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 $d$ with random on-site energies $\epsilon_{\vec r}$ and critical long-ranged hoppings decaying typically as $V^{typ}(r) \sim V/r^d$, we show that the strong multifractality regime corresponding to small $V$ can be studied via the standard perturbation theory for eigenvectors in quantum mechanics. The Inverse Participation Ratios $Y_q(L)$, 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 $\tau_{typ}(q)$ and $\tau_{av}(q)$ as a function of $q$. For $q<1/2$, we obtain the non-vanishing limiting spectrum $\tau_{typ}(q)=\tau_{av}(q)=d(2q-1)$ as $V \to 0^+$. For $q>1/2$, this method yields the same disorder-averaged spectrum $\tau_{av}(q)$ of order $O(V)$ 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 $O(V)$, but with a different $q$-dependence $\tau_{typ}(q) \ne \tau_{av}(q)$ for all $q>q_c=1/2$. As a consequence, we find that the corresponding singularity spectra $f_{typ}(\alpha)$ and $f_{av}(\alpha)$ differ even in the positive region $f>0$, and vanish at different values $\alpha_+^{typ} > \alpha_+^{av}$, in contrast to the standard picture. We also obtain that the saddle value $\alpha_{typ}(q)$ of the Legendre transform reaches the termination point $\alpha_+^{typ}$ where $f_{typ}(\alpha_+^{typ})=0$ only in the limit $q \to +\infty$.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 $d=2,3$-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