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.

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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



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