0
SPECIAL ISSUE PAPERS

Continuation Method on Cumulant Neglect Equations

[+] Author and Article Information
Edmon Perkins

Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: edmon@auburn.edu

Tim Fitzgerald

Department of Mechanical Engineering,
Gonzaga University,
Spokane, WA 99258
e-mail: fitzgeraldt@gonzaga.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 30, 2017; final manuscript received October 30, 2017; published online July 26, 2018. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 13(9), 090913 (Jul 26, 2018) (3 pages) Paper No: CND-17-1332; doi: 10.1115/1.4038895 History: Received July 30, 2017; Revised October 30, 2017

For stochastic systems, the Fokker–Planck equation (FPE) is used to describe the system dynamics. The FPE is a partial differential equation, which is a function of all the variables in state space and of time. To solve the FPE, several methods are used, including finite elements, moment neglect methods, and cumulant neglect methods. This paper will study the cumulant neglect equations, which are derived from the FPE. It will be shown that the cumulant neglect method, while being a useful and popular tool for studying the system response, introduces several nonphysical artifacts.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Socha, L. , 2007, Linearization Methods for Stochastic Dynamic Systems, Vol. 730, Springer, New York.
Daqaq, M. F. , 2012, “ On Intentional Introduction of Stiffness Nonlinearities for Energy Harvesting Under White Gaussian Excitations,” Nonlinear Dyn., 69(3), pp. 1063–1079. [CrossRef]
Floris, C. , 2015, “ Mean Square Stability of a Second-Order Parametric Linear System Excited by a Colored Gaussian Noise,” J. Sound Vib., 336, pp. 82–95. [CrossRef]
Perkins, E. , and Balachandran, B. , 2015, “ Effects of Phase Lag on the Information Rate of a Bistable Duffing Oscillator,” Phys. Lett. A, 379(4), pp. 308–313. [CrossRef]
Su, Z. , Falzarano, J. M. , and Su, Z. , 2011, “ Gaussian and Non Gaussian Response of Ship Rolling in Random Beam Waves,” 12th International Ship Stability Workshop (ISSW), Washington, DC, June 12–15, pp. 189–193. http://www.shipstab.org/files/Proceedings/ISSW/ISSW_2011_Washington_USA/Session_05_Stochastic_Dynamics/05_6_Su_Falzarano.pdf
Spencer , B., Jr. , and Bergman, L. , 1993, “ On the Numerical Solution of the Fokker-Planck Equation for Nonlinear Stochastic Systems,” Nonlinear Dyn., 4(4), pp. 357–372. [CrossRef]
Gammaitoni, L. , Hänggi, P. , Jung, P. , and Marchesoni, F. , 1998, “ Stochastic Resonance,” Rev. Mod. Phys., 70(1), pp. 223–287. [CrossRef]
Stocks, N. , Stein, N. , and McClintock, P. , 1993, “ Stochastic Resonance in Monostable Systems,” J. Phys. A, 26(7), p. L385. [CrossRef]
Benzi, R. , Sutera, A. , and Vulpiani, A. , 1981, “ The Mechanism of Stochastic Resonance,” J. Phys. A, 14(11), p. L453. [CrossRef]
Fan, F. , and Ahmadi, G. , 1990, “ On Loss of Accuracy and Non-Uniqueness of Solutions Generated by Equivalent Linearization and Cumulant-Neglect Methods,” J. Sound Vib., 137(3), pp. 385–401. [CrossRef]
Bergman, L. , Spencer, B. , Wojtkiewicz, S. , and Johnson, E. , 1996, “ Robust Numerical Solution of the Fokker-Planck Equation for Second Order Dynamical Systems Under Parametric and External White Noise Excitations,” Nonlinear Dyn. Stochastic Mech., 9, pp. 23–37. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.31.156&rep=rep1&type=pdf
Bergman, L. , Wojtkiewicz, S. , Johnson, E. , and Spencer , B., Jr. , 1995, “ Some Reflections on the Efficacy of Moment Closure Methods,” Second International Conference on Computational Stochastic Mechanics, Athens, Greece, June 12–15, pp. 87–95.
Morillo, M. , Gómez-Ordóñez, J. , and Casado, J. M. , 2014, “ Checking the Validity of Truncating the Cumulant Hierarchy Description of a Small System,” Localized Excitations in Nonlinear Complex Systems, Springer International Publishing, Cham, Switzerland, pp. 377–387. [CrossRef]
Nayfeh, A. H. , and Mook, D. T. , 2008, Nonlinear Oscillations, John Wiley & Sons, Hoboken, NJ.
Chorin, A. , and Hald, O. , 2009, Stochastic Tools in Mathematics and Science, Vol. 1, Springer, New York. [CrossRef]
Gardiner, C. , 1985, Stochastic Methods, Springer-Verlag, Berlin.
Allgower, E. L. , and Georg, K. , 2003, Introduction to Numerical Continuation Methods, Society for Industrial and Applied Mathematics, Philadelphia, PA. [CrossRef] [PubMed] [PubMed]
Dhooge, A. , Govaerts, W. , and Kuznetsov, Y. A. , 2003, “ MATCONT: A MATLAB Package for Numerical Bifurcation Analysis of ODEs,” ACM Trans. Math. Software, 29(2), pp. 141–164. [CrossRef]
Perkins, E. , 2017, “ Effects of Noise on the Frequency Response of the Monostable Duffing Oscillator,” Phys. Lett. A, 381(11), pp. 1009–1013. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

The frequency–response relationship for Eq. (3) is given by Eq. (10). This hysteresis curve may be used as a reference for the results of the continuation method. The solid curves represent stable fixed points, while the dashed curve represents unstable fixed points.

Grahic Jump Location
Fig. 2

The frequency response curves of μ1 as Ω̂ is varied from 6 to near 0 for a range of noise amplitudes σ̂

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In