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We review some basic theoretical ideas and revealing experimental observations on stochastic resonance, a phenomenon in which the signal and/or signalto-noise ratio in a nonlinear system increases with increasing intensity of noise. These ideas and results are related to those known from earlier work in condensed matter physics.

The change of the character of the response of a nonlinear system to a low-frequency periodic field induced by external noise is analyzed by means of analog electronic simulation and theoretically. In general, noise of sufficient intensity linearizes the response. For certain parameter ranges, however, an increase in the noise intensity can sometimes have the opposite effect and is shown to de-linearize the response. The physical origins of these contrary behaviors are discussed.

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, tuneable, range of high frequencies in highly underdamped SQUIDs that need not necessarily be bistable.

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.

The effect of fluctuations on the nonlinear response of an underdamped oscillator to an external periodic field at a subharmonic frequency has been investigated theoretically, numerically, and with an analog electronic circuit model. The system studied has often been analyzed in nonlinear optics in the context of two-photon absorption and second-harmonic generation. We consider its nonlinear spectroscopy. Its resonant nonlinear response is described over a broad range of the fluctuation intensities. It is shown that the fluctuation intensity can be used to "tune" the oscillator so as to maximize the nonlinear response. The dependence of the absorption cross section on the fluctuation intensity displays a clearly resolved maximum. If the eigenfrequency of the oscillator is a nonmonotonic function of its energy, the signal at the second harmonic displays a resonant peak at one of two different frequencies, depending on the noise intensity.

A new form of heterodyning in an all-optical bistable system, related to stochastic resonance,is reported. It is showntheoretically and experimentally that the heterodyne signal and signal-to-noiseratio can be enhanced by the addition of noise to the input signal.

Stochastic resonance, in which the signal and/or signal/noise ratio in a nonlinear system can be enhanced by the addition of random fluctuations (noise) of appropriate intensity, is discussed. By revealing the relationship of stochastic resonance to earlier research, and especially to work by Debye in the 1920s. the phenomenon is set in a broad physical context. It is shown that the traditional techniques of statistical physics, for example linear response theory, are applicable to stochastic resonance and their implications for its range of occurrence are discussed.

The response of an overdamped system to quasi-monochromatic (high-frequency narrow band) noise (QMN) has been investigated by means of analogue electronic experiments. Measurements of the stationary and prehistory probability distributions demonstrate that large occasional fluctuations are markedly different in character from those that occur when the system is driven by white or exponentially correlated noise. In particular, we report the fIrst experimental observation of a switching point, a critical value of the coordinate x that separates regions which, with an overwhelming probability, are accessed by the fluctuating system along topologically different paths. The results are discussed in terms of, and shown to be consistent with an existing theoretical description of the QMN-driven system.

The effect of fluctuations in an underdamped nonlinear oscillator on its absorption of an external periodic field at a subharmonic frequency has been investigated theoretically and with an analogue electronic circuit model. It is shown that the absorption is at its most pronounced when the eigenfrequency dispersion has an extremum. In such cases "resonances" with respect to noise intensity occur in two different frequency ranges of the external field, corresponding to two different ranges of noise intensity.

We outline the historical development of stochastic resonance (SR), a phenomenon in which the signal and/or the signal-to-noise ratio in a nonlinear system increases with increasing intensity of noise. We discuss basic theoreticak ideas explaining and describing SR, and we review some revealing experimental data that place SR within the wider context of statistical physics. We emphasize the close relationship of SR to some effects that are well known in condensed-matter physics.

We review stochastic resonance (SR), a counter-intuitive phenomenon in which the signal due to a weak periodic force in a nonlinear system can be enhanced by the addition of external noise. A theoretical approach based on linear response theory (LRn 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. We outline the application of LRT to overdamped motion in a bistable potential, the most commonly studied form of SR. Two new forms of SR, predicted on the basis of LRT and subsequently observed in analogue electronic experiments, are described.

A new form of heterodyning is reported in which a heterodyne signal at the difference frequency between an input signal and a reference signal can be enhanced by adding noise. The underlying mechanism is closely related to that of stochastic resonance. The dependences of the heterodyne signal and the signal-to-noise ratio on the amplitudes and frequencies of the input and reference signals have been investigated. Noise-induced enhancement of the heterodyning has been demonstrated both for white noise and for high-frequency noise (with the power spectrum centered at the frequency of the reference signal) added at the input.

The kinetics of a periodically driven nonlinear oscillator, bistable in a nearly resonant field, has been investigated theoretically and through analog experiments. An activation dependence of the probabilities of fluctuational transitions between the coexisting attractors has been observed, and the activation energies of the transitions have been calculated and measured for a wide range of parameters. The position of the kinetic phase transition (KPT) at which the populations of the attractors are equal has been established. A range of critical phenomena is shown to arise in the vicinity of the KPT including, in particular, the appearance of a supernarrow peak in the spectral density of the fluctuations, and the occurrence of high-frequency stochastic resonance (HFSR). The experimental measurements of the transition probabilities, the KPT line, the multipeaked spectral densities, the strength of the supernarrow peak, and the HFSR effect are shown to be in good agreement with the theoretical predictions.

We analyse a relaxation process from an unstable state during which transient multimodality occurs. This phenomenon is investigated experimentally on an electronic analogue circuit which mimics an overdamped bistable oscillator described by a sixth--order polynomial U(x) and is driven by a Gaussian white noise. The measured times and positions at which new maxima appear in the probability distribution function agree well with the theoretical predictions. It is shown that the shape of the potential guarantees the existence of two different time scales, allowing for the coexistence of three probability distribution peaks during a sizeable interval of time, even though there is no long "flat" region in the potential where U'(x) is very small. Finally, the concept of marginality with reference to unsteady states is discussed.

It is found that the response of a nonlinear dynamical system can be linearised, and its frequency dispersion diminished, by the addition of external noise of sufficient intensity. Taking as an example an overdamped bistable system driven by a low-frequency periodic field, this noise-induced linearisation is investigated through analogue electronic experiments. The wider implications are considered.

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 which is bistable in a nearly resonant field of frequency omegaF. It is shown that, with increasing noise intensity, the signal-to-noise ratio (SNR) for a signal due to a weak trial force of frequency Omega approximately equal to omegaF at first decreases, then increases, and finally decreases again at higher noise intensities: The behavior is 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 |Omega-2omegaF| has also been observed, in accordance with theoretical predictions. Relationships with phenomena in nonlinear optics are discussed.

The response of a bistable system to a low-frequency driving force is demonstrated to display a giant nonlinearity: a sinusoidal force can induce a nearly rectangular signal in the system even when its amplitude is small. The nonlinearity comes about through the force modulating the effective activation energies of fluctuational transitions between the stable states, which in turn can give rise to strong modulation of the transition probabilities and the populations of the states for low noise intensities. Theoretical results are shown to be in good agreement with the results of analog and digital simulations.

The motion of an overdamped particle in a bistable potential U(x), driven quasimonochromatic noise (high-frequency, narrow-band noise), has been investigated by means of analog simulation. The escape rate from one potential well to another was found to be exponentially small compared to the reciprocal mean first-passage time to the top of the potential barrier. The logarithm of the quasistationary probability distribution was observed to fall extremely sharply at a particular value of x, quite close to the equilibrium position. Theory describing the nonanalytic dependence of this logarithm on the bandwidth of the noise is presented and shown to be in good agreement with experiment. Data are also presented for a symmetric monostable potential. In a certain parameter range, the quasistationary distribution is demonstrated to be independent of the form of such a potential.

Zero dispersion peaks (ZDPs), which can arise in the fluctuation spectra of noise-driven underdamped oscillators for which the dependence of the eigenfrequency omega on energy E possesses a maximum or minimum, have been investigated by means of analogue electronic experiments. Two different model systems were studied: a tilted Duffing oscillator (TDO); and a bistable superconducting quantum interference device (SQUID) model. It is demonstrated experimentally for the first time that, for strong enough intensity $T$ of Gaussian pseudo-white noise (equivalent to temperature in a thermal system), the shape of the ZDP becomes universal, independent of the system under investigation. Its evolution with T is also shown to exhibit universal features, being governed by a single parameter provided that $T$ exceeds a critical value Tc, below which the ZDP disappears abruptly. The hierarchy of universalities connected to particular types of extrema in omega (E) is discussed. The results are of relevance to underdamped SQUIDs and, in particular, to the recently discovered phenomenon of zero-dispersion stochastic resonance.

The response of a bistable system to a weak periodic force is investigated using linear response theory (LRT) and by analogue electronic experiment. For quasithermal systems the response, and in particular its increase with increasing noise intensity D, are described by the fluctuation dissipation relations. For small D the low-frequency susceptibility of the system chi(omega) has been found in explicit form allowing for both forced oscillations about the states and periodic modulation of the probabilities of fluctuational transitions between the states. It is shown, both theoretically and experimentally, that a phase lag phi between the force and the response passes through a maximum when D is tuned through the range where stochastic resonance (SR) occurs. A giant nonlinearity of the response is shown to arise for small D and small frequencies of the driving force. It results in the signal induced by a sinusoidal force being nearly rectangular. The range of applicability of LRT is established.

A new form of heterodyning is reported, related to stochastic resonance, in which a heterodyne signal can be enhanced by adding noise.

It is argued, on the basis of linear response theory (LRT), that new types of stochastic resonance (SR) are to be anticipated in diverse systems, quite different from the one most commonly studied to date, which has a static double-well potential and is driven by a net force equal to the sum of periodic and stochastic terms. On this basis, three new nonconventional forms of SR are predicted, sought, found, and investigated both theoretically and by analogue electronic experiment: (a) in monostable systems; (b) in bistable systems with periodically modulated noise; and (c) in a system with coexisting periodic attractors. In each case, it is shown that LRT can provide a good quantitative description of the experimental results for sufficiently weak driving fields. It is concluded that SR is a much more general phenomenon than has hitherto been appreciated.

The susceptibility of an overdamped Markov system fluctuating in a bistable potential of general form is obtained by analytic solution of the Fokker-Planck equation (FPE) for low noise intensities. The results are discussed in the context of the LRT theory of stochastic resonance. They go over into recent results of Hu et al. [Phys. Lett. A 172 (1992) 21] obtained from the FPE for the case of a symmetrical potential, and they coincide with the LRT results of Dykman et al. [Phys. Rev. Lett. 65 (1990) 2606; JETP Lett. 52 (1990) 144; Phys. Rev. Lett. 68 (1992) 2985] obtained for the general case of bistable systems.

The first observations of noise-induced enhancements and phase shifts of a weak periodic signal - characteristic signatures of stochastic resonance (SR) - are reported for a monostabable system. The results arc shown to be in good agreement with a theoretical description based on linear-response theory and the fluctuation dissipation theorem. It is argued that SR is a general phenomenon that can in principle occur for any underdamped nonlinear oscillator.

Fluctuation-induced transitions between coexisting periodic attractors in a periodically driven nonlinear oscillator have been investigated theoretically and by analogue electronic experiment. Calculations and measurements of the corresponding activation energies are in good agreement, and have enabled the position of the kinetic phase transition (KPT) line to be established over its full range.

The response of an all-optical bistable system and of an analog model of the Brownian motion in the symmetric Duffing polential to a weak periodic force in the presence of noise is investigated. The appearance of a stochastic resonance in the both cases is explained in the theory of a linear response.

A new form of stochastic resonance (SR), recently discovered in underdamped nonlinear oscillators for which the dependence of eigenfrequency upon energy has an extremum, is investigated. Its charaneristic features are identified and discussed on the basis of linear response theory and the fluctuation dissipation theorem. In common with conventional SR (in bistable Wlems), sharp increases in the nesponse 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 strongiy resonant.

A prehistory problem is formulated for large occasional fluctuations in noise-driven systems. It has been studied theoretically and experimentally, thereby illuminating the concept of optimal paths and making it possible to visualize and investigate them. The prehistory probability distribution measured for a white-noise-driven system, taken as an example, is shown to be in agreement with the theory.

Fluctuation phenomena, to be anticipated in all systems for which the internal fluctuating variable is a multivalued function of the observable quantity, are reported. Analog electronic experiments on a model optically bistable Fabry-Perot cavity have shown that the observable may be described in terms of a Boltzmann distribution with a multibranch effective potential. The distribution was observed to possess up to four peaks (two twin peaks) within the range of bistability, and to become singular at outer boundaries corresponding to branching points of the potential. The experimental results are compared with theory.

A different form of stochastic resonance, in which the weak periodic force is applied multiplicatively (rather than additively) in the noise, has been investigated for a Brownian particle moving in a double-well potential. A regular periodic signal whose amplitude increases sharply with increasing noise intensity is shown to arise when the potential is asymmetric. The experimental measurements are in good agreement with a theoretical analysis.

The influence of noise on the Lorenz model has been studied in parameter ranges where it is characterized by pairs of nonsymmetrical periodic attractors. The investigations, based on electronic analog and digital techniques, have shown that the noise can give rise to pronounced zero-frequency peaks of complicated shape and to chaos.

The effects of Gaussian quasi-monochromatic (narrow-band) noise have been investigated with an analogue electronic circuit model. The escape probability and the reciprocal mean time to reach the top of a potential barrier were found to differ exponentially strongly. An extremely steep statistical distribution was observed and is discussed. For systems fluctuating in a symmetric single-well potential the distribution was found to be independent of the shape of the potential.