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Mechanical resonators are widely used as precision clocks and sensitive detectors that rely on the stability of their eigenfrequencies. The phase noise is determined by different factors ranging from thermal noise and frequency noise of the resonator to noise in the feedback circuitry. Increasing the vibration amplitude can mitigate some of these effects but the improvements are limited by nonlinearities that are particularly strong for miniaturized micro- and nano-mechanical systems. Here we design a micromechanical resonator with non-monotonic dependence of the frequency of eigenoscillations on energy. Near the extremum, where the dispersion of the eigenfrequency is zero, the system regains certain characteristics of a linear resonator, albeit at large vibration amplitudes. The spectral peak undergoes counter-intuitive narrowing when the noise intensity is increased. With the resonator serving as the frequency determining element in a feedback loop, the phase noise at the extremum amplitude is three times smaller than the conventional nonlinear regime. Zero dispersion phenomena open new opportunities for improving resonant sensors and frequency references.

We address the enhancement of electron drift in semiconductor superlattices of nanometre scale that occurs in combined electric and tilted magnetic fields if Bloch oscillations become resonant with cyclotron rotation in the transverse plane. We uncover the true dynamical mechanism of the phenomenon: the electron dynamics at relevant time-scales is regular or almost regular, contrary to the widespread belief that the enhancement arises through chaotic diffusion between collisions. The theory provides an accurate description of earlier numerical simulations, predicts new remarkable features verified by simulations, and suggests new ways of controlling resonant transport., Comment: 6 pages of the main Letter and 23 pages of the Supplementary Material

Stochastic webs were discovered, first by Arnold for multi-dimensional Hamiltonian systems, and later by Chernikov et al. for the low-dimensional case. Generated by weak perturbations, they consist of thread-like regions of chaotic dynamics in phase space. Their importance is that, in principle, they enable transport from small energies to high energies. In this introductory review, we concentrate on low-dimensional stochastic webs and on their applications to quantum transport in semiconductor superlattices subject to electric and magnetic fields. We also describe a recently-suggested modification of the stochastic web to enhance chaotic transport through it and we discuss its possible applications to superlattices. keywords: stochastic webs; quantum transport; superlattices; separatrix chaos, Comment: Review, to be published in "Contemporary Physics"

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 PRE

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., Comment: 83 pages, review - submitted as a chapter for the book dedicated to George Zaslavsky; several figs. are of reduced quality, original files are available on request from authors

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 press

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 PRE

The most probable transition path in a one-dimensional overdamped system is rigorously proved to possess less than two turning points. The proof is valid for any potentials, transition times, initial and final transition points.

We show for the first time that a {\it weak} perturbation in a Hamiltonian system may lead to an arbitrarily {\it wide} chaotic layer and {\it fast} chaotic transport. This {\it generic} effect occurs in any spatially periodic Hamiltonian system subject to a sufficiently slow ac force. We explain it and develop an explicit theory for the layer width, verified in simulations. Chaotic spatial transport as well as applications to the diffusion of particles on surfaces, threshold devices and others are discussed., Comment: 4 pages including 3 EPS figures, this is an improved version of the paper (accepted to PRL, 2005)

The Chirikov resonance-overlap criterion predicts the onset of global chaos if nonlinear resonances overlap in energy, which is conventionally assumed to require a non-small magnitude of perturbation. We show that, for a time-periodic perturbation, the onset of global chaos may occur at unusually {\it small} magnitudes of perturbation if the unperturbed system possesses more than one separatrix. The relevant scenario is the combination of the overlap in the phase space between resonances of the same order and their overlap in energy with chaotic layers associated with separatrices of the unperturbed system. One of the most important manifestations of this effect is a drastic increase of the energy range involved into the unbounded chaotic transport in spatially periodic system driven by a rather {\it weak} time-periodic force, provided the driving frequency approaches the extremal eigenfrequency or its harmonics. We develop the asymptotic theory and verify it in simulations., Comment: 5 pages, 4 figures, LaTeX, to appear PRL

The main subject of the paper is an escape from a multi-well metastable potential on a time-scale of a formation of the quasi-equilibrium between the wells. The main attention is devoted to such ranges of friction in which an external saddle does not belong to a basin of attraction of an initial attractor. A complete rigorous analysis of the problem for the most probable escape path is presented and a corresponding escape rate is calculated with a logarithmic accuracy. Unlike a conventional rate for a quasi-stationary flux, the rate on shorter time-scales strongly depends on friction, moreover, it may undergo oscillations in the underdamped range and a cutoff in the overdamped range. A generalization of the results for inter-attractor transitions in stable potentials with more than two wells is also presented as well as a splitting procedure for a phenomenological description of inter-attractor transitions is suggested. Applications to such problems as a dynamics of escape on time-scales shorter than an optimal fluctuation duration, prehistory problem, optimal control of fluctuations, fluctuational transport in ratchets, escapes at a periodic driving and transitions in biased Josephson junctions and ionic channels are briefly discussed., Comment: It is scheduled to be published in J.Stat.Phys. v.97, No. 3/4 (November, 1999), 76pp, 11 figures

High frequency stochastic resonance (SR) phenomena, associated with fluctuational transitions between coexisting periodic attractors, have been investigated experimentally in an electronic model of a single-well Duffing oscillator bistable in a nearly resonant field of frequency $\omega_F$. It is shown that, with increasing noise intensity, the signal/noise ratio (SNR) for a signal due to a weak trial force of frequency $\Omega \sim \omega_F$ at first decreases, then {\it increases}, and finally decreases again at higher noise intensities: behaviour similar to that observed previously for conventional (low frequency) SR in systems with static bistable potentials. The stochastic enhancement of the SNR of an additional signal at the mirror-reflected frequency $\vert \Omega - 2 \omega_F \vert$ is also observed, in accordance with theoretical predictions. Relationships with phenomena in nonlinear optics are discussed., Comment: 10 pages, no special macro. Figures on request

Stochastic resonance (SR) - a counter-intuitive phenomenon in which the signal due to a weak periodic force in a nonlinear system can be {\it enhanced} by the addition of external noise - is reviewed. A theoretical approach based on linear response theory (LRT) is described. It is pointed out that, although the LRT theory of SR is by definition restricted to the small signal limit, it possesses substantial advantages in terms of simplicity, generality and predictive power. The application of LRT to overdamped motion in a bistable potential, the most commonly studied form of SR, is outlined. Two new forms of SR, predicted on the basis of LRT and subsequently observed in analogue electronic experiments, are described., Comment: 9 pages, no special macro. Figures request to PVE McClintock