Abstract
We investigate a Duffing oscillator driven by random noise which is assumed to be a harmonic function of the Wiener process. We show that the correlation time of the noise has a strong effect on the form of the response stationary probability density functions. It represents the so-called reentrance transitions, i.e. for the same noise intensity the probability density function has an identical modality for both the small and the large correlation time but a different modality for the moderate correlation time. The transitions are observed for both the single-well and twin-well potential case. A new approach is used to study the response probability density function. It is based on analysis of hyperbolic systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)
Moon, F.: Chaotic Vibrations. Wiley, New York (1992)
Lin, Y.K., Cai, G.K.: Probabilistic Structural Dynamics: Advanced Theory and Applications. McGraw-Hill, New York (1995)
Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences. Springer-Verlag, Berlin (1985)
Dimentberg, M., Cai, G.Q., Lin, Y.K.: Application of quasi-conservative averaging to a non-linear system under non-white excitation. Int. J. Non-Linear Mech. 30, 677–685 (1995)
Di Paola, M., Ricciardi, G., Vasta, M.: A method for probabilistic analysis of nonlinear systems. Probabilistic Eng. Mech. 10, 1–10 (1995)
Wojtkiewicz, S.F., Spencer, B.F., Bergman, L.A.: On the cumulant-neglect closure method in stochastic dynamics. Int. J. Non-Linear Mech. 31, 657–684 (1996)
Sobczyk, K., Trebicki, J.: Approximate probability distributions for stochastic systems: maximum entropy method. Comput. Methods Appl. Mech. Eng. 168, 91–111 (1998)
Bontempi, F., Faravelli, L.: Lagranganian/Eulerian description of the dynamical systems. ASCE J. Eng. Mech. 124, 901–911 (1998)
Landa, P.S., McClintock, P.V.E.: Changes in the dynamical behavior of nonlinear systems induced by noise. Phys. Rep. 323, 1–80 (2000)
von Wagner, U., Wedig, W.V.: On the calculation of stationary solutions of multi-dimensional Fokker-Planck equations by orthogonal functions. Nonlinear Dyn. 21, 289–306 (2000)
Naess, A., Moe, V.: Efficient path integration methods for nonlinear dynamic systems. Probabilistic Eng. Mech. 15, 221–231 (2000)
Pradlwarter, H.J.: Non-linear stochastic response distributions by local stochastic linearization. Int. J. Non-Linear Mech. 36, 1135–1151 (2001)
Haiwu, R., Wu, X., Guang, F., Tong, F.: Response of a Duffing oscillator to combined deterministic harmonic and random excitations. J. Sound Vib. 242, 362–368 (2001)
Cai, G.K., Lin, Y.K.: Random vibration of strongly nonlinear systems. Nonlinear Dyn. 24, 3–15 (2001)
Crandall, S.H.: On using non-Gaussian distributions to perform statistical linearization. Int. J. Non-Linear Mech. 39, 1395–1406 (2004)
Caughey, T.K.: Nonlinear theory of random vibration. In: Yin, C.S. (ed.) Advances in Applied Mechanics, Vol. 11, pp. 209–253. Academic Press, New York (1971)
Castro, F., Sanchez, A.D., Wio, H.S.: Reentrance phenomena in noise induced transitions. Phys. Rev. Lett. 75, 1691–1694 (1995)
Wio, H.S., Toral, R.: Effect of non-Gaussian noise sources in a noise-induced transition. Physica D 193, 161–168 (2004)
Dimentberg, M.F.: Stability and subcritical dynamics of structures with spatially disordered travelling parametric excitation. Probabilistic Eng. Mech. 7, 131–134 (1992)
Wedig, W.V.: Iterative schemes for stability problems with non-singular Fokker-Planck equations. Int. J. Non-Linear Mech. 31, 707–715 (1996)
Pandey, M.D., Ariaratnam, S.T.: Stability analysis of wind-induced torsional motion of slender bridges. Struct. Safety 20, 379–389 (1998)
Haiwu, R., Wei, X., Xiangdon, W., Guang, M., Tong, F.: Response statistics of two-degree-of freedom non-linear system to narrow-band random excitation. Int. J. Non-Linear Mech. 37, 1017–1028 (2002)
Huang, Z.L., Zhu, W.Q.: Stochastic averaging of quasi-integrable Hamiltonian systems under bounded noise excitations. Probabilistic Eng. Mech. 19, 219–228 (2004)
van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1981)
Cameron, R.H., Martin, W.T.: Transformations of Wiener integrals under translations. Annals Math. 45, 386–396 (1944)
Yeh, J.: Stochastic Processes and the Wiener Integral. M. Dekker, New York (1973)
Bobryk, R.V., Stettner, L.: A closure method for randomly perturbed linear systems. Demonstratio Mathematica 34, 415–424 (2001)
Bobryk, R.V., Stettner, L.: Moment stability for linear systems with a random parametric excitation. Syst. Control Lett. 54, 781–786 (2005)
Bobryk, R.V., Chrzeszczyk, A.: Transitions induced by bounded noise. Physica A 358, 263–272 (2005)
Vishik, M.I., Lyusternik, L.A.: Regular degeneration and boundary layer for linear differential equations with small parameter. Uspekhi Matematicheskikh Nauk 12, 3–122 (1957) (in Russian)
Trenogin, V.A.: The development and application of the asymptotic method of Lyusternik and Vishik. Russian Math. Surv. 25, 119–156 (1970)
Vasil'eva, A.B., Butuzov, V.F., Kalachev, L.V.: The Boundary Function Method for Singular Perturbation Problems. SIAM, Philadelphia (1995)
Bobryk, R.V.: Closure method and asymptotic expansions for linear stochastic systems. J. Math. Anal. Appl. 329, 703–711 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bobryk, R.V., Chrzeszczyk, A. Transitions in a Duffing oscillator excited by random noise. Nonlinear Dyn 51, 541–550 (2008). https://doi.org/10.1007/s11071-007-9243-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-007-9243-0