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We consider the problem of probabilistic quantification of dynamical systemsthat have heavy-tailed characteristics. These heavy-tailed features areassociated with rare transient responses due to the occurrence of internalinstabilities. Here we develop a computational method, a probabilisticdecomposition-synthesis technique, that takes into account the nature ofinternal instabilities to inexpensively determine the non-Gaussian probabilitydensity function for any arbitrary quantity of interest. Our approach relies onthe decomposition of the statistics into a `non-extreme core', typicallyGaussian, and a heavy-tailed component. This decomposition is in fullcorrespondence with a partition of the phase space into a `stable' region wherewe have no internal instabilities, and a region where non-linear instabilitieslead to rare transitions with high probability. We quantify the statistics inthe stable region using a Gaussian approximation approach, while thenon-Gaussian distributions associated with the intermittently unstable regionsof the phase space are inexpensively computed through order-reduction methodsthat take into account the strongly nonlinear character of the dynamics. Theprobabilistic information in the two domains is analytically synthesizedthrough a total probability argument. The proposed approach allows for theaccurate quantification of non-Gaussian tails at more than 10 standarddeviations, at a fraction of the cost associated with the direct Monte-Carlosimulations. We demonstrate the probabilistic decomposition-synthesis methodfor rare events for two dynamical systems exhibiting extreme events: atwo-degree-of-freedom system of nonlinearly coupled oscillators, and in anonlinear envelope equation characterizing the propagation of unidirectionalwater waves.

A probabilistic decomposition-synthesis method for the quantification of rare events due to internal instabilities