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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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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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  1. Grigoriu, M.: Applied Non-Gaussian Processes: Examples, Theory, Simulation, Linear Random Vibration, and Matlab Solutions. Prentice Hall, Englewood Cliffs (1995)

    MATH  Google Scholar 

  2. Spanos, P.D., Zeldin, B.A.: Monte Carlo treatment of random fields: a broad perspective. Appl. Mech. Rev. 51(3), 219–237 (1998)

    Article  Google Scholar 

  3. Vanmarcke, E.: Random Fields: Analysis and Synthesis (Revised and Expanded New Edition). World Scientific, Singapore (2010)

    Book  MATH  Google Scholar 

  4. Roberts, J.B., Spanos, P.D.: Random Vibration and Statistical Linearization. Courier Corporation, New York (2003)

    MATH  Google Scholar 

  5. Li, J., Chen, J.: Stochastic Dynamics of Structures. Wiley, New York (2009)

    Book  MATH  Google Scholar 

  6. Grigoriu, M.: Stochastic Systems: Uncertainty Quantification and Propagation. Springer, London (2012)

    Book  MATH  Google Scholar 

  7. Wiener, N.: The average of an analytic functional and the Brownian movement. Proc. Natl. Acad. Sci. 7(10), 294–298 (1921)

    Article  MATH  Google Scholar 

  8. Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20(2), 367–387 (1948)

    MathSciNet  Article  MATH  Google Scholar 

  9. Di Matteo, A., Kougioumtzoglou, I.A., Pirrotta, A., Spanos, P.D., Di Paola, M.: Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral. Probab. Eng. Mech. 38, 127–135 (2014)

    Article  Google Scholar 

  10. Petromichelakis, I., Psaros, A.F., Kougioumtzoglou, I.A.: Stochastic response determination and optimization of a class of nonlinear electromechanical energy harvesters: a Wiener path integral approach. Probab. Eng. Mech. 53, 116–125 (2018)

    Article  Google Scholar 

  11. Psaros, A.F., Brudastova, O., Malara, G., Kougioumtzoglou, I.A.: Wiener path integral based response determination of nonlinear systems subject to non-white, non-Gaussian, and non-stationary stochastic excitation. J. Sound Vib. 433, 314–333 (2018)

    Article  Google Scholar 

  12. Psaros, A.F., Kougioumtzoglou, I.A., Petromichelakis, I.: Sparse representations and compressive sampling for enhancing the computational efficiency of the Wiener path integral technique. Mech. Syst. Signal Process. 111, 87–101 (2018)

    Article  Google Scholar 

  13. Meimaris, A.T., Kougioumtzoglou, I.A., Pantelous, A.A.: A closed form approximation and error quantification for the response transition probability density function of a class of stochastic differential equations. Probab. Eng. Mech. 54, 87–94 (2018)

    Article  Google Scholar 

  14. Meimaris, A.T., Kougioumtzoglou, I.A., Pantelous, A.A.: Approximate analytical solutions for a class of nonlinear stochastic differential equations. Eur. J. Appl. Math. 1–17 (2018) (In Press)

  15. Grigoriu, M.: Stochastic Calculus: Applications in Science and Engineering. Springer, New York (2002)

    Book  MATH  Google Scholar 

  16. Gardiner, C.: Stochastic Methods: A Handbook for the Natural and Social Sciences. Springer, New York (2009)

    MATH  Google Scholar 

  17. Risken, H.: The Fokker–Planck Equation: Methods of Solution and Applications. Springer, New York (1996)

    Book  MATH  Google Scholar 

  18. Chaichian, M., Demichev, A.: Path Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics. CRC Press, Bath (2001)

    Book  MATH  Google Scholar 

  19. Kougioumtzoglou, I.A., Spanos, P.D.: An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators. Probab. Eng. Mech. 28, 125–131 (2012)

    Article  Google Scholar 

  20. Naess, A., Moe, V.: Stationary and non-stationary random vibration of oscillators with bilinear hysteresis. Int. J. Non-Linear Mech. 31(5), 553–562 (1996)

    Article  MATH  Google Scholar 

  21. Wehner, M.F., Wolfer, W.G.: Numerical evaluation of path-integral solutions to Fokker–Planck equations. II. Restricted stochastic processes. Phys. Rev. A 28(5), 3003–3011 (1983)

    Article  Google Scholar 

  22. Naess, A., Johnsen, J.M.: Response statistics of nonlinear, compliant offshore structures by the path integral solution method. Probab. Eng. Mech. 8(2), 91–106 (1993)

    Article  Google Scholar 

  23. Alevras, P., Yurchenko, D.: GPU computing for accelerating the numerical Path integration approach. Comput. Struct. 171, 46–53 (2016)

    Article  Google Scholar 

  24. Ewing, G.M.: Calculus of Variations with Applications. Dover Publications, New York (1969)

    MATH  Google Scholar 

  25. Kougioumtzoglou, I.A., Spanos, P.D.: Nonstationary stochastic response determination of nonlinear systems: a Wiener path integral formalism. ASCE J. Eng. Mech. 140(9), 04014064: 1–14 (2014)

    Article  Google Scholar 

  26. Kougioumtzoglou, I.A.: A Wiener path integral solution treatment and effective material properties of a class of one-dimensional stochastic mechanics problems. ASCE J. Eng. Mech. 143(6), 04017014: 1–12 (2017)

    Article  Google Scholar 

  27. Kougioumtzoglou, I.A., Di Matteo, A., Spanos, P.D., Pirrotta, A., Di Paola, M.: An efficient Wiener path integral technique formulation for stochastic response determination of nonlinear MDOF systems. J. Appl. Mech. 82(10), 101005: 1–7 (2015)

    Article  Google Scholar 

  28. Psaros, A.F., Petromichelakis, I., Kougioumtzoglou, I.A.: Wiener path integrals and multi-dimensional global bases for non-stationary stochastic response determination of structural systems. Mech. Syst. Signal Process. (Under Review) (2019)

  29. Steele, J.M.: The Cauchy–Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press, New York (2004)

    Book  MATH  Google Scholar 

  30. Roberts, J.B., Spanos, P.D.: Stochastic averaging: an approximate method of solving random vibration problems. Int. J. Non-Linear Mech. 21(2), 314–333 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  31. Spanos, P.D., Kougioumtzoglou, I.A., dos Santos, K.R.M., Beck, A.T.: Stochastic averaging of nonlinear oscillators: Hilbert transform perspective. ASCE J. Eng. Mech. 144(2), 04017173: 1–9 (2018)

    Article  Google Scholar 

  32. Kougioumtzoglou, I.A., Spanos, P.D.: An approximate approach for nonlinear system response determination under evolutionary stochastic excitation. Curr. Sci. 97(8), 1203–1211 (2009)

    MathSciNet  Google Scholar 

  33. Forsgren, A., Philip, E., Gill, P.E., Wright, M.H.: Interior methods for nonlinear optimization. SIAM Rev. 44(4), 525–597 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  34. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

  35. Macki, J.W., Nistri, P., Zecca, P.: Mathematical models for hysteresis. SIAM Rev. 35(1), 94–123 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  36. Bertotti, G., Mayergoyz, I.D.: The Science of Hysteresis: Mathematical Modeling and Applications, vol. I. Elsevier, New York (2003)

    MATH  Google Scholar 

  37. Ktena, A., Fotiadis, D.I., Spanos, P.D., Massalas, C.V.: A Preisach model identification procedure and simulation of hysteresis in ferromagnets and shape-memory alloys. Phys. B Condens. Matter 306(1–4), 84–90 (2001)

    Article  Google Scholar 

  38. Spanos, P.D., Cacciola, P., Red-Horse, J.: Random vibration of SMA systems via Preisach formalism. Nonlinear Dyn. 36(2–4), 405–419 (2004)

    Article  MATH  Google Scholar 

  39. Mayergoyz, I.D.: Mathematical Models of Hysteresis and Their Applications. Elsevier, New York (2003)

    Google Scholar 

  40. Ni, Y.Q., Ying, Z.G., Ko, J.M.: Random response analysis of Preisach hysteretic systems with symmetric weight distribution. ASME J. Appl. Mech. 69(2), 171–178 (2002)

    Article  MATH  Google Scholar 

  41. Spanos, P.D., Cacciola, P., Muscolino, G.: Stochastic averaging of Preisach hysteretic systems. ASCE J. Eng. Mech. 130(11), 1257–1267 (2004)

    Article  Google Scholar 

  42. Wang, Y., Ying, Z.G., Zhu, W.Q.: Stochastic averaging of energy envelope of Preisach hysteretic systems. J. Sound Vib. 321(3–5), 976–993 (2009)

    Article  Google Scholar 

  43. Kougioumtzoglou, I.A., Spanos, P.D.: Response and first-passage statistics of nonlinear oscillators via a numerical path integral approach. ASCE J. Eng. Mech. 139, 1207–1217 (2013)

    Article  Google Scholar 

  44. Kougioumtzoglou, I.A.: Stochastic joint time-frequency response analysis of nonlinear structural systems. J. Sound Vib. 332, 7153–7173 (2013)

    Article  Google Scholar 

  45. Spanos, P.D., Kougioumtzoglou, I.A.: Survival probability determination of nonlinear oscillators subject to evolutionary stochastic excitation. ASME J. Appl. Mech. 81, 051016: 1–9 (2014)

    Article  Google Scholar 

  46. Di Matteo, A., Spanos, P.D., Pirrotta, A.: Approximate survival probability determination of hysteretic systems with fractional derivative elements. Probab. Eng. Mech. 54, 138–146 (2018)

    Article  Google Scholar 

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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

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

Equation (58) can be further manipulated into

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

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

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

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

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

Next, taking into account that

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

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

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

Further, it can be readily verified that

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

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

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

or, alternatively,

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

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

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

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

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

Next, integrating Eq. (68) leads to

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

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

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

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

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

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

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

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

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

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

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


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

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

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

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).

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