197 research outputs found
Separatrix chaos: new approach to the theoretical treatment
We develop a new approach to the theoretical treatment of the separatrix
chaos, using a special analysis of the separatrix map. The approach allows us
to describe boundaries of the separatrix chaotic layer in the Poincar\'{e}
section and transport within the layer. We show that the maximum which the
width of the layer in energy takes as the perturbation frequency varies is much
larger than the perturbation amplitude, in contrast to predictions by earlier
theories suggesting that the maximum width is of the order of the amplitude.
The approach has also allowed us to develop the self-consistent theory of the
earlier discovered (PRL 90, 174101 (2003)) drastic facilitation of the onset of
global chaos between adjacent separatrices. Simulations agree with the theory.Comment: 10 pages, 4 figures, proceedings of the conference "Chaos, Complexity
and Transport" (Marseille, 5-9 June 2007), in pres
New approach to the treatment of separatrix chaos and its application to the global chaos onset between adjacent separatrices
We have developed the {\it general method} for the description of {\it
separatrix chaos}, basing on the analysis of the separatrix map dynamics.
Matching it with the resonant Hamiltonian analysis, we show that, for a given
amplitude of perturbation, the maximum width of the chaotic layer in energy may
be much larger than it was assumed before. We apply the above theory to explain
the drastic facilitation of global chaos onset in time-periodically perturbed
Hamiltonian systems possessing two or more separatrices, previously discovered
(PRL 90, 174101 (2003)). The theory well agrees with simulations. We also
discuss generalizations and applications. Examples of applications of the
facilitation include: the increase of the DC conductivity in spatially periodic
structures, the reduction of activation barriers for noise-induced transitions
and the related acceleration of spatial diffusion, the facilitation of the
stochastic web formation in a wave-driven or kicked oscillator.Comment: 29 pages, 16 figures (figs. are of reduced quality, original files
are available on request from authors), paper has been significantly revised
and resubmitted to PR
Broken space-time symmetries and mechanisms of rectification of ac fields by nonlinear (non)adiabatic response
We consider low-dimensional dynamical systems exposed to a heat bath and to
additional ac fields. The presence of these ac fields may lead to a breaking of
certain spatial or temporal symmetries which in turn cause nonzero averages of
relevant observables. Nonlinear (non)adiabatic response is employed to explain
the effect. We consider a case of a particle in a periodic potential as an
example and discuss the relevant symmetry breakings and the mechanisms of
rectification of the current in such a system.Comment: 11 pages, 10 figure
Resonant ratcheting of a Bose-Einstein condensate
We study the rectification process of interacting quantum particles in a
periodic potential exposed to the action of an external ac driving. The
breaking of spatio-temporal symmetries leads to directed motion already in the
absence of interactions. A hallmark of quantum ratcheting is the appearance of
resonant enhancement of the current (Europhys. Lett. 79 (2007) 10007 and Phys.
Rev. A 75 (2007) 063424). Here we study the fate of these resonances within a
Gross-Pitaevskii equation which describes a mean field interaction between many
particles. We find, that the resonance is i) not destroyed by interactions, ii)
shifting its location with increasing interaction strength. We trace the
Floquet states of the linear equations into the nonlinear domain, and show that
the resonance gives rise to an instability and thus to the appearance of new
nonlinear Floquet states, whose transport properties differ strongly as
compared to the case of noninteracting particles
AC-driven quantum spins: resonant enhancement of transverse DC magnetization
We consider s=1/2 spins in the presence of a constant magnetic field in
z-direction and an AC magnetic field in the x-z plane. A nonzero DC
magnetization component in y direction is a result of broken symmetries. A
pairwise interaction between two spins is shown to resonantly increase the
induced magnetization by one order of magnitude. We discuss the mechanism of
this enhancement, which is due to additional avoided crossings in the level
structure of the system.Comment: 7 pages, 7 figure
Anderson localization transition with long-ranged hoppings : analysis of the strong multifractality regime in terms of weighted Levy sums
For Anderson tight-binding models in dimension with random on-site
energies and critical long-ranged hoppings decaying
typically as , we show that the strong multifractality
regime corresponding to small can be studied via the standard perturbation
theory for eigenvectors in quantum mechanics. The Inverse Participation Ratios
, which are the order parameters of Anderson transitions, can be
written in terms of weighted L\'evy sums of broadly distributed variables (as a
consequence of the presence of on-site random energies in the denominators of
the perturbation theory). We compute at leading order the typical and
disorder-averaged multifractal spectra and as a
function of . For , we obtain the non-vanishing limiting spectrum
as . For , this method
yields the same disorder-averaged spectrum of order as
obtained previously via the Levitov renormalization method by Mirlin and Evers
[Phys. Rev. B 62, 7920 (2000)]. In addition, it allows to compute explicitly
the typical spectrum, also of order , but with a different -dependence
for all . As a consequence, we find
that the corresponding singularity spectra and
differ even in the positive region , and vanish at
different values , in contrast to the standard
picture. We also obtain that the saddle value of the Legendre
transform reaches the termination point where
only in the limit .Comment: 13 pages, 2 figures, v2=final versio
A new approach to the treatment of Separatrix Chaos and its applications
We consider time-periodically perturbed 1D Hamiltonian systems possessing one or more separatrices. If the perturbation is weak, then the separatrix chaos is most developed when the perturbation frequency lies in the logarithmically small or moderate ranges: this corresponds to the involvement of resonance dynamics into the separatrix chaos. We develop a method matching the discrete chaotic dynamics of the separatrix map and the continuous regular dynamics of the resonance Hamiltonian. The method has allowed us to solve the long-standing problem of an accurate description of the maximum of the separatrix chaotic layer width as a function of the perturbation frequency. It has also allowed us to predict and describe
new phenomena including, in particular: (i) a drastic facilitation of the onset of global chaos between neighbouring separatrices, and (ii) a huge increase in the
size of the low-dimensional stochastic web
Matching of separatrix map and resonant dynamics, with application to global chaos onset between separatrices
We have developed a general method for the description of separatrix chaos, based on the analysis of the separatrix map dynamics. Matching it with the resonant Hamiltonian analysis, we show that, for a given amplitude of perturbation, the maximum width of the chaotic layer in energy may be much larger than it was assumed before. We use the above method to explain the drastic facilitation of global chaos onset in time-periodically perturbed Hamiltonian systems possessing two or more separatrices, previously discovered [S. M. Soskin, O. M. Yevtushenko, and R. Mannella, Phys. Rev. Lett. 90, 174101 (2003)]. The theory well agrees with simulations. We also discuss generalizations and applications. The method may be generalized for single-separatrix cases. The facilitation of global chaos onset may be relevant to a variety of systems, e.g., optical lattices, magnetic and semiconductor superlattices, meandering flows in the ocean, and spinning pendulums. Apart from dynamical transport, it may facilitate noise-induced transitions and the stochastic web formation
Vortex and translational currents due to broken time-space symmetries
We consider the classical dynamics of a particle in a -dimensional
space-periodic potential under the influence of time-periodic external fields
with zero mean. We perform a general time-space symmetry analysis and identify
conditions, when the particle will generate a nonzero averaged translational
and vortex currents. We perform computational studies of the equations of
motion and of corresponding Fokker-Planck equations, which confirm the symmetry
predictions. We address the experimentally important issue of current control.
Cold atoms in optical potentials and magnetic traps are among possible
candidates to observe these findings experimentally.Comment: 4 pages, 2 figure
- …