Two-State Theory of Nonlinear Stochastic Resonance

Chinese Physics, 2005

The phenomenon of stochastic resonance (SR) in a new asymmetric bistable model is investigated. Firstly, a new asymmetric bistable model with an asymmetric term is proposed based on traditional bistable model and the influence of system parameters on the asymmetric bistable potential function is studied. Secondly, the signal-to-noise ratio (SNR) as the index of evaluating the model are researched. Thirdly, Applying the two-state theory and the adiabatic approximation theory, the analytical expressions of SNR is derived for the asymmetric bistable system driven by a periodic signal, unrelated multiplicative and additive Gaussian noise. Finally, the asymmetric bistable stochastic resonance (ABSR) is applied to the bearing fault detection and compared with classical bistable stochastic resonance (CBSR) and classical tri-stable stochastic resonance (CTSR). The numerical computations results show that:(1) the curve of SNR as a function of the additive Gaussian noise and multiplicative Gaussian noise first increased and then decreased with the different influence of the parameters a, b, r and A; This demonstrates that the phenomenon of SR can be induced by system parameters; (2) by parameter compensation method, the ABSR performs better in bearing fault detection than the CBSR and CTSR with merits of higher output SNR, better anti-noise and frequency response capability.

Physical Review E, 2004

Conventional stochastic resonance can be viewed as an amplitude effect, in which a small ͑subthreshold͒ input signal receives assistance from noise to trigger a stronger response from a nonlinear system. We demonstrate another mechanism of improvement by the noise, which is more of a temporal effect. An intrinsically slow system has difficulty to respond to a fast ͑suprathreshold͒ input, and the noise plays a constructive role by spurring the system for a more efficient response. The possibility of this form of stochastic resonance is established and studied here in a double-well bistable dynamic system, driven by a suprathreshold random binary signal, with the noise accelerating the switching between wells.

Physical Review E, 2004

We analyze the phenomenon of nonlinear stochastic resonance (SR) in noisy bistable systems driven by pulsed time periodic forces. The driving force contains, within each period, two pulses of equal constant amplitude and duration but opposite signs. Each pulse starts every half-period and its duration is varied. For subthreshold amplitudes, we study the dependence of the output signal-to-noise ratio (SNR) and the SR gain on the noise strength and the relative duration of the pulses. We find that the SR gains can reach values larger than unity, with maximum values showing a nonmonotonic dependence on the duration of the pulses.

Reviews of Modern Physics, 1998

Physics Letters A, 2006

We carry out a detailed numerical investigation of stochastic resonance in underdamped systems in the non-perturbative regime. We point out that an important distinction between stochastic resonance in overdamped and underdamped systems lies in the lack of dependence of the amplitude of the noise-averaged trajectory on the noise strength, in the latter case. We provide qualitative explanations for the observed behavior and show that signatures such as the initial decay and long-time oscillatory behaviour of the temporal correlation function and peaks in the noise and phase averaged power spectral density, clearly indicate the manifestation of resonant behaviour in noisy, underdamped bistable systems in the weak to moderate noise regime.

Physical Review E, 2004

Two methods of realizing aperiodic stochastic resonance (ASR) by adding noise and tuning system parameters in a bistable system, after a scale transformation, can be compared in a real parameter space. In this space, the resonance point of ASR via adding noise denotes the extremum of a line segment, whereas the method of tuning system parameters presents the extrema of a parameter plane. We demonstrate that, in terms of the system performance, the method of tuning system parameters takes the precedence of the approach of adding noise for an adjustable bistable system. Besides, adding noise can be viewed as a specific case of tuning system parameters. Further research shows that the optimal system found by tuning system parameters may be subthreshold or suprathreshold, and the conventional ASR effects might not occur in some suprathreshold optimal systems.

Chaos Solitons & Fractals, 2002

A thorough evaluation of stochastic resonance with tuning system parameters in bistable systems is presented as a nonlinear signal processor. It is shown that the output signal-to-noise ratio obtained by adjusting systems parameters can exceed that by tuning noise intensity, especially when the input noise intensity is already beyond the resonance region. It is demonstrated that the theory and the method presented here can markedly improve the output signal-to-noise ratio, and minimize phase lag as well as the distortion of the system output signal with multi-frequency. Ó

Journal of Statistical Physics, 1993

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.

Physical Review A, 1991

Stochastic resonance is a cooperative effect of noise and periodic driving in bistable systems. It can be used for the detection and amplification of weak signals embedded within a large noise background. In doing so, the noise triggers the transfer of power to the signal. In this paper we first present general properties of periodically driven Brownian motion, such as the long-time behavior of correlation functions and the existence of a "supersymmetric" partner system. Within the framework of nonstationary stochastic processes, we present a careful numerical study of the stochastic resonance effect, without restrictions on the modulation amplitude and frequency. In particular, in the regime of intermediate driving frequencies which has not yet been covered by theories, we have discovered a secondary resonance at smaller values of the noise strength.

Noise in Complex Systems and Stochastic Dynamics II, 2004

We study an extended system that without noise shows a monostable dynamics, but when submitted to an adequate multiplicative noise, an effective bistable dynamics arises. The stochastic resonance between the attractors of the noise-sustained dynamics is investigated theoretically in terms of a two-state approximation. The knowledge of the exact nonequilibrium potential allows us to obtain the output signal-to-noise ratio. Its maximum is predicted in the symmetric case for which both attractors have the same nonequilibrium potential value.

Circuits and Systems …, 1999

Stochastic resonance (SR), a phenomenon in which a periodic signal in a nonlinear system can be amplified by added noise, is introduced and discussed. Techniques for investigating SR using electronic circuits are described in practical terms. The physical nature of SR, and the explanation of weak-noise SR as a linear response phenomenon, are considered. Conventional SR, for systems characterized by static bistable potentials, is described together with examples of the data obtainable from the circuit models used to test the theory.

Journal of Physics A: Mathematical and General, 1981

It is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance (peak in the power spectrum) which is absent when either the forcing or the perturbation is absent.

IEEE Transactions …, 2002

Physics Letters A, 2008

In an uncoupled parallel array of bistable dynamical elements subject to a common noisy subthreshold rectangular signal, the signal-to-noise ratio (SNR) gain can be improved by tuning the internally added array noise. A SNR gain above unity is observed in certain regions of the array noise intensity. The maximum SNR gain is obtained as the size of the array goes to infinity. This form of stochastic resonance (SR), i.e., array SR, is analytically described by introducing a quasi-stationary probability density model, yielding expressions for the transition probability and the stationary autocorrelation function. The analytical results have interpretative value for the mechanism of array SR, and may be of heuristic interest for applying array SR to related signal processing problems. Additionally, the analytical descriptions of an isolated bistable system agree well with numerical results of both conventional SR for a subthreshold rectangular input and residual SR for a slightly suprathreshold signal.

Physical Review E, 2000

The concept of controlling stochastic resonance has been recently introduced ͓L. Gammaitoni et al., Phys. Rev. Lett. 82, 4574 ͑1999͔͒ to enhance or suppress the spectral response to threshold-crossing events triggered by a time-periodic signal in background noise. Here, we develop a general theoretical framework, based on a rate equation approach. This generic two-state theory captures the essential features observed in our experiments and numerical simulations.