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Suppression of explosion by polynomial noise for nonlinear differential systems

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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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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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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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  • explosion suppression
  • noise
  • non-autonomous differential system
  • growth with general rate
  • decay with general rate
  • general polynomial growth condition