Mohamad,
M. A.
, and
Sapsis,
T. P.
, 2016, “
Probabilistic Response and Rare Events in Mathieu's Equation Under Correlated Parametric Excitation,” Ocean Eng.,
120, pp. 289–297.

[CrossRef]
Belenky,
V. L.
, and
Sevastianov,
N. B.
, 2007, Stability and Safety of Ships: Risk of Capsizing,
The Society of Naval Architects and Marine Engineers, Alexandria, VA.

Cousins,
W.
, and
Sapsis,
T. P.
, 2016, “
Reduced Order Precursors of Rare Events in Unidirectional Nonlinear Water Waves,” J. Fluid Mech.,
790, pp. 368–388.

[CrossRef]
Cousins,
W.
, and
Sapsis,
T. P.
, 2014, “
Quantification and Prediction of Extreme Events in a One-Dimensional Nonlinear Dispersive Wave Model,” Physica D,
280–281, pp. 48–58.

[CrossRef]
Riley,
M. R.
,
Coats,
T.
,
Haupt,
K.
, and
Jacobson,
D.
, 2011, “
Ride Severity Index a New Approach to Quantifying the Comparison of Acceleration Responses of High-Speed Craft,” 11th International Conference on Fast Sea Transportation, Honolulu, HI, Sept. 26–29, pp. 693–699.

Riley,
M. R.
, and
Coats,
T. W.
, 2012, “
A Simplified Approach for Analyzing Accelerations Induced by Wave-Impacts in High-Speed Planing Craft,” Third Chesapeake Power Boat Symposium, Annapolis, MD, June 15–16, pp. 14–15.

Abou-Rayan,
A.
, and
Nayfeh,
A.
, 1993, “
Stochastic Response of a Buckled Beam to External and Parametric Random Excitations,” AIAA Paper No. 93-1425-CP.

Lin,
Y. K.
, and
Cai,
C. Q.
, 1995, Probabilistic Structural Dynamics,
McGraw-Hill, New York.

Lin,
Y. K.
, 1963, “
Application of Nonstationary Shot Noise in the Study of System Response to a Class of Nonstationary Excitations,” ASME J. Appl. Mech.,
30(4), pp. 555–558.

[CrossRef]
Branstetter,
L. J.
,
Jeong,
G. D.
,
Yao,
J. T.
,
Wen,
Y.
, and
Lin,
Y.
, 1988, “
Mathematical Modelling of Structural Behaviour During Earthquakes,” Probab. Eng. Mech.,
3(3), pp. 130–145.

[CrossRef]
Lin,
Y.
, 1996, “
Stochastic Stability of Wind-Excited Long-Span Bridges,” Probab. Eng. Mech.,
11(4), pp. 257–261.

[CrossRef]
Spence,
S.
, and
Gioffrè,
M.
, 2012, “
Large Scale Reliability-Based Design Optimization of Wind Excited Tall Buildings,” Probab. Eng. Mech.,
28, pp. 206–215.

[CrossRef]
Soong,
T. T.
, and
Grigoriu,
M.
, 1993, Random Vibration of Mechanical and Structural Systems,
Prentice Hall,
Englewood Cliffs, NJ, p. 14690.

Sobczyk,
K.
, 2001, Stochastic Differential Equations: With Applications to Physics and Engineering, Vol.
40,
Springer, Berlin.

Sapsis,
T. P.
, and
Athanassoulis,
G. A.
, 2008, “
New Partial Differential Equations Governing the Joint, Response–Excitation, Probability Distributions of Nonlinear Systems, Under General Stochastic Excitation,” Probab. Eng. Mech.,
23(2–3), pp. 289–306.

[CrossRef]
Venturi,
D.
,
Sapsis,
T. P.
,
Cho,
H.
, and
Karniadakis,
G. E.
, 2012, “
A Computable Evolution Equation for the Joint Response-Excitation Probability Density Function of Stochastic Dynamical Systems,” Proc. R. Soc. A,
468(2139), p. 759.

[CrossRef]
Joo,
H. K.
, and
Sapsis,
T. P.
, 2016, “
A Moment-Equation-Copula-Closure Method for Nonlinear Vibrational Systems Subjected to Correlated Noise,” Probab. Eng. Mech.,
46, pp. 120–132.

Athanassoulis,
G.
,
Tsantili,
I.
, and
Kapelonis,
Z.
, 2016, “
Beyond the Markovian Assumption: Response-Excitation Probabilistic Solution to Random Nonlinear Differential Equations in the Long Time,” Proc. R. Soc. A,
471, p. 201505.

Beran,
M.
, 1968, Statistical Continuum Theories,
Interscience Publishers, Geneva, Switzerland.

Wu,
W. F.
, and
Lin,
Y. K.
, 1984, “
Cumulant-Neglect Closure for Non-Linear Oscillators Under Random Parametric and External Excitations,” Int. J. Non Linear Mech.,
19(4), pp. 349–362.

[CrossRef]
Athanassoulis,
G. A.
, and
Gavriliadis,
P. N.
, 2002, “
The Truncated Hausdorff Moment Problem Solved by Using Kernel Density Functions,” Probab. Eng. Mech.,
17(3), pp. 273–291.

[CrossRef]
Xiu,
D.
, and
Karniadakis,
G.
, 2002, “
The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations,” SIAM J. Sci. Comput.,
24(2), pp. 619–644.

[CrossRef]
Xiu,
D.
, and
Karniadakis,
G.
, 2003, “
Modeling Uncertainty in Flow Simulations Via Generalized Polynomial Chaos,” J. Comput. Phys,
187(1), pp. 137–167.

[CrossRef]
Majda,
A. J.
, and
Branicki,
M.
, 2012, “
Lessons in Uncertainty Quantification for Turbulent Dynamical Systems,” Discrete Contin. Dyn. Syst.,
32(9), pp. 3133–3221.

[CrossRef]
Meerschaert,
M. M.
, 2016, Fractional Calculus, Anomalous Diffusion, and Probability,
World Scientific, Singapore, pp. 265–284.

Klafter,
J.
,
Lim,
S. C.
, and
Metzler,
R.
, eds., 2016, Fractional Dynamics,
World Scientific, Singapore.

[CrossRef]
Aban,
I. B.
,
Meerschaert,
M. M.
, and
Panorska,
A. K.
, 2006, “
Parameter Estimation for the Truncated Pareto Distribution,” J. Am. Stat. Assoc.,
101(473), pp. 270–277.

[CrossRef]
Liang,
Y.
, and
Chen,
W.
, 2015, “
A Relative Entropy Method to Measure Non-Exponential Random Data,” Phys. Lett. A,
379(3), pp. 95–99.

[CrossRef]
Liang,
Y.
, and
Chen,
W.
, 2015, “
Reliability Analysis for Sluice Gate Anti-Sliding Stability Using Lévy Stable Distributions,” Signal Process.,
107, pp. 425–432.

[CrossRef]
Liang,
Y.
, and
Chen,
W.
, 2016, “
A Regularized Miner's Rule for Fatigue Reliability Analysis With Mittag-Leffler Statistics,” Int. J. Damage Mech.,
25(5), pp. 691–704.

[CrossRef]
Vasta,
M.
, 1995, “
Exact Stationary Solution for a Class of Non-Linear Systems Driven by a Non-Normal Delta-Correlated Process,” Int. J. Non-Linear Mech.,
30(4), pp. 407–418.

[CrossRef]
Köylüoglu,
H. U.
,
Nielsen,
S. R. K.
, and
Iwankiewicz,
R.
, 1995, “
Response and Reliability of Poisson-Driven Systems by Path Integration,” J. Eng. Mech.,
121(1) pp. 117–130.

[CrossRef]
Iwankiewicz,
R.
, and
Nielsen,
S. R. K.
, 2000, “
Solution Techniques for Pulse Problems in Non-Linear Stochastic Dynamics,” Probab. Eng. Mech.,
15(1), pp. 25–36.

[CrossRef]
Barone,
G.
,
Navarra,
G.
, and
Pirrotta,
A.
, 2008, “
Probabilistic Response of Linear Structures Equipped With Nonlinear Damper Devices (PIS Method),” Probab. Eng. Phys.,
23(2–3), pp. 125–133.

Zhu,
W. Q.
, 1988, “
Stochastic Averaging Methods in Random Vibration,” Appl. Mech. Rev.,
41(5), pp. 189–199.

[CrossRef]
Zeng,
Y.
, and
Zhu,
W. Q.
, 2011, “
Stochastic Averaging of Strongly Nonlinear Oscillators Under Poisson White Noise Excitation,” IUTAM Symposium on Nonlinear Stochastic Dynamics and Control, Hangzhou, China, May 10–14, pp. 147–155.

Masud,
A.
, and
Bergman,
L. A.
, 2005, “
Solution of the Four Dimensional Fokker-Planck Equation: Still a Challenge,” ICOSSAR 2005, Rome, Italy, June 19–23, pp. 1911–1916.

http://web.engr.illinois.edu/~amasud/Papers/Masud-Bergman-ICOSSAR-OS0506.pdf
Di Matteo,
A.
,
Di Paola,
M.
, and
Pirrotta,
A.
, 2014, “
Probabilistic Characterization of Nonlinear Systems Under Poisson White Noise Via Complex Fractional Moments,” Nonlinear Dyn.,
77(3), pp. 729–738.

[CrossRef]
Mohamad,
M. A.
, and
Sapsis,
T. P.
, 2015, “
Probabilistic Description of Extreme Events in Intermittently Unstable Systems Excited by Correlated Stochastic Processes,” SIAM/ASA J. Uncertain. Quantif.,
3(1), pp. 709–736.

[CrossRef]
Mohamad,
M. A.
,
Cousins,
W.
, and
Sapsis,
T. P.
, 2016, “
A Probabilistic Decomposition-Synthesis Method for the Quantification of Rare Events Due to Internal Instabilities,” J. Comput. Phys.,
322, pp. 288–308.

[CrossRef]
Joo,
H. K.
,
Mohamad,
M. A.
, and
Sapsis,
T. P.
, “
Extreme Events and Their Optimal Mitigation in Nonlinear Structural Systems Excited by Stochastic Loads: Application to Ocean Engineering Systems,” Ocean Eng.,
142, pp. 145–160.

[CrossRef]
Srirangarajan,
S.
,
Allen,
M.
,
Preis,
A.
,
Iqbal,
M.
,
Lim,
H. B.
, and
Whittle,
A. J.
, 2013, “
Wavelet-Based Burst Event Detection and Localization in Water Distribution Systems,” J. Signal Process. Syst.,
72(1), pp. 1–16.

[CrossRef]
Langley,
R. S.
, 1986, “
On Various Definitions of the Envelope of a Random Process,” J. Sound Vib.,
105(3), pp. 503–512.

[CrossRef]
Ochi,
M.,K.
, 1990, Applied Probability and Stochastic Processes: In Engineering and Physical Sciences, Vol.
226,
Wiley-Interscience, Hoboken, NJ.

Karadeniz,
H.
, 2012, Stochastic Analysis of Offshore Steel Structures: An Analytical Appraisal,
Springer Science & Business Media, London.