Enlargement of a low-dimensional stochastic web

We consider an archetypal example of a low-dimensional stochastic web, arising in a 1D oscillator driven by a plane wave of a frequency equal or close to a multiple of the oscillator’s natural frequency. We show that the web can be greatly enlarged by the introduction of a slow, very weak, modulation of the wave angle. Generalizations are discussed. An application to electron transport in a nanometre-scale semiconductor superlattice in electric and magnetic fields is suggested

The Kramers problem:beyond quasi-stationarity

Noise-induced escape from a metastable potential is considered on time-scales preceding the formation of quasi-equilibrium within the metastable part of the potential. It is shown that the escape flux may then depend exponentially strongly, and in a complicated manner, on time and friction

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

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

Zero-dispersion nonlinear resonance in dissipative systems.

It is shown theoretically and by analog electronic experiment that, in dissipative oscillatory systems for which the frequency of eigenoscillation displays an extremum as a function of energy, the dynamics of nonlinear resonance can differ markedly from the conventional case. Transitions between the conventional and novel types of nonlinear resonance, as parameters vary, correspond to changes in the topology of basins of attraction. With added noise, they can result in drastic changes in fluctuational transition rates between small- and large-amplitude regimes

Kramers problem for a multiwell potential.

Fluctuational escape from a multiwell potential is shown to display new features, as compared to the conventional single-well case. The flux J may depend on friction Gamma exponentially strongly, over an exponentially long period; for small enough temperatures, J(Gamma) undergoes marked oscillations in the range of small Gamma, and the time evolution of J changes drastically as Gamma exceeds a critical value

Comment on "Nonlinear resonance and chaos in the relativistic phase space for driven nonlinear systems".

Kim and Lee (Phys. Rev. E 52, 473; 1995) report relativity-induced resonances in periodically driven oscillators. We comment that zero-dispersion nonlinear resonance (ZDNR) will occur in some of the systems considered, outline the physical origins of the ZDNR, and propose an explanation of a discrepancy noted by Kim and Lee between their theoretical and numerical values of the energy at the stationary stable points of Poincare sections

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-l 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

A new approach to the treatment of separatrix chaos

We review an approach to separatrix chaos that has allowed us to solve some significant problems by: (i) finding analytically the maximum width of the chaotic layer, a problem that lay unsolved for 40 years, and showing that the maximum may be much larger than had previously been assumed; (ii) describing the drastic facilitation of the onset of global chaos between neighboring separatrices, a phenomenon discovered eight years ago

Zero-dispersion stochastic resonance in a model for a superconducting quantum interference device

It is demonstrated that the signal-to-noise ratio for a weak periodic signal in a superconductive loop with a Josephson junction (a superconducting quantum interference device, or SQUID) can be substantially enhanced, over a wide range of frequencies, by the addition of noise. This manifestation of zero-dispersion stochastic resonance (ZDSR) is shown to occur for a wide variety of loop parameters and signal frequencies. Unlike most earlier examples of stochastic resonance, ZDSR does not depend on fluctuational transitions between coexisting stable states. Rather, it exploits the noise-enhanced susceptibility that arises in underdamped nonlinear oscillators for which the oscillation eigenfrequency possesses one or more extrema as a function of energy. The phenomenon is investigated theoretically, and by means of analog and digital simulations. It is suggested that ZDSR could be used to enhance the sensitivity of radio-frequency SQUIDs and other SQUID-based devices. In the course of the work, two additional useful results were obtained: (a) an asymptotic expression describing ZDSR for the general case in the limit of weak dissipation; (b) a method for the numerical calculation of fluctuation spectra in bistable or multistable underdamped systems