Stochastic resonance: theory and numerics

Enhanced fault diagnosis via stochastic resonance in a piecewise asymmetric bistable system

Weak fault signals are often overwhelmed by strong noise or interference. The key issue in fault diagnosis is to accurately extract useful fault characteristics. Stochastic resonance is an important signal processing method that utilizes noise to enhance weak signals. In this paper, to address the issues of output saturation and imperfect optimization of potential structure models in classical bistable stochastic resonance (CBSR), we propose a piecewise asymmetric stochastic resonance system. A two-state model is used to theoretically derive the output signal-to-noise ratio (SNR) of the bistable system under harmonic excitations, which is compared with the SNR of CBSR to demonstrate the superiority of the method. The method is then applied to fault data. The results indicate that it can achieve a higher output SNR and higher spectral peaks at fault characteristic frequencies/orders, regardless of whether the system operates under fixed or time-varying speed conditions. This study provides new ideas and theoretical guidance for improving the accuracy and reliability of fault diagnosis technology.

A New Piecewise Nonlinear Asymmetry Bistable Stochastic Resonance Model for Weak Fault Extraction

In order to solve output saturation problems found in traditional stochastic resonance methods and to improve the diagnosis ability of weak faults, a new piecewise nonlinear asymmetric bistable stochastic resonance (PNABSR) method is proposed. This model uses a left and right potential function with an asymmetrical shape, which makes it easier to induce stochastic resonance phenomena. Based on the PNABSR model, the expression of the signal-to-noise ratio (SNR) is derived, and the changes in the SNR with different parameters in the PNABSR model are analyzed. Then, the parameters in the PNABSR model are optimized using the adaptive intelligent algorithm to enhance the diagnostic ability. The diagnosis properties of the weak fault are compared between the PNABSR model and the classical bistable stochastic resonance model (CBSR). The experimental results prove that the PNABSR model can effectively extract the weak fault characteristic frequency under a strong noise background, verifying the effectiveness of this method.

Acoustic vibrational resonance in a Rayleigh-Plesset bubble oscillator

The phenomenon of vibrational resonance (VR) has been investigated in a Rayleigh-Plesset oscillator for a gas bubble oscillating in an incompressible liquid while driven by a dual-frequency force consisting of high-frequency, amplitude-modulated, weak, acoustic waves. The complex equation of the Rayleigh-Plesset bubble oscillator model was expressed as the dynamics of a classical particle in a potential well of the Liénard type, thus allowing us to use both numerical and analytic approaches to investigate the occurrence of VR. We provide clear evidence that an acoustically-driven bubble oscillates in a time-dependent single or double-well potential whose properties are determined by the density of the liquid and its surface tension. We show both theoretically and numerically that, besides the VR effect facilitated by the variation of the parameters on which the high-frequency depends, amplitude modulation, the properties of the liquid in which the gas bubble oscillates contribute significantly to the occurrence of VR. In addition, we discuss the observation of multiple resonances and their origin for the double-well case, as well as their connection to the low frequency, weak, acoustic force field.

Quantum vibrational resonance in a dual-frequency-driven Tietz-Hua quantum well

We investigate the response of a quantum particle in the Tietz-Hua quantum potential driven by biharmonic fields: a low-frequency force and a very high frequency force. The response is characterized by the occurrence of a maximum in the first-order transition probability amplitude |s|^{2} under the influence of the applied fields. It is shown that in the absence of the high-frequency component of the applied fields, |s|^{2} shows a distinct sequence of resonances, whereas an increase in the amplitude of the high-frequency field induces minima in |s|^{2}. However, the |s|^{2} maximum occurs in the low-frequency regime where it may be considered otherwise weak in the presence of a single harmonic force.

Vibrational resonance in an inhomogeneous medium with periodic dissipation

The role of nonlinear dissipation in vibrational resonance (VR) is investigated in an inhomogeneous system characterized by a symmetric and spatially periodic potential and subjected to nonuniform state-dependent damping and a biharmonic driving force. The contributions of the parameters of the high-frequency signal to the system's effective dissipation are examined theoretically in comparison to linearly damped systems, for which the parameter of interest is the effective stiffness in the equation of slow vibration. We show that the VR effect can be enhanced by varying the nonlinear dissipation parameters and that it can be induced by a parameter that is shared by the damping inhomogeneity and the system potential. Furthermore, we have apparently identified the origin of the nonlinear-dissipation-enhanced response: We provide evidence of its connection to a Hopf bifurcation, accompanied by monotonic attractor enlargement in the VR regime.

Analysis of vibrational resonance in bi-harmonically driven plasma

The phenomenon of vibrational resonance (VR) is examined and analyzed in a bi-harmonically driven two-fluid plasma model with nonlinear dissipation. An equation for the slow oscillations of the system is analytically derived in terms of the parameters of the fast signal using the method of direct separation of motion. The presence of a high frequency externally applied electric field is found to significantly modify the system's dynamics, and consequently, induce VR. The origin of the VR in the plasma model has been identified, not only from the effective plasma potential but also from the contributions of the effective nonlinear dissipation. Beside several dynamical changes, including multiple symmetry-breaking bifurcations, attractor escapes, and reversed period-doubling bifurcations, numerical simulations also revealed the occurrence of single and double resonances induced by symmetry breaking bifurcations.

Numerical analysis of pulse signal restoration by stochastic resonance in a buckled microcavity

A novel scheme is proposed to restore weak pulse signals immersed in noise by stochastic resonance based on photothermal-effect-induced optical bistability in a buckled dome microcavity. The bistable properties of the dome microcavity are analyzed with different initial detuning wavelengths and effective cavity lengths, and bistable transmission can be obtained for input powers in submilliwatt range. A theoretical model is derived to interpret the nonlinear process of pulse signal recovery through double-well potential theory. The cross-correlation coefficient between output signals and pure input pulses is calculated to quantitatively analyze the influence of noise intensity on stochastic resonance. A cross-correlation gain of 7 is obtained, and the noise-hidden signal can be recovered effectively though the buckled dome microcavity with negligible distortion. The simulation results show the potential of using this structure to restore low-level or noise-hidden pulse signals in all-optical integrated systems.

General approach for dealing with dynamical systems with spatiotemporal periodicities

Dynamical systems often contain oscillatory forces or depend on periodic potentials. Time or space periodicity is reflected in the properties of these systems through a dependence on the parameters of their periodic terms. In this paper we provide a general theoretical framework for dealing with these kinds of systems, regardless of whether they are classical or quantum, stochastic or deterministic, dissipative or nondissipative, linear or nonlinear, etc. In particular, we are able to show that simple symmetry considerations determine, to a large extent, how their properties depend functionally on some of the parameters of the periodic terms. For the sake of illustration, we apply this formalism to find the functional dependence of the expectation value of the momentum of a Bose-Einstein condensate, described by the Gross-Pitaewskii equation, when it is exposed to a sawtooth potential whose amplitude is periodically modulated in time. We show that, by using this formalism, a small set of measurements is enough to obtain the functional form for a wide range of parameters. This can be very helpful when characterizing experimentally the response of systems for which performing measurements is costly or difficult.

Improving the detection sensitivity of chromatography by stochastic resonance

This review aims to provide readers with an overview of various methodologies and approaches used to improve sensitivity through stochastic resonance (SR) methods, with special emphasis on applications to improve the detectability of analytes in chromatographic signals.

The unsaturated bistable stochastic resonance system

We investigated the characteristics of the output saturation of the classical continuous bistable system (saturation bistable system) and its impact on stochastic resonance (SR). We further proposed a piecewise bistable SR system (unsaturated bistable system) and developed the expression of signal-to-noise ratio (SNR) using the adiabatic approximation theory. Compared with the saturation bistable system, the SNR is significantly improved in our unsaturated bistable SR system. The numerical simulation showed that the unsaturated bistable system performed better in extracting weak signals from strong background noise than the saturation bistable system.

Enhancement and weakening of stochastic resonance for a coupled system

In the paper, we investigate the phenomenon of stochastic resonance of a system with finite locally coupled linear elements driven by multiplicative dichotomous noise and temporal periodic signal. It is shown that, for some suitably selected values of the parameters, with increasing the size of the system or the coupling among the nearest elements, the stochastic resonance phenomenon can be enhanced; while for some other suitably selected parameters' values, with the increase of the size or the coupling, the phenomenon of stochastic resonance can be weakened. Our results can provide some useful insights for the investigation of the stochastic resonance phenomenon of the systems with locally (or globally) coupled finite (or infinite) elements.

Classical-quantum crossover in magnetic stochastic resonance in uniaxial superparamagnets

It is shown that the signal-to-noise ratio of the magnetic moment fluctuations in the magnetic stochastic resonance of a quantum uniaxial paramagnet of arbitrary spin value S subjected to a weak probing ac field H(t) = H cos Ωt and a dc bias magnetic field H(0) displays a pronounced dependence on S. The dependence arises from the quantum dynamics of spins which differs markedly from the magnetization dynamics of classical superparamagnets. In the large spin limit, S --> ∞, the quantum solutions reduce to those for a classical uniaxial superparamagnet.

Very large stochastic resonance gains in finite sets of interacting identical subsystems driven by subthreshold rectangular pulses

We study the phenomenon of nonlinear stochastic resonance (SR) in a complex noisy system formed by a finite number of interacting subunits driven by rectangular pulsed time periodic forces. We find that very large SR gains are obtained for subthreshold driving forces with frequencies much larger than the values observed in simpler one-dimensional systems. These effects are explained using simple considerations.

The mechanism for stochastic resonance enhancement of mammalian auditory information processing

Background

In a mammalian auditory system, when intrinsic noise is added to a subthreshold signal, not only can the resulting noisy signal be detected, but also the information carried by the signal can be completely recovered. Such a phenomenon is called stochastic resonance (SR). Current analysis of SR commonly employs the energies of the subthreshold signal and intrinsic noise. However, it is difficult to explain SR when the energy addition of the signal and noise is not enough to lift the subthreshold signal over the threshold. Therefore, information modulation has been hypothesized to play a role in some forms of SR in sensory systems. Information modulation, however, seems an unlikely mechanism for mammalian audition, since it requires significant a priori knowledge of the characteristics of the signal.

Results

We propose that the analysis of SR cannot rely solely on the energies of a subthreshold signal and intrinsic noise or on information modulation. We note that a mammalian auditory system expends energy in the processing of a noisy signal. A part of the expended energy may therefore deposit into the recovered signal, lifting it over threshold. We propose a model that in a rigorous mathematical manner expresses this new theoretical viewpoint on SR in the mammalian auditory system and provide a physiological rationale for the model.

Conclusion

Our result indicates that the mammalian auditory system may be more active than previously described in the literature. As previously recognized, when intrinsic noise is used to generate a noisy signal, the energy carried by the noise is added to the original subthreshold signal. Furthermore, our model predicts that the system itself should deposit additional energy into the recovered signal. The additional energy is used in the processing of the noisy signal to recover the original subthreshold signal.

Quantum stochastic synchronization

We study, within the spin-boson dynamics, the synchronization of a quantum tunneling system with an external, time-periodic driving signal. As a main result, we find that at a sufficiently large system-bath coupling strength (i.e., for a friction strength alpha > 1) the thermal noise plays a constructive role in yielding forced synchronization. This noise-induced synchronization can occur when the driving frequency is larger than the zero-temperature tunneling rate. As an application evidencing the effect, we consider the charge transfer dynamics in molecular complexes.

Stochastic resonance of collective variables in finite sets of interacting identical subsystems

We explore stochastic resonance effects in the response of a complex stochastic system formed by a finite number of interacting, identical subunits driven by a time-periodic force. The driving force alone cannot induce sustained oscillations between the different attractors of the dynamics in the absence of noise. We focus on a global stochastic variable defined as the arithmetic mean of the relevant stochastic variable of each subunit. We construct numerical approximations to its first two long time cumulant moments and its long time correlation function. We also compute the output signal-to-noise ratio and the stochastic resonance gain, for a wide range of parameter values and several types of driving forces. The coupling between the subsystems leads, within adequate ranges of the parameter values, to global outputs with very large signal-to-noise ratios. We have also observed gains larger than unity in the global response to subthreshold sinusoidal driving forces.