Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes
In this work, we consider systems that are subjected to intermittent instabilities due to external stochastic excitation. These intermittent instabilities, though rare, have a large impact on the probabilistic response of the system and give rise to heavy-tailed probability distributions. By making...
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| Language: | en_US |
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Society of Industrial and Applied Mathematics
2017
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| Online Access: | http://hdl.handle.net/1721.1/108788 https://orcid.org/0000-0001-9666-4810 https://orcid.org/0000-0003-0302-0691 |
| _version_ | 1826188491616157696 |
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| author | Mohamad, Mustafa A. Sapsis, Themistoklis P. |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Mohamad, Mustafa A. Sapsis, Themistoklis P. |
| author_sort | Mohamad, Mustafa A. |
| collection | MIT |
| description | In this work, we consider systems that are subjected to intermittent instabilities due to external stochastic excitation. These intermittent instabilities, though rare, have a large impact on the probabilistic response of the system and give rise to heavy-tailed probability distributions. By making appropriate assumptions on the form of these instabilities, which are valid for a broad range of systems, we formulate a method for the analytical approximation of the probability distribution function (pdf) of the system response (both the main probability mass and the heavy-tail structure). In particular, this method relies on conditioning the probability density of the response on the occurrence of an instability and the separate analysis of the two states of the system, the unstable and stable states. In the stable regime we employ steady state assumptions, which lead to the derivation of the conditional response pdf using standard methods for random dynamical systems. 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. The method we present allows us to capture the statistics associated with the dynamics that give rise to heavy-tails in the system response, and the analytical approximations compare favorably with direct Monte Carlo simulations, which we illustrate for two prototype intermittent systems: an intermittently unstable mechanical oscillator excited by correlated multiplicative noise and a complex mode in a turbulent signal with fixed frequency, where nonlinear mode interaction terms are replaced by a stochastic drag and additive white noise forcing. |
| first_indexed | 2024-09-23T08:00:27Z |
| format | Article |
| id | mit-1721.1/108788 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T08:00:27Z |
| publishDate | 2017 |
| publisher | Society of Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | mit-1721.1/1087882022-09-30T01:40:49Z Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes Mohamad, Mustafa A. Sapsis, Themistoklis P. Massachusetts Institute of Technology. Department of Mechanical Engineering Mohamad, Mustafa A. Sapsis, Themistoklis P. In this work, we consider systems that are subjected to intermittent instabilities due to external stochastic excitation. These intermittent instabilities, though rare, have a large impact on the probabilistic response of the system and give rise to heavy-tailed probability distributions. By making appropriate assumptions on the form of these instabilities, which are valid for a broad range of systems, we formulate a method for the analytical approximation of the probability distribution function (pdf) of the system response (both the main probability mass and the heavy-tail structure). In particular, this method relies on conditioning the probability density of the response on the occurrence of an instability and the separate analysis of the two states of the system, the unstable and stable states. In the stable regime we employ steady state assumptions, which lead to the derivation of the conditional response pdf using standard methods for random dynamical systems. 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. The method we present allows us to capture the statistics associated with the dynamics that give rise to heavy-tails in the system response, and the analytical approximations compare favorably with direct Monte Carlo simulations, which we illustrate for two prototype intermittent systems: an intermittently unstable mechanical oscillator excited by correlated multiplicative noise and a complex mode in a turbulent signal with fixed frequency, where nonlinear mode interaction terms are replaced by a stochastic drag and additive white noise forcing. Massachusetts Institute of Technology. Naval Engineering Education Center (Grant 3002883706) United States. Office of Naval Research (Grant ONR N00014-14-1-0520) 2017-05-09T19:35:33Z 2017-05-09T19:35:33Z 2015-08 2015-06 Article http://purl.org/eprint/type/JournalArticle 2166-2525 http://hdl.handle.net/1721.1/108788 Mohamad, Mustafa A., and Themistoklis P. Sapsis. “Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes.” SIAM/ASA Journal on Uncertainty Quantification 3.1 (2015): 709–736. © 2015 Society for Industrial and Applied Mathematics https://orcid.org/0000-0001-9666-4810 https://orcid.org/0000-0003-0302-0691 en_US http://dx.doi.org/10.1137/140978235 SIAM/ASA Journal on Uncertainty Quantification Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Society of Industrial and Applied Mathematics SIAM |
| spellingShingle | Mohamad, Mustafa A. Sapsis, Themistoklis P. Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes |
| title | Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes |
| title_full | Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes |
| title_fullStr | Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes |
| title_full_unstemmed | Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes |
| title_short | Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes |
| title_sort | probabilistic description of extreme events in intermittently unstable dynamical systems excited by correlated stochastic processes |
| url | http://hdl.handle.net/1721.1/108788 https://orcid.org/0000-0001-9666-4810 https://orcid.org/0000-0003-0302-0691 |
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