength. Time dependence of the flow caused by vortex motion gives rise to chaotic tracer trajectories, which fill parts of the flow plane referred to as mixing regions. For general vortex positions, a large connected mixing region (chaotic sea) is formed around vortices. It comprises a number of coherent fluid patches (islands), which do not mix with the rest of the chaotic sea, inside them particle motion is predominantly regular; three near-circular islands surrounding vortices are distinguished by their robust nature. Tracers in the chaotic sea rotate around the center of vorticity in an irregular way. Their trajectories are intermittent, long flights of almost regular motion are caused by trappings in the boundary regions of regular islands. The statistics of tracer rotation exhibits anomalous features, such as faster than linear growth of tracer ensemble variance and asymmetric probability distribution with long power tails. Exponent of the variance growth power law is different for different time ranges. Central part of the tracer distribution and its low (noninteger) moments evolve in a self-similar way, characterized by an exponent, which is different from that of the variance, and contrary to the latter is constant in time. Algebraic tails of the tracer distribution, controlling the behavior of the variance, are responsible for this effect. Long correlations in tracer motion lead to non-Poissonian distribution of Poincare recurrences in the mixing region. Analysis of long recurrences proves, that they are caused by tracer trappings inside boundary layers of islands of regular motion, which always exist inside the mixing region. Statistics of Poincare recurrences and trapping times exhibit power-law decay, indicating absence of a characteristic relaxation time. Values of the decay exponent for recurrences and for escape from the analyzed traps are very close to each other; long correlations are not dominated by a single trap, but are a cumulative effect of all of them, relative importance of a trap is determined by its size, and by its rotation frequency with respect to the background.

Anomalous transport due to Levy-type flights in quantum kicked systems is studied. These systems are kicked rotor and kicked Harper model. It is confirmed for a kicked rotor that there exist special "magic" values of a control parameter of chaos K=K(*)=6.908 745 em leader for which an essential increasing of a localization length is obtained. Functional dependence of the localization length on both parameter of chaos and quasiclassical parameter h is studied. We also observe immense delocalization of the order of 10(9) for a kicked Harper model when a control parameter K is taken to be K(*)=6.349 972. This "magic" value corresponds to special phase space topology in the classical limit, when a hierarchical self-similar set of sticky islands emerges. The origin of the effect is of the general nature and similar immense delocalization as well as increasing of localization length can be found in other systems. (c) 2000 American Institute of Physics.

Hamiltonian chaotic dynamics is not ergodic due to the infinite number of islands imbedded in the stochastic sea. Stickiness of the islands' boundaries makes the wandering process very erratic with multifractal space-time structure. This complication of the chaotic process can be described on the basis of fractional kinetics. Anomalous properties of the chaotic transport become more transparent when there exists a set of islands with a hierarchical structure. Different consequences of the described phenomenon are discussed: a distribution of Poincare recurrences, characteristic exponents of transport, nonuniversality of transport, log periodicity, and chaos erasing. (c) 2000 American Institute of Physics.

A ray-based approach has been considered for evaluation of the coarse-grained Wigner function. From the viewpoint of wave propagation theory this function represents the local spectrum of the wave field smoothed over some spatial and angular scales. A very simple formula has been considered which expresses the smoothed Wigner function through parameters of ray trajectories. Although the formula is ray-based, it nevertheless has no singularities at caustics and its numerical implementation does not require looking for eigenrays. These advantages are especially important under conditions of ray chaos when fast growing numbers of eigenrays and caustics are the important factors spoiling applicability of standard semiclassical approaches already at short ranges. Similar factors restrict applicability of some semiclassical predictions in quantum mechanics at times exceeding the so-called "logarithm break time." Numerical calculations have been carried out for a particular model of range-dependent waveguide where ray trajectories exhibit chaotic motion. These calculations have confirmed our conjecture that by choosing large enough smoothing scales, i.e., by sacrificing small details of the interference pattern, one can substantially enhance the validity region of ray theory. (c) 2000 American Institute of Physics.

We observe immense delocalization of the order of 10(9) for a kicked Harper model when a control parameter K is taken to be K*=6.349 972. This "magic" value corresponds to special phase space topology in the classical limit, when a hierarchical self-similar set of sticky islands emerges. The origin of the effect is of the general nature and similar immense delocalization can be found in other systems.

To explore the character of underlying transport in a sandpile, we have followed the motion of tracer particles. Moments of the distribution function of the particle positions, =D(0)t(nnu(n)), are determined as a function of the elapsed time. The numerical results show that the transport mechanism for distances less than the sandpile length is superdiffusive with an exponent nu(n) close to 0.75, for n<1.

Compressible helical flow with div v not equal to 0 drastically increases the area of chaotic dynamics and mixing properties when the helicity parameter is spatially dependent. We show that the density dependence on the z coordinate can be incorporated in new variables in a way that leads to a Hamiltonian formulation of the system. This permits the application of various important results like the Kolmogorov-Arnold-Moser theory and, particularly, an understanding of why and in which sense the compressible helical flow is "more chaotic" than the incompressible one. Simulation demonstrates this property for an analog of the ABC flow. An interesting type of the dynamical system with "dense" island chains is described.

The dispersion of a wave packet in an acoustic medium is considered in the paraxial wave approximation, where the effective potential, due to variation of the speed of propagation, varies both with depth and propagation distance. The analysis of the resulting parabolic equation, similar to the Schrodinger equation, clearly demonstrates the role of ray chaos in enhancing the dispersion of the initial packet. However, wave coherence effects are also seen that suppress the effects of the ray chaos in a manner analogous to the effects of quantum chaos. (c) 1999 American Institute of Physics.

We present evidence that anomalous transport in the classical standard map results in strong enhancement of fluctuations in the localization length of quasienergy states in the corresponding quantum dynamics. This generic effect occurs even far from the semiclassical limit and reflects the interplay of local and global quantum suppression mechanisms of classically chaotic dynamics. Possible experimental scenarios are also discussed.

Anomalous transport is investigated near threshold in the standard map. Very long time flights, and a large anomaly in the transport, are shown to be associated with a new form of multi-island structures causing orbit sticking. The phase space structure of these traps, and the exponents of the characteristic long time tails associated with them are determined. In general these structures are very complex, but some cases, consisting of layers of islands, allow simple modeling. (c) 1998 American Institute of Physics.

Fractional generalization of the diffusion equation includes fractional derivatives with respect to time and coordinate. It had been introduced to describe anomalous kinetics of simple dynamical systems with chaotic motion. We consider a symmetrized fractional diffusion equation with a source and find different asymptotic solutions applying a method which is similar to the method of separation of variables. The method has a clear physical interpretation presenting the solution in a form of decomposition of the process of fractal Brownian motion and Levy-type process. Fractional generalization of the Kolmogorov-Feller equation is introduced and its solutions are analyzed. (c) 1997 American Institute of Physics.

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.

A detailed description of fractional kinetics is given in connection to islands' topology in the phase space of a system. The method of renormalization group is applied to the fractional kinetic equation in order to obtain characteristic exponents of the fractional space and time derivatives, and an analytic expression for the transport exponents. Numerous simulations for the web-map and standard map demonstrate different results of the theory. Special attention is applied to study the singular zone, a domain near the island boundary with a self-similar hierarchy of subislands. The birth and collapse of islands of different types are considered. (c) 1997 American Institute of Physics.

The problem of the existence of Maxwell's Demon (MD) is formulated for systems with dynamical chaos. Property of stickiness of individual trajectories, anomalous distribution of the Poincare recurrence time, and anomalous (non-Gaussian) transport for a typical system with Hamiltonian chaos results in a possibility to design a situation equivalent to the MD operation. A numerical example demonstrates a possibility to set without expenditure of work a thermodynamically non-equilibrium state between two contacted domains of the phase space lasting for an arbitrarily long time. This result offers a new view of the Hamiltonian chaos and its role in the foundation of statistical mechanics. (c) 1995 American Institute of Physics.

Exponentially small splitting of the separatrix has been calculated for a high frequency large amplitude perturbation and the correspondent correction to the width of the stochastic layer is obtained. The result can be applied to the large amplitude perturbation.

We present a general scheme to describe particle kinetics in the case of incomplete Hamiltonian chaos when a set of islands of stability forms a complicated fractal space-time dynamics and when there is orbit stickiness to the islands' boundary. This kinetics is alternative to the "normal" Fokker-Planck-Kolmogorov equation. A new kinetic equation describes random wandering in the fractal space-time. Critical exponents of the anomalous kinetics are expressed through dynamical characteristics of a Hamiltonian using the renormalization group approach. Renormalization transformation has been applied simultaneously for space and time and fractional calculus has been exploited.