294 research outputs found
Maximal width of the separatrix chaotic layer
The main goal of the paper is to find the {\it absolute maximum} of the width
of the separatrix chaotic layer as function of the frequency of the
time-periodic perturbation of a one-dimensional Hamiltonian system possessing a
separatrix, which is one of the major unsolved problems in the theory of
separatrix chaos. For a given small amplitude of the perturbation, the width is
shown to possess sharp peaks in the range from logarithmically small to
moderate frequencies. These peaks are universal, being the consequence of the
involvement of the nonlinear resonance dynamics into the separatrix chaotic
motion. Developing further the approach introduced in the recent paper by
Soskin et al. ({\it PRE} {\bf 77}, 036221 (2008)), we derive leading-order
asymptotic expressions for the shape of the low-frequency peaks. The maxima of
the peaks, including in particular the {\it absolute maximum} of the width, are
proportional to the perturbation amplitude times either a logarithmically large
factor or a numerical, still typically large, factor, depending on the type of
system. Thus, our theory predicts that the maximal width of the chaotic layer
may be much larger than that predicted by former theories. The theory is
verified in simulations. An application to the facilitation of global chaos
onset is discussed.Comment: 18 pages, 16 figures, submitted to PR
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
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
Tangled nonlinear driven chain reactions of all optical singularities
Dynamics of polarization optical singularities chain reactions in generic
elliptically polarized speckle fields created in photorefractive crystal LiNbO3
was investigated in details Induced speckle field develops in the tens of
minutes scale due to photorefractive 'optical damage effect' induced by
incident beam of He-Ne laser. It was shown that polarization singularities
develop through topological chain reactions of developing speckle fields driven
by photorefractive nonlinearities induced by incident laser beam. All optical
singularities (C points, optical vortices, optical diabolos,) are defined by
instantaneous topological structure of the output wavefront and are tangled by
singular optics lows. Therefore, they have develop in tangled way by six
topological chain reactions driven by nonlinear processes in used nonlinear
medium (photorefractive LiNbO3:Fe in our case): C-points and optical diabolos
for right (left) polarized components domains with orthogonally left (right)
polarized optical vortices underlying them. All elements of chain reactions
consist from loop and chain links when nucleated singularities annihilated
directly or with alien singularities in 1:9 ratio. The topological reason of
statistics was established by low probability of far enough separation of born
singularities pair from existing neighbor singularities during loop
trajectories. Topology of developing speckle field was measured and analyzed by
dynamic stokes polarimetry with few seconds' resolution. The hierarchy of
singularities govern scenario of tangled chain reactions was defined. The
useful space-time data about peculiarities of optical damage evolution were
obtained from existence and parameters of 'islands of stability' in developing
speckle fields.Comment: 11 pages, 12 figure
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
Asymmetric vortex solitons in nonlinear periodic lattices
We reveal the existence of asymmetric vortex solitons in ideally symmetric
periodic lattices, and show how such nonlinear localized structures describing
elementary circular flows can be analyzed systematically using the
energy-balance relations. We present the examples of rhomboid, rectangular, and
triangular vortex solitons on a square lattice, and also describe novel
coherent states where the populations of clockwise and anti-clockwise vortex
modes change periodically due to a nonlinearity-induced momentum exchange
through the lattice. Asymmetric vortex solitons are expected to exist in
different nonlinear lattice systems including optically-induced photonic
lattices, nonlinear photonic crystals, and Bose-Einstein condensates in optical
lattices.Comment: 4 pages, 5 figure
Melting of Discrete Vortices via Quantum Fluctuations
We consider nonlinear boson states with a nontrivial phase structure in the
three-site Bose-Hubbard ring, {\em quantum discrete vortices} (or {\em
q-vortices}), and study their "melting" under the action of quantum
fluctuations. We calculate the spatial correlations in the ground states to
show the superfluid-insulator crossover and analyze the fidelity between the
exact and variational ground states to explore the validity of the classical
analysis. We examine the phase coherence and the effect of quantum fluctuations
on q-vortices and reveal that the breakdown of these coherent structures
through quantum fluctuations accompanies the superfluid-insulator crossover.Comment: Revised version, 4 pages, 5 figures, Accepted for publication in
Physical Review Letter
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