Analytical Approximation of the Heavy-Tail Structure for Intermittently Unstable Complex Modes

In this work, we consider systems that are subjected to intermittent instabilities due to external, correlated stochastic excitation. These intermittent instabilities, though rare, give rise to heavy-tailed probability distribution functions (pdf). By making appropriate assumptions on the form of these instabilities, we formulate a method for the analytical approximation of the pdf of the system response. This method relies on conditioning the pdf of the response on the occurrence of an instability and the separate analysis of the two states of the system, the unstable and stable state. In the stable regime we employ steady state assumptions, which lead to the derivation of the conditional response pdf using standard methods. The unstable regime is inherently transient and in order to analyze this regime we characterize the statistics under the assumption of an exponential growth phase and a subsequent decay phase until the system is brought back to the stable attractor. We illustrate our method to a prototype intermittent system, a complex mode in a turbulent signal, and show that the analytic results compare favorably with direct Monte Carlo simulations for a broad range of parameters.

[1]  Andrew J. Majda,et al.  Lessons in uncertainty quantification for turbulent dynamical systems , 2012 .

[2]  Andrew J. Majda,et al.  Improving filtering and prediction of spatially extended turbulent systems with model errors through stochastic parameter estimation , 2010, J. Comput. Phys..

[3]  T. T. Soong,et al.  Random Vibration of Mechanical and Structural Systems , 1992 .

[4]  Andrew J. Majda,et al.  Test models for improving filtering with model errors through stochastic parameter estimation , 2010, J. Comput. Phys..

[5]  Harald E. Krogstad,et al.  Oceanic Rogue Waves , 2008 .

[6]  Marie F. Kratz.  Level crossings and other level functionals of stationary Gaussian processes , 2006 .

[7]  G. Karniadakis,et al.  A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[8]  Kazimierz Sobczyk,et al.  Stochastic Differential Equations , 1991 .

[9]  Andrew J. Majda,et al.  Filtering skill for turbulent signals for a suite of nonlinear and linear extended Kalman filters , 2012, J. Comput. Phys..

[10]  Guangyu Wu,et al.  Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution , 2013, Journal of Fluid Mechanics.

[11]  Themistoklis P. Sapsis,et al.  Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model , 2014, 1401.3397.

[12]  T. DelSole,et al.  Stochastic Models of Quasigeostrophic Turbulence , 2004 .

[13]  Michael J. Rycroft,et al.  Storms in Space , 2004 .

[14]  Gerassimos A. Athanassoulis,et al.  New partial differential equations governing the joint, response–excitation, probability distributions of nonlinear systems, under general stochastic excitation , 2008 .

[15]  R. Salmon,et al.  Geophysical Fluid Dynamics , 2019, Classical Mechanics in Geophysical Fluid Dynamics.

[16]  Andrew J. Majda,et al.  Filtering Complex Turbulent Systems , 2012 .

[17]  S. O. Rice,et al.  Distribution of the duration of fades in radio transmission: Gaussian noise model , 1958 .

[18]  Andrew J. Majda,et al.  Information theory and stochastics for multiscale nonlinear systems , 2005 .

[19]  Ian F. Blake,et al.  Level-crossing problems for random processes , 1973, IEEE Trans. Inf. Theory.

[20]  Andrew J. Majda,et al.  Quantifying uncertainty for predictions with model error in non-Gaussian systems with intermittency , 2012 .

[21]  Andrew J. Majda,et al.  A one-dimensional model for dispersive wave turbulence , 1997 .

[22]  Andrew J. Majda,et al.  Predicting the cloud patterns of the Madden‐Julian Oscillation through a low‐order nonlinear stochastic model , 2014 .