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.

References

References
1.
Socha
,
L.
,
2007
,
Linearization Methods for Stochastic Dynamic Systems
, Vol.
730
,
Springer
, New York.
2.
Daqaq
,
M. F.
,
2012
, “
On Intentional Introduction of Stiffness Nonlinearities for Energy Harvesting Under White Gaussian Excitations
,”
Nonlinear Dyn.
,
69
(
3
), pp.
1063
1079
.
3.
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
.
4.
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
.
5.
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
6.
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
.
7.
Gammaitoni
,
L.
,
Hänggi
,
P.
,
Jung
,
P.
, and
Marchesoni
,
F.
,
1998
, “
Stochastic Resonance
,”
Rev. Mod. Phys.
,
70
(
1
), pp.
223
287
.
8.
Stocks
,
N.
,
Stein
,
N.
, and
McClintock
,
P.
,
1993
, “
Stochastic Resonance in Monostable Systems
,”
J. Phys. A
,
26
(
7
), p.
L385
.
9.
Benzi
,
R.
,
Sutera
,
A.
, and
Vulpiani
,
A.
,
1981
, “
The Mechanism of Stochastic Resonance
,”
J. Phys. A
,
14
(11), p.
L453
.
10.
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
.
11.
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
12.
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
.
13.
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
.
14.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
2008
,
Nonlinear Oscillations
, John Wiley & Sons, Hoboken, NJ.
15.
Chorin
,
A.
, and
Hald
,
O.
,
2009
,
Stochastic Tools in Mathematics and Science
, Vol.
1
, Springer, New York.
16.
Gardiner
,
C.
,
1985
,
Stochastic Methods
, Springer-Verlag, Berlin.
17.
Allgower
,
E. L.
, and
Georg
,
K.
,
2003
,
Introduction to Numerical Continuation Methods
, Society for Industrial and Applied Mathematics, Philadelphia, PA.
18.
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
.
19.
Perkins
,
E.
,
2017
, “
Effects of Noise on the Frequency Response of the Monostable Duffing Oscillator
,”
Phys. Lett. A
,
381
(
11
), pp.
1009
1013
.
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