Abstract
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.
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Acknowledgments
This research has been partially supported by the Naval Engineering Education Center (NEEC) grant 3002883706 and by the Office of Naval Research (ONR) grant ONR N00014-14-1-0520. The authors thank Dr. Craig Merrill (NEEC Technical Point of Contact), Dr. Vadim Belenky, and Prof. Andrew Majda for numerous stimulating discussions.
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Mohamad, M.A., Sapsis, T.P. (2015). Analytical Approximation of the Heavy-Tail Structure for Intermittently Unstable Complex Modes. In: Ravela, S., Sandu, A. (eds) Dynamic Data-Driven Environmental Systems Science. DyDESS 2014. Lecture Notes in Computer Science(), vol 8964. Springer, Cham. https://doi.org/10.1007/978-3-319-25138-7_14
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DOI: https://doi.org/10.1007/978-3-319-25138-7_14
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