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Suppression of explosion by polynomial noise for nonlinear differential systems

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Abstract

In this paper, we study the problem of the suppression of explosion by noise for nonlinear non-autonomous differential systems. For a deterministic non-autonomous differential system dx(t) = f(x(t), t)dt, which can explode at a finite time, we introduce polynomial noise and study the perturbed system dx(t) = f(x(t), t)dt + h(t) 12 |x(t)|ßAx(t)dB(t). We demonstrate that the polynomial noise can not only guarantee the existence and uniqueness of the global solution for the perturbed system, but can also make almost every path of the global solution grow at most with a certain general rate and even decay with a certain general rate (including super-exponential, exponential, and polynomial rates) under specific weak conditions.

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Acknowledgements

This work was partially supported by National Natural Science Foundation of China (Grant No. 11571024), China Postdoctoral Science Foundation (Grant No. 2017M621588), Natural Science Foundation of Hebei Province of China (Grant No. A2015209229), Science and Technology Research Foundation of Higher Education Institutions of Hebei Province of China (Grant No. QN2017116), Grant From the Simons Foundation (Grant No. 429343, Renming Song), and Graduate Foundation of the North China University of Science and Technology (Grant No. K1603).

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Authors and Affiliations

  1. College of Science, North China University of Science and Technology, Tangshan, 063210, China

    Lichao Feng & Yemo Li

  2. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, China

    Lichao Feng, Shoumei Li & Renming Song

  3. School of Mathematics, and Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing, 210096, China

    Lichao Feng

  4. Department of Mathematics, University of Illinois, Urbana, 61801, USA

    Renming Song

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  1. Lichao Feng
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  2. Shoumei Li
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  3. Renming Song
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  4. Yemo Li
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Correspondence to Shoumei Li.

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Feng, L., Li, S., Song, R. et al. Suppression of explosion by polynomial noise for nonlinear differential systems. Sci. China Inf. Sci. 61, 70215 (2018). https://doi.org/10.1007/s11432-017-9340-4

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  • DOI: https://doi.org/10.1007/s11432-017-9340-4

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