Frequency stabilization and noise-induced spectral narrowing in resonators with zero dispersion.

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 including thermal noise, frequency noise of the resonator and 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 eigenfrequency 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 amplitudes. The spectral peak undergoes narrowing when the noise intensity is increased. With the resonator serving as the frequency-selecting element in a feedback loop, the phase noise at the extremum amplitude is ~3 times smaller than the minimal noise in the conventional nonlinear regime.

Regular rather than chaotic origin of the resonant transport in superlattices.

We address the enhancement of electron transport in semiconductor superlattices that occurs in combined electric and magnetic fields when cyclotron rotation becomes resonant with Bloch oscillations. We show that the phenomenon is regular in origin, contrary to the widespread belief that it arises through chaotic diffusion. The theory verified by simulations provides an accurate description of earlier numerical results and suggests new ways of controlling resonant transport.

Maximal width of the separatrix chaotic layer.

The main goal of the paper is to find the 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 [Phys. Rev. E 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 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.

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.

Divergence of the chaotic layer width and strong acceleration of the spatial chaotic transport in periodic systems driven by an adiabatic ac force.

We show for the first time that a weak perturbation in a Hamiltonian system may lead to an arbitrarily wide chaotic layer and fast chaotic transport. This 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.

Drastic facilitation of the onset of global chaos.

We show that the onset of global chaos in a time periodically perturbed Hamiltonian system may occur at unusually 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. We develop the asymptotic theory and verify it in simulations.

Comment on "monostable array-enhanced stochastic resonance".

Lindner et al. [Phys. Rev. E 63, 051107 (2001)] have reported multiple stochastic resonances (SRs) in an array of underdamped monostable nonlinear oscillators. This is in contrast to the single SR observed earlier in a similar but isolated oscillator. Though the idea that such an effect might occur is intuitively reasonable, the notation and the interpretation of some of the major results seem confusing. These issues are identified and some of them are clarified. In addition, comments are made on two possible extensions of the central idea of Lindner et al.: one of these promises to provide much more striking manifestations of multiple SR in arrays; the other significantly widens the range of systems in which multiple SRs may be observed.

Strong enhancement of noise-induced escape by nonadiabatic periodic driving due to transient chaos.

We have found a mechanism by which a moderately weak nonadiabatic periodic driving may significantly facilitate noise-induced interwell transitions in an underdamped multiwell system. The mechanism is associated with the onset of a homoclinic tangle in the noise-free system: if the ratio of the driving amplitude A to the damping gamma exceeds a critical value approximately 1, then the basins of attraction of the linear responses related to different wells are mixed in a complex manner in some layer associated with the separatrix of the undriven nondissipative system, and the minimal energy in such layer is lower than the top of the barrier. Thus the energy to which the system needs to be activated by the noise, to be able to make a transition, is lower than the top of the barrier.

Short time scales in the Kramers problem: a stepwise growth of the escape flux.

We prove rigorously and demonstrate in simulations that, for a potential system staying initially at the bottom of a well, the escape flux over the barrier grows on times of the order of a period of eigenoscillation in a stepwise manner, provided that friction is small or moderate. If the initial state is not at the bottom of the well, then, typically, some of the steps transform into oscillations. The stepwise/oscillatory evolution at short times appears to be a generic feature of a noise-induced flux.

Noise-induced escape on time scales preceding quasistationarity: New developments in the Kramers problem.

Noise-induced escape from the metastable part of a potential is considered on time scales preceding the formation of quasiequilibrium within that part of the potential. It is shown that, counterintuitively, the escape flux may then depend exponentially strongly, and in a complicated manner, on time and friction. (c) 2001 American Institute of Physics.