The 2019 work of He, Hu, Zhang and Lu derived, via the small-time-delay approximation and time-delayed Fokker–Planck equations, the output signal-to-noise ratio (SNR) of an asymmetric bistable system with delayed feedback and delayed random force under cross-correlated multiplicative and additive Gaussian white noise, and applied this asymmetric time-delayed bistable stochastic resonance (ATDBSR) method to bearing fault detection. Deriving an SNR theory of this system valid beyond the small-delay/adiabatic regime remains an open problem. The problem asks for a quantitative SR theory of the ATDBSR system valid for general delay, without the small-time-delay/adiabatic restriction; generalizing the noise statistics and turning parameter tuning into theory-guided selection are related directions that such a theory would enable, not completion requirements of this problem.

Background

Stochastic resonance (SR) denotes the counterintuitive cooperation of noise and nonlinearity in which a weak periodic input is amplified in a noisy threshold system because random forcing drives transitions between states that are otherwise too rare; the standard measure of the effect is the output signal-to-noise ratio (SNR), the ratio of the spectral line height at the driving frequency to the background noise floor. In the classic bistable setting, an overdamped particle moves in a symmetric double-well potential U(x)=−ax2/2+bx4/4U(x)=-ax^{2}/2+bx^{4}/4 driven by a weak periodic force and noise, and the SNR is computed from the noise-driven inter-well transition rates in a two-state picture (McNamara and Wiesenfeld, Phys. Rev. A 39, 4854, 1989). Successive generalizations introduced (a) an asymmetry or tilt parameter rr that makes the two wells inequivalent, (b) multiplicative noise, whose intensity multiplies a state-dependent coefficient, alongside additive noise of intensity QQ (multiplicative noise intensity DD), with a cross-correlation strength λ\lambda between the two noise sources, and (c) a time-delayed feedback term of strength ε\varepsilon that evaluates the state at x(t−τ)x(t-\tau), τ\tau being the delay, so that the dynamics become a stochastic delay differential equation. Because a delayed trajectory depends on its own history, the process is non-Markovian and no exact Fokker–Planck equation (the partial differential equation governing the probability density of a Markovian diffusion) is available for nonlinear delayed drift; the standard tool is the small-time-delay approximation (STDA) of Guillouzic, L'Heureux and Longtin (Phys. Rev. E 59, 3970, 1999), which expands the delayed term to derive an approximate time-delayed Fokker–Planck equation, valid only for τ\tau small relative to the relaxation times of the system. Within such approximations, SNR formulas were derived for the symmetric delayed bistable system in the adiabatic limit, i.e. for signal periods long compared with the inter-well dwell times (Shao and Chen, Physica A 388, 977, 2009), for optimal SNR in symmetric stochastic time-delayed bistable systems (Gao, Eur. Phys. J. B 2016), and for an asymmetric delayed bistable system with a mixed periodic signal (Guo, Xu and Wang, Chin. Phys. B 19, 040503, 2010); colored noise combined with delayed feedback was treated for the symmetric bistable case (Gu, Eur. Phys. J. D 2012), and asymmetric bistable systems driven by colored and non-Gaussian noise without any delay were analyzed by Zhou and Xu (Chaos Solitons Fractals 2008) and Liu, Cao, Wang and Hu (Physica A 2019). Closure schemes reaching beyond the small-delay regime exist for other classes of delayed Langevin systems, e.g. the force-linearization closure of Loos and Klapp (Phys. Rev. E 96, 012106, 2017), but have not been developed into an SR/SNR theory for the asymmetric bistable case. The engineering motivation is bearing fault detection: early localized bearing defects imprint weak periodic impulses that are buried in measurement noise, and SR-based methods use noise constructively to enhance these fault signatures; He, Hu, Zhang and Lu (Mod. Phys. Lett. B 33, 1950341, 2019) combined all the above ingredients — asymmetry, delayed feedback including a delay in the random force, and cross-correlated multiplicative and additive Gaussian white noise — derived the output SNR using the STDA and time-delayed Fokker–Planck equations, analyzed its dependence on the full parameter set a,b,r,τ,ε,D,Q,λa,b,r,\tau,\varepsilon,D,Q,\lambda, and validated the resulting ATDBSR method on laboratory bearing tests. The derivation, however, rests on the small-delay/adiabatic approximation and on Gaussian white noise with the correlation cases ∣λ∣<1|\lambda|<1 and λ=±1\lambda=\pm1.

Problem Statement

For the asymmetric time-delayed bistable stochastic system studied by He, Hu, Zhang and Lu (2019) — an overdamped particle in an asymmetric double-well potential (system parameters a,ba,b; asymmetry parameter rr) whose drift contains a time-delayed feedback term of strength ε\varepsilon acting on the delayed state x(t−τ)x(t-\tau), with the delay also entering the random force, driven by a weak periodic signal and by mutually correlated multiplicative and additive Gaussian white noises of intensities DD and QQ with correlation strength λ\lambda, the precise equations being those given in the cited work — establish a quantitative stochastic-resonance theory without the small-time-delay/adiabatic restriction: derive the output signal-to-noise ratio of this system valid for general delay τ\tau and feedback strength ε\varepsilon outside the adiabatic (slow-driving) limit, the result being exact or a controlled approximation whose approximation error is quantified over that same full general-delay domain and which recovers the published small-time-delay, adiabatic SNR formula in the limit where both approximations hold, beyond the small-delay/adiabatic regime in which the published SNR formula was derived. Extensions of such a delay-valid theory from Gaussian white noises with constant correlation cases ∣λ∣<1|\lambda|<1 and λ=±1\lambda=\pm1 to colored, non-Gaussian, or time-correlated noise, and the theory-guided characterization of SNR-optimal parameter selection over the full parameter vector (a,b,r,τ,ε,D,Q,λ)(a,b,r,\tau,\varepsilon,D,Q,\lambda) — the goal motivating the ATDBSR method in bearing fault diagnosis, where parameter tuning is currently heuristic — are related directions that such a theory would enable; they are background and motivation, not completion requirements of this problem.

References

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