Beyond the Markovian assumption: response–excitation probabilistic solution to random nonlinear differential equations in the long time

Uncertainty quantification for dynamical systems under non-white excitation is a difficult problem encountered across many scientific and engineering disciplines. Difficulties originate from the lack of Markovian character of system responses. The response–excitation (RE) theory, recently introduced by Sapsis & Athanassoulis (Sapsis & Athanassoulis 2008Probabilistic Eng. Mech. 23, 289–306 (doi:10.1016/j.probengmech.2007.12.028 )) and further studied by Venturiet al. (Venturiet al. 2012Proc. R. Soc. A 468, 759–783 (doi:10.1098/rspa.2011.0186 )), is a new approach, based on a simple differential constraint which is exact but non-closed. The evolution equation obtained for the RE probability density function (pdf) has the form of a generalized Liouville equation, with the excitation time frozen in the time-derivative term. In this work, the missing information of the RE differential constraint is identified and a closure scheme is developed for the long-time, stationary, limit-state of scalar nonlinear random differential equations (RDEs) under coloured excitation. The closure scheme does not alter the RE evolution equation, but collects the missing information through the solution of local statistically linearized versions of the nonlinear RDE, and interposes it into the solution scheme. Numerical results are presented for two examples, and compared with Monte Carlo simulations.