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An approximate technique for determining in closed form the response transition probability density function of diverse nonlinear/hysteretic oscillators

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Abstract

An approximate analytical technique is developed for determining, in closed form, the transition probability density function (PDF) of a general class of first-order stochastic differential equations (SDEs) with nonlinearities both in the drift and in the diffusion coefficients. Specifically, first, resorting to the Wiener path integral most probable path approximation and utilizing the Cauchy–Schwarz inequality yields a closed-form expression for the system response PDF, at practically zero computational cost. Next, the accuracy of this approximation is enhanced by proposing a more general PDF form with additional parameters to be determined. This is done by relying on the associated Fokker–Planck operator to formulate and solve an error minimization problem. Besides the mathematical merit of the derived closed-form approximate PDFs, an additional significant advantage of the technique relates to the fact that it can be readily coupled with a stochastic averaging treatment of second-order SDEs governing the dynamics of diverse stochastically excited nonlinear/hysteretic oscillators. In this regard, it is shown that the technique is capable of determining approximately the response amplitude transition PDF of a wide range of nonlinear oscillators, including hysteretic systems following the Preisach versatile modeling. Several numerical examples are considered for demonstrating the reliability and computational efficiency of the technique. Comparisons with pertinent Monte Carlo simulation data are provided as well.

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Acknowledgements

I. A. Kougioumtzoglou gratefully acknowledges the support through his CAREER award by the CMMI Division of the National Science Foundation, USA (Award No. 1748537)

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Appendix A: Derivation of Eq. (14)

Appendix A: Derivation of Eq. (14)

Employing the Wiener path integral approximate solution technique and substituting the associated Lagrangian function of Eq. (8) into the E–L Eq. (11) yields

$$\begin{aligned} \begin{aligned}&\frac{\ddot{x}_\mathrm{c}-\frac{\partial \mu (x_\mathrm{c})}{\partial x_\mathrm{c}}{\dot{x}}_\mathrm{c}}{\sigma (x_\mathrm{c})^2}-2\frac{\left( {\dot{x}}_\mathrm{c}-\mu (x_\mathrm{c})\right) \frac{\partial \sigma (x_\mathrm{c})}{\partial x_\mathrm{c}}{\dot{x}}_\mathrm{c}}{\sigma (x_\mathrm{c})^3} \\&\quad =-\frac{\left( {\dot{x}}_\mathrm{c}-\mu (x_\mathrm{c})\right) \frac{\partial \mu (x_\mathrm{c})}{\partial x_\mathrm{c}}}{\sigma (x_\mathrm{c})^2} -\frac{\left( {\dot{x}}_\mathrm{c}-\mu (x_\mathrm{c})\right) ^2\frac{\partial \sigma (x_\mathrm{c})}{\partial x_\mathrm{c}}}{\sigma (x_\mathrm{c})^3}. \end{aligned} \end{aligned}$$
(58)

Equation (58) can be further manipulated into

$$\begin{aligned} \ddot{x}_\mathrm{c}-\mu (x_\mathrm{c}) \frac{\partial \mu (x_\mathrm{c})}{\partial x_\mathrm{c}}=\frac{\frac{\partial \sigma (x_\mathrm{c})}{\partial x_\mathrm{c}}}{\sigma (x_\mathrm{c})}\left( {{\dot{x}}_\mathrm{c}}^2 - {\mu (x_\mathrm{c})} ^2 \right) , \end{aligned}$$
(59)

in conjunction with the boundary conditions \(x_\mathrm{c}(t_i)=x_i\), \(x_\mathrm{c}(t_\mathrm{f})=x_\mathrm{f}\). Equivalently, Eq. (59) can be cast into the form

$$\begin{aligned} \begin{aligned}&\ddot{x}_\mathrm{c}-\frac{\frac{\partial \sigma (x_\mathrm{c})}{\partial x_\mathrm{c}}}{\sigma (x_\mathrm{c})}{{\dot{x}}_\mathrm{c}}^2=\mu (x_\mathrm{c}) \frac{\partial \mu (x_\mathrm{c})}{\partial x_\mathrm{c}}-\frac{{\mu (x_\mathrm{c})} ^2\frac{\partial \sigma (x_\mathrm{c})}{\partial x_\mathrm{c}}}{\sigma (x_\mathrm{c})}\\&\quad =\frac{\mu (x_\mathrm{c})}{\sigma (x_\mathrm{c})} \left( \sigma (x_\mathrm{c}) \frac{\partial \mu (x_\mathrm{c})}{\partial x_\mathrm{c}} -\mu (x_\mathrm{c}) \frac{\partial \sigma (x_\mathrm{c})}{\partial x_\mathrm{c}}\right) , \end{aligned} \end{aligned}$$
(60)

and multiplying both sides by \(\frac{2{\dot{x}}_\mathrm{c}}{\sigma (x_\mathrm{c})^2}\) yields

$$\begin{aligned} \begin{aligned}&\frac{2{\dot{x}}_\mathrm{c}\ddot{x}_\mathrm{c}}{{\sigma (x_\mathrm{c})}^2}-\frac{2\frac{\partial \sigma (x_\mathrm{c})}{\partial x_\mathrm{c}}}{{\sigma (x_\mathrm{c})}^3}{{\dot{x}}_\mathrm{c}}^3\\&\quad =2\frac{\mu (x_\mathrm{c})}{\sigma (x_\mathrm{c})} \left( \frac{ \sigma (x_\mathrm{c}) \frac{\partial \mu (x_\mathrm{c})}{\partial x_\mathrm{c}} -\mu (x_\mathrm{c}) \frac{\partial \sigma (x_\mathrm{c})}{\partial x_\mathrm{c}}}{{\sigma (x_\mathrm{c})}^2}\right) {\dot{x}}_\mathrm{c}. \end{aligned} \end{aligned}$$
(61)

Next, taking into account that

$$\begin{aligned} \begin{aligned}&\frac{\partial }{\partial x_\mathrm{c}}\left( \left( \frac{\mu (x_\mathrm{c})}{\sigma (x_\mathrm{c})} \right) ^2 \right) \\&\quad =2\frac{\mu (x_\mathrm{c})}{\sigma (x_\mathrm{c})} \left( \frac{ \sigma (x_\mathrm{c}) \frac{\partial \mu (x_\mathrm{c})}{\partial x_\mathrm{c}} -\mu (x_\mathrm{c}) \frac{\partial \sigma (x_\mathrm{c})}{\partial x_\mathrm{c}}}{{\sigma (x_\mathrm{c})}^2}\right) , \end{aligned} \end{aligned}$$
(62)

in conjunction with the chain rule of differentiation, i.e., \(\frac{\mathrm{d}}{\mathrm{d}t}\left( \left( \frac{\mu (x_\mathrm{c})}{\sigma (x_\mathrm{c})} \right) ^2 \right) =\frac{\partial }{\partial x_\mathrm{c}}\left( \left( \frac{\mu (x_\mathrm{c})}{\sigma (x_\mathrm{c})} \right) ^2 \right) {\dot{x}}_\mathrm{c}\), Eq. (61) becomes

$$\begin{aligned} \frac{2{\dot{x}}_\mathrm{c}\ddot{x}_\mathrm{c}}{{\sigma (x_\mathrm{c})}^2}-\frac{2\frac{\partial \sigma (x_\mathrm{c})}{\partial x_\mathrm{c}}}{{\sigma (x_\mathrm{c})}^3}{{\dot{x}}_\mathrm{c}}^3=\frac{\mathrm{d} }{\mathrm{d}t}\left( \left( \frac{\mu (x_\mathrm{c})}{\sigma (x_\mathrm{c})} \right) ^2 \right) . \end{aligned}$$
(63)

Further, it can be readily verified that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{{{\dot{x}}_\mathrm{c}}^2}{{\sigma (x_\mathrm{c})}^2} \right) =\frac{2{\dot{x}}_\mathrm{c}\ddot{x}_\mathrm{c}}{{\sigma (x_\mathrm{c})}^2}-\frac{2\frac{\partial \sigma (x_\mathrm{c})}{\partial x_\mathrm{c}}}{{\sigma (x_\mathrm{c})}^3}{{\dot{x}}_\mathrm{c}}^3. \end{aligned}$$
(64)

Utilizing Eq. (64), Eq. (63) becomes

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{{{\dot{x}}_\mathrm{c}}^2}{{\sigma (x_\mathrm{c})}^2} \right) =\frac{\mathrm{d}}{\mathrm{d}t}\left( \left( \frac{\mu (x_\mathrm{c})}{\sigma (x_\mathrm{c})} \right) ^2 \right) , \end{aligned}$$
(65)

or, alternatively,

$$\begin{aligned} \frac{{{\dot{x}}_\mathrm{c}}^2}{{\sigma (x_\mathrm{c})}^2}=\left( \frac{\mu (x_\mathrm{c})}{\sigma (x_\mathrm{c})} \right) ^2+b, \end{aligned}$$
(66)

where b is a constant, dependent on the boundary conditions, i.e., \(x_\mathrm{c}(t_i)=x_i\), \(x_\mathrm{c}(t_\mathrm{f})=x_\mathrm{f}\). Considering next Eq. (8), and expanding, leads to

$$\begin{aligned} L(x_\mathrm{c},{\dot{x}}_\mathrm{c})=\frac{1}{2}\left( \frac{{{\dot{x}}_\mathrm{c}}^{2}-2{\dot{x}}_\mathrm{c}\mu (x_\mathrm{c}) +{\mu (x_\mathrm{c})}^2}{{\sigma (x_\mathrm{c})}^2} \right) , \end{aligned}$$
(67)

whereas substituting Eq. (66) into Eq. (67) yields

$$\begin{aligned} L(x_\mathrm{c},{\dot{x}}_\mathrm{c})=\frac{1}{2}\left( \frac{2{{\dot{x}}_\mathrm{c}}^{2}-2{\dot{x}}_\mathrm{c}\mu (x_\mathrm{c})}{{\sigma (x_\mathrm{c})}^2} -b \right) . \end{aligned}$$
(68)

Next, integrating Eq. (68) leads to

$$\begin{aligned} \begin{aligned}&\int _{t_i}^{t_\mathrm{f}}L(x_\mathrm{c},{\dot{x}}_\mathrm{c})\mathrm{d}t\\&\qquad =\frac{1}{2}\left( 2\int _{t_i}^{t_\mathrm{f}}\frac{{{\dot{x}}_\mathrm{c}}^2}{{\sigma (x_\mathrm{c})}^2}\mathrm{d}t -\int _{t_i}^{t_\mathrm{f}}\frac{2{\dot{x}}_\mathrm{c}\mu (x_\mathrm{c})}{{\sigma (x_\mathrm{c})}^2}\mathrm{d}t\right. \\&\qquad \left. -\,b\left( t_\mathrm{f}-t_i\right) \right) . \end{aligned} \end{aligned}$$
(69)

Furthermore, for arbitrary functions \(f(\cdot )\), \(g(\cdot )\), the Cauchy–Schwarz inequality (e.g., [29]) states that

$$\begin{aligned} \left( {\int _a^b f(t) g(t) \mathrm{d}t}\right) ^2 \le \int _a^b f(t)^2 \mathrm{d}t \int _a^b g(t)^2 \mathrm{d}t. \end{aligned}$$
(70)

Clearly, setting \(f\equiv 1\) yields the special case

$$\begin{aligned} \int _a^b g(t)^2 \mathrm{d}t \ge \frac{1}{b-a}\left( {\int _a^b g(t) \mathrm{d}t}\right) ^2. \end{aligned}$$
(71)

Next, denoting by \({\mathcal {M}}(\cdot )\) an antiderivative of \(\frac{2\mu (\cdot )}{{\sigma (\cdot )}^2}\) and by \({\mathcal {R}}(\cdot )\) an antiderivative of \(\frac{1}{{\sigma (\cdot )}}\), and applying Eq. (71) to the term \(2\int _{t_i}^{t_\mathrm{f}}\frac{{{\dot{x}}_\mathrm{c}}^2}{{\sigma (x_\mathrm{c})}^2}\mathrm{d}t\) in Eq. (69) yields

$$\begin{aligned} \begin{aligned} 2\int _{t_i}^{t_\mathrm{f}}\frac{{{\dot{x}}_\mathrm{c}}^2}{{\sigma (x_\mathrm{c})}^2}\mathrm{d}t&\ge \int _{t_i}^{t_\mathrm{f}}\frac{{{\dot{x}}_\mathrm{c}}^2}{{\sigma (x_\mathrm{c})}^2}\mathrm{d}t \\&\ge \frac{\left( \int _{t_i}^{t_\mathrm{f}}\frac{{\dot{x}}_\mathrm{c}}{\sigma (x_\mathrm{c})}\mathrm{d}t \right) ^2}{t_\mathrm{f}-t_i}\\&=\frac{\left( {\mathcal {R}}(x_\mathrm{f}) - {\mathcal {R}}(x_i) \right) ^2}{t_\mathrm{f}-t_i}. \end{aligned} \end{aligned}$$
(72)

Considering Eq. (72), Eq. (69) becomes

$$\begin{aligned} \begin{aligned}&\int _{t_i}^{t_\mathrm{f}}L(x_\mathrm{c},{\dot{x}}_\mathrm{c})\mathrm{d}t\ge -\frac{b\left( t_\mathrm{f}-t_i\right) }{2} \\&\quad +\frac{1}{2}\left( \frac{\left( {\mathcal {R}}(x_\mathrm{f}) - {\mathcal {R}}(x_i) \right) ^2}{t_\mathrm{f}-t_i} -\left( {\mathcal {M}}(x_\mathrm{f}) - {\mathcal {M}}(x_i)\right) \right) . \end{aligned} \end{aligned}$$
(73)

Thus, taking into account Eqs. (12) and (73) an approximation for the response transition PDF of Eq. (13) is given by

$$\begin{aligned} {\hat{p}}\left( x_\mathrm{f},t_\mathrm{f}|x_i,t_i \right) = {\mathcal {N}}\left( t_\mathrm{f}|x_i,t_i \right) \exp \left( - G \left( x_\mathrm{f},t_\mathrm{f}|x_i,t_i \right) \right) , \end{aligned}$$
(74)

where

$$\begin{aligned} \begin{aligned}&G \left( x_\mathrm{f},t_\mathrm{f}|x_i,t_i \right) \\&\quad = \frac{1}{2}\left( \frac{\left( {\mathcal {R}}(x_\mathrm{f}) - {\mathcal {R}}(x_i) \right) ^2}{t_\mathrm{f}-t_i} -\left( {\mathcal {M}}(x_\mathrm{f}) - {\mathcal {M}}(x_i)\right) \right) , \end{aligned} \end{aligned}$$
(75)

and \({\mathcal {N}}\) in Eq. (74) serves as the normalization constant, which is determined as

$$\begin{aligned} {\mathcal {N}}\left( t_\mathrm{f}|x_i,t_i \right) =\left( \int _{{\mathcal {D}}}\exp \left( - G \left( z,t_\mathrm{f}|x_i,t_i \right) \right) \mathrm{d}z\right) ^{-1}, \end{aligned}$$
(76)

where \({\mathcal {D}}\) denotes the domain of integration, accounting for any restrictions that \({\mathcal {M}}(\cdot )\) and \({\mathcal {R}}(\cdot )\) may impose.

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Meimaris, A.T., Kougioumtzoglou, I.A., Pantelous, A.A. et al. An approximate technique for determining in closed form the response transition probability density function of diverse nonlinear/hysteretic oscillators. Nonlinear Dyn 97, 2627–2641 (2019). https://doi.org/10.1007/s11071-019-05152-w

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Keywords

  • Nonlinear stochastic dynamics
  • Path integral
  • Cauchy–Schwarz inequality
  • Fokker–Planck equation
  • Stochastic differential equations