We propose a method to find an approximate theoretical solution to the mean first exit time (MFET) of a one-dimensional bistable kinetic system subjected to additive Poisson white noise, by extending an earlier method used to solve stationary probability density function. Based on the Dynkin formula and the properties of Markov processes, the equation of the mean first exit time is obtained. It is an infinite-order partial differential equation that is rather difficult to solve theoretically. Hence, using the non-Gaussian property of Poisson white noise to truncate the infinite-order equation for the mean first exit time, the analytical solution to the mean first exit time is derived by combining perturbation techniques with Laplace integral method. Monte Carlo simulations for the bistable system are applied to verify the validity of our approximate theoretical solution, which shows a good agreement with the analytical results.
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September 2017
Research-Article
The Estimates of the Mean First Exit Time of a Bistable System Excited by Poisson White Noise
Yong Xu,
Yong Xu
Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China;
Potsdam Institute for Climate Impact Research,
Potsdam 14412, Germany;
Department of Physics,
Humboldt University of Berlin,
Berlin 12489, Germany
e-mail: hsux3@nwpu.edu.cn
Northwestern Polytechnical University,
Xi'an 710072, China;
Potsdam Institute for Climate Impact Research,
Potsdam 14412, Germany;
Department of Physics,
Humboldt University of Berlin,
Berlin 12489, Germany
e-mail: hsux3@nwpu.edu.cn
Search for other works by this author on:
Hua Li,
Hua Li
Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China
Northwestern Polytechnical University,
Xi'an 710072, China
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Haiyan Wang,
Haiyan Wang
School of Marine Sciences,
Northwestern Polytechnical University,
Xi'an 710072, China
Northwestern Polytechnical University,
Xi'an 710072, China
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Wantao Jia,
Wantao Jia
Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China
Northwestern Polytechnical University,
Xi'an 710072, China
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Xiaole Yue,
Xiaole Yue
Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China
Northwestern Polytechnical University,
Xi'an 710072, China
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Jürgen Kurths
Jürgen Kurths
Potsdam Institute for Climate Impact Research,
Potsdam 14412, Germany;
Department of Physics,
Humboldt University of Berlin,
Berlin 12489, Germany
Potsdam 14412, Germany;
Department of Physics,
Humboldt University of Berlin,
Berlin 12489, Germany
Search for other works by this author on:
Yong Xu
Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China;
Potsdam Institute for Climate Impact Research,
Potsdam 14412, Germany;
Department of Physics,
Humboldt University of Berlin,
Berlin 12489, Germany
e-mail: hsux3@nwpu.edu.cn
Northwestern Polytechnical University,
Xi'an 710072, China;
Potsdam Institute for Climate Impact Research,
Potsdam 14412, Germany;
Department of Physics,
Humboldt University of Berlin,
Berlin 12489, Germany
e-mail: hsux3@nwpu.edu.cn
Hua Li
Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China
Northwestern Polytechnical University,
Xi'an 710072, China
Haiyan Wang
School of Marine Sciences,
Northwestern Polytechnical University,
Xi'an 710072, China
Northwestern Polytechnical University,
Xi'an 710072, China
Wantao Jia
Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China
Northwestern Polytechnical University,
Xi'an 710072, China
Xiaole Yue
Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China
Northwestern Polytechnical University,
Xi'an 710072, China
Jürgen Kurths
Potsdam Institute for Climate Impact Research,
Potsdam 14412, Germany;
Department of Physics,
Humboldt University of Berlin,
Berlin 12489, Germany
Potsdam 14412, Germany;
Department of Physics,
Humboldt University of Berlin,
Berlin 12489, Germany
1Corresponding author.
Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 27, 2017; final manuscript received June 19, 2017; published online July 12, 2017. Assoc. Editor: Walter Lacarbonara.
J. Appl. Mech. Sep 2017, 84(9): 091004 (8 pages)
Published Online: July 12, 2017
Article history
Received:
February 27, 2017
Revised:
June 19, 2017
Citation
Xu, Y., Li, H., Wang, H., Jia, W., Yue, X., and Kurths, J. (July 12, 2017). "The Estimates of the Mean First Exit Time of a Bistable System Excited by Poisson White Noise." ASME. J. Appl. Mech. September 2017; 84(9): 091004. https://doi.org/10.1115/1.4037158
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