Canonical stochastic web map.

In many Hamiltonian systems subjected to a time (or space) -periodic perturbation with a broad spectrum an approach is widely used of replacing the system by periodically delta -kicked dynamical systems, which allows us to reduce them to symplectic mappings. In this paper it is shown that this approach has a fundamental failure, and the corresponding mapping does not correctly describe the continuous original system. It is demonstrated on the example of the stochastic web map obtained by this approach to describe a periodically driven harmonic oscillator which exhibits non-Kolmogorov-Arnold-Moser chaos, particularly, the formation of stochastic webs. A correct canonical map corresponding to this system is obtained using a recently developed method based on the canonical transformation of variables [S. S. Abdullaev, J. Phys. A 35, 2811 (2002)]. Using a direct numerical integration of the system it is shown that the canonical map correctly describes the periodically driven harmonic oscillator with a finite number of spectrum modes.

Asymptotical forms of canonical mappings near separatrix of Hamiltonian systems.

Asymptotical behavior of canonical mappings near the separatrix of Hamiltonian systems subjected to time-periodic perturbations is studied. Based on general forms of these mappings [S. S. Abdullaev, Phys. Rev. E 70, 046202 (2004)] it is shown that the Melnikov-type integrals determining their generating functions can be presented as a sum of regular, R(reg)(h), and oscillatory, R(osc)(h), parts. General asymptotical formulas for R(osc)(h) are derived. The oscillatory parts have zeros at primary resonant values of energy. Conditions are found at which the oscillatory parts, R(osc)(h), can be neglected in the generating functions thus allowing us to obtain simplified mappings depending only the regular parts R(reg)(h). Since the latter are smooth functions of energy h this allows us also to justify the widely used conventional separatrix mapping determined by R(reg)(h) at the separatrix h = 0. A theory is illustrated for a specific example of a Hamiltonian system, a particle dynamics in periodically perturbed double-well potential.

Canonical maps near separatrix in Hamiltonian systems.

A systematic and rigorous method to construct symplectic maps near separatrix of generic Hamiltonian systems subjected to time-periodic perturbations is developed. It is based on the method of canonical transformation of variables to construct Hamiltonian maps [J. Phys. A 35, 2811 (2002)]]. Using canonical transformation of variables and the first-order approximation for the generating function, the general form of mapping in terms of time and energy variables is obtained. Different limiting cases of the mapping are considered. The method is illustrated for simple Hamiltonian systems with one and a large number of saddle points. It is also applied to derive mappings for the periodic-driven Morse oscillator describing the process of stochastic excitation and dissociation of diatomic molecules. The so-called canonical Kepler map is derived for the one-dimensional hydrogen atom in a microwave field.

Ray dynamics of the propagation of sound in nonuniform moving media.

This paper presents a new ray theory for the propagation of sound waves in nonuniformly moving media. It is found that the ray equations in weakly inhomogeneous and slowly moving media are analogous to the equations of motion of charged particles in nonuniform electric and magnetic fields. The adiabatic approximation is used to study the problem of the propagation of sound rays in a model of near-ocean-bottom waveguide with horizontal flow and slowly varying parameters along the direction of propagation of the wave. A general formula is derived that describes the transverse displacement of the trajectory of the ray relative to the direction of propagation of the wave.

Classical chaos and nonlinear dynamics of rays in inhomogeneous media.

An investigation is made of the classical nonlinear resonance and the classical stochastic dynamics of rays in waveguide media with irregular inhomogeneities. Analytic and numerical methods are used to study the characteristics of the ray trajectories, their confinement in a nonlinear resonance, and the development of chaotic behavior in waveguides with longitudinal periodic inhomogeneities. It is established that the localization of the rays has fractal properties; in particular, the cycle length of a ray and the time and velocity of propagation of a signal depend on the initial parameters of the ray in the form of a "devil's staircase." A waveguide with an inhomogeneous index of refraction and a periodically corrugated wall is considered.

Classical chaotic dynamics of sound rays in 3-D inhomogeneous waveguide media.

The chaotic dynamics of sound rays in a near-bottom waveguide channel is studied on the basis of the Hamiltonian dynamics of nonparaxial rays in inhomogeneous moving media. The bottom is assumed to have a two-dimensional roughness. The mapping of the coordinates of the rays upon reflection from the rough bottom is derived through a solution of the corresponding ray equations in an unperturbed waveguide with a horizontal bottom. A numerical analysis of the mapping reveals that a chaotic instability of rays which start out at small angles from the horizontal develops at short distances from the source. Because of this instability, the path segments of a ray along the horizontal coordinates and the signal passage time along a ray are random functions of the angle at which the ray emerges from the source. Upon a further reflection of rays from the rough bottom, there is a diffusion of rays in a stochastic ring which forms in the plane of horizontal ray directions as a result of the overlap and intersection of resonance curves. A qualitative analysis of this effect is carried out. This effect leads to a nearly isotropic distribution of ray directions.

Two-dimensional model of a kicked oscillator: Motion with intermittency.

The phenomenon of intermittency, which arises near a point of degeneracy of an unperturbed Hamiltonian under the influence of a discontinuous perturbation function, is studied in the example of a two-dimensional (2-D) model of a kicked oscillator. This example describes the dynamics of a particle in a cylindrically symmetric potential well subjected to radial kicks which occur periodically in time. The problem is reduced to a Hamiltonian system with N=3/2 degrees of freedom, whose unperturbed Hamiltonian has a degeneracy point. The intermittency is studied numerically and analytically.

Chaotic transmission of waves and "cooling" of signals.

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.

Chaotic transport in hamiltonian systems perturbed by a weak turbulent wave field.

Chaotic transport in a hamiltonian system perturbed by a weak turbulent wave field is studied. It is assumed that a turbulent wave field has a wide spectrum containing up to thousands of modes whose phases are fluctuating in time with a finite correlation time. To integrate the hamiltonian equations a fast symplectic mapping is derived. It has a large time-step equal to one full turn in angle variable. It is found that the chaotic transport across tori caused by the interactions of small-scale resonances have a fractal-like structure with the reduced or zero values of diffusion coefficients near low-order rational tori thereby forming transport barriers there. The density of rational tori is numerically calculated and its properties are investigated. It is shown that the transport barriers are formed in the gaps of the density of rational tori near the low-order rational tori. The dependencies of the depth and width of transport barriers on the wave field spectrum and the correlation time of fluctuating turbulent field (or the Kubo number) are studied. These numerical findings may have importance in understanding the mechanisms of transport barrier formation in fusion plasmas.

Suppression of runaway electrons by resonant magnetic perturbations in TEXTOR disruptions.

The generation of runaway electrons in the international fusion experiment ITER disruptions can lead to severe damage at plasma facing components. Massive gas injection might inhibit the generation process, but the amount of gas needed can affect, e.g., vacuum systems. Alternatively, magnetic perturbations can suppress runaway generation by increasing the loss rate. In TEXTOR disruptions runaway losses were enhanced by the application of resonant magnetic perturbations with toroidal mode number n=1 and n=2. The disruptions are initiated by fast injection of about 3x10{21} argon atoms, which leads to a reliable generation of runaway electrons. At sufficiently high perturbation levels a reduction of the runaway current, a shortening of the current plateau, and the suppression of high energetic runaways are observed. These findings indicate the suppression of the runaway avalanche during disruptions.

Improved confinement due to open ergodic field lines imposed by the dynamic ergodic divertor in TEXTOR.

The ergodization of the magnetic field lines imposed by the dynamic ergodic diverter (DED) in TEXTOR can lead both to confinement improvement and to confinement deterioration. The cases of substantial improvement are in resonant ways related to particular conditions in which magnetic flux tubes starting at the X points of induced islands are connected with the wall. This opening process is connected with a characteristic modification of the heat deposition pattern at the divertor target plate and leads to a substantial increase and steepening of the core plasma density and pressure. The improvement is tentatively attributed to a modification of the electric potential in the plasma carried by the open field lines. The confinement improvement bases on a spontaneous density built up due to the application of the DED and is primarily a particle confinement improvement.

Change of the magnetic-field topology by an ergodic divertor and the effect on the plasma structure and transport.

The magnetic-field perturbation produced by the dynamic ergodic divertor in TEXTOR changes the topology of the magnetic field in the plasma edge, creating an open chaotic system. The perturbation spectrum contains only a few dominant harmonics and therefore it can be described by an analytical model. The modeling is performed in the vacuum approximation without assuming a backreaction of the plasma and does not rely on any experimentally obtained parameters. It is shown that this vacuum approximation predicts in many details the experimentally observed plasma structure. Several experiments have been performed to prove that the plasma edge behavior is defined mostly by the magnetic topology of the perturbed volume. The change in the transport can be explained with the knowledge of only the magnetic structures; i.e., the ergodic pattern dominates the plasma properties.

Toroidal plasma rotation induced by the dynamic ergodic divertor in the TEXTOR tokamak.

The first results of the Dynamic Ergodic Divertor in TEXTOR, when operating in the m/n=3/1 mode configuration, are presented. The deeply penetrating external magnetic field perturbation of this configuration increases the toroidal plasma rotation. Staying below the excitation threshold for the m/n=2/1 tearing mode, this toroidal rotation is always in the direction of the plasma current, even if the toroidal projection of the rotating magnetic field perturbation is in the opposite direction. The observed toroidal rotation direction is consistent with a radial electric field, generated by an enhanced electron transport in the ergodic layers near the resonances of the perturbation. This is an effect different from theoretical predictions, which assume a direct coupling between rotating perturbation and plasma to be the dominant effect of momentum transfer.

Rescaling invariance and anomalous transport in a stochastic layer.

The anomalous chaotic transport in a one-degree-of-freedom Hamiltonian system subjected to a small time-periodic perturbation is investigated. Strong quasiperiodic dependencies of the statistical properties of the motion on log epsilon are found, where epsilon is a perturbation parameter. The period log lambda depends on the rescaling parameter lambda, which is determined only by the frequency of perturbation and behavior of unperturbed Hamiltonian near a saddle point. The results confirm and generalize a recently established new universal rescaling property of perturbed motion near a saddle point.