Stanislav M. Soskin

Zero-dispersion phenomena in oscillatory systems

Phenomena occurring in a particular class of nonlinear oscillatory systems—zero-dispersion systems—are reviewed for cases with and without damping while the system is driven either by random fluctuations (noise), or by a periodic force, or by both together. Zero-dispersion (ZD) systems are those whose frequency of oscillation ω possesses an extremum as a function of energy E. Oscillations at energies close to the extremal energy Em, where the “frequency dispersion” dω/dE is equal to zero, correlate with each other for very long times, to some extent like in a harmonic oscillator. But unlike the latter, the correlation time decreases as the energy shifts away from Em. It is the combination of this local harmonicity, with the fact that a perturbation can cause transitions between strongly and weakly correlated behaviour, that gives rise to the rich manifold of interesting ZD phenomena that are reviewed. A diverse range of physical systems may be expected to exhibit ZD behaviour under particular circumstances. Examples considered in detail include superconducting quantum interference devices, the 2D electron gas in a magnetic superlattice, axial molecules, electrical circuits, particle accelerators, impurities in lattices, relativistic oscillators, and the Harper oscillator. The ZD effects to be anticipated in quantum systems are also discussed. Each section ends with a suggested outlook for future research.

High-frequency stochastic resonance in SQUIDs

Abstract It is shown theoretically and by analogue electronic experiment that stochastic resonance (SR), in which a weak periodic signal can be optimally enhanced by the addition of noise of appropriate intensity, is to be anticipated in underdamped SQUIDs (superconducting quantum interference devices). It manifests under conditions quite unlike those needed for classical SR, which is restricted to low frequencies and confined to systems that are both overdamped and bistable. The zero-dispersion SR reported here can be expected over a vastly wider, tunable, range of high frequencies in highly underdamped SQUIDs that need not necessarily be bistable.

Zero-dispersion stochastic resonance

A new form of stochastic resonance (SR), discovered in underdamped nonlinear oscillators for which the dependence of eigenfrequency upon energy has an extremum, is investigated. Its characteristic features are identified and discussed on the basis of linear response theory and the fluctuation dissipation theorem. In common with conventional SR (in bistable systems), sharp increases in the response to a weak periodic force, and in the signal/noise ratio, occur with increasing intensity of external noise (temperature) within a certain range. Unlike conventional SR, however, the dependence of the response on frequency is strongly resonant.

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.

Observation of zero-dispersion peaks in the fluctuation spectrum of an underdamped single-well oscillator.

It is demonstrated by analogue electronic experiment that the fluctuation spectra of single-well oscillators can sometimes display a twin-peaked structure. The results for the particular system considered confinn and illustrate for the fIrst time the earlier general prediction of zero-dispersion (additional) peaks in the spectra of underdamped systems possessing an extremum in the dependence of their eigenfrequency on energy. The corresponding large, noise-enhanced, susceptibility implies the existence of stochastic resonance in such systems.

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.

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.

Bifurcation Analysis of Zero-Dispersion Nonlinear Resonance

The problem of zero-dispersion nonlinear resonance — a phenomenon that can occur in a periodically-driven nonlinear oscillator whose eigenfrequency as a function of energy possesses an extremum — has been formulated in general for both the dissipative and nondissipative situations. A complete bifurcation analysis and classification of period-1 orbits is presented. The significance of bifurcations for the onset of chaos in the system, and for fluctuations in the presence of external noise, is discussed.

Stochastic webs and quantum transport in superlattices: an introductory review

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.

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.