Stochastic resonance in an asymmetric tri-stable system driven by correlated noises and periodic signal

Abstract

This paper proposes an Asymmetric Tri-stable Stochastic Resonance (ATSSR) system that is driven by a periodic signal and a combination of correlated non-Gaussian noise and Gaussian white noise. The authors obtain the Markov process using the unified color noise approximation method and derive analytical expressions for the steady-state probability density, the Mean First-Pass Time, and the spectral amplification under the adiabatic approximation limit. Afterwards, the effects of various system parameters on them are analyzed, and the results show that both non-Gaussian noise and Gaussian white noise can induce stochastic resonance, with stronger resonance occurring when the two types of noise are correlated. Then, a periodic attenuated pulse signal and a harmonic vibration signal are constructed, which are applied in simulated experiments to detect fault signals using the ATSSR system. The experimental results demonstrate the outstanding performances in detecting fault signals and confirm its the feasibility for this purpose.

Appendix A

R is the transition matrix, which is:

$$ R = \left( {\begin{array}{*{20}c} { - r_{1,2} } & {r_{2,1} } & 0 \\ {r_{1,2} } & { - \left( {r_{2,1} + r_{2,3} } \right)} & {r_{3,2} } \\ 0 & {r_{2,3} } & { - r_{3,2} } \\ \end{array} } \right) $$

(A.1)

In the above formula, r_i,j represents the transition probability of Brownian particles from s_i to s_j. The specific expression of r_i,j is:

$$ \begin{aligned} r_{{m,m + 1}} & = \frac{{\sqrt {\left| {U^{{\prime \prime }} \left( {s_{m} } \right)U^{{\prime \prime }} \left( {u_{m} } \right)} \right|} }}{{2\pi }}\exp \left[ { - \frac{{\tilde{U}\left( {u_{m} ,t} \right) - \tilde{U}\left( {s_{m} ,t} \right)}}{{D_{{{\text{eff}}}} }}} \right], \\ m & = 1,2 \\ \end{aligned} $$

(A.2a)

$$ \begin{aligned} r_{{n,n - 1}} & = \frac{{\sqrt {\left| {U^{{\prime \prime }} \left( {s_{n} } \right)U^{{\prime \prime }} \left( {u_{{n - 1}} } \right)} \right|} }}{{2\pi }}\exp \left[ { - \frac{{\tilde{U}\left( {u_{{n - 1}} ,t} \right) - \tilde{U}\left( {s_{n} ,t} \right)}}{{D_{{{\text{eff}}}} }}} \right], \\ n & = 2,3 \\ \end{aligned} $$

(A.2b)

From Eq. (13), by expanding Eq. (A.2a) and Eq. (A.2b) to the first-order term of $\cos \left( {2\pi f_{0} t} \right)$, it can be obtained:

$$ r_{m,m + 1} = r_{{_{m,m + 1} }}^{\left( 0 \right)} + A_{0} D_{{{\text{eff}}}}^{ - 1} r_{{_{m,m + 1} }}^{\left( 0 \right)} \Delta g_{m,m + 1} \cos \left( {2\pi f_{0} t} \right) $$

(A.3a)

$$ r_{n,n - 1} = r_{{_{n,n - 1} }}^{\left( 0 \right)} + A_{0} D_{{{\text{eff}}}}^{ - 1} r_{{_{n,n - 11} }}^{\left( 0 \right)} \Delta g_{n,n - 1} \cos \left( {2\pi f_{0} t} \right) $$

(A.3b)

With

$$ r_{{_{m,m + 1} }}^{\left( 0 \right)} = \frac{{\sqrt {\left| {U^{\prime \prime } \left( {s_{m} } \right)U^{\prime \prime } \left( {u_{m} } \right)} \right|} }}{2\pi }\exp \left[ { - \frac{{U_{0} \left( {u_{m} } \right) - U_{0} \left( {s_{m} } \right)}}{{D_{{{\text{eff}}}} }}} \right] $$

(A.4a)

$$ r_{{_{n,n - 1} }}^{\left( 0 \right)} = \frac{{\sqrt {\left| {U^{\prime \prime } \left( {s_{n} } \right)U^{\prime \prime } \left( {u_{n - 1} } \right)} \right|} }}{2\pi }\exp \left[ { - \frac{{U_{0} \left( {u_{n - 1} } \right) - U_{0} \left( {s_{n} } \right)}}{{D_{{{\text{eff}}}} }}} \right] $$

(A.4b)

$$ \Delta g_{m,m + 1} = g\left( {u_{m} ,s_{m} } \right),\Delta g_{n,n - 1} = g\left( {u_{n - 1} ,s_{n} } \right) $$

(A.4c)