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SCIENCE CHINA Information Sciences, Volume 61, Issue 7: 070215(2018) https://doi.org/10.1007/s11432-017-9340-4

Suppression of explosion by polynomial noise for nonlinear differential systems

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  • ReceivedSep 24, 2017
  • AcceptedJan 8, 2018
  • PublishedJun 4, 2018

Abstract

In this paper, we study the problem of the suppression of explosion by noise fornonlinear non-autonomous differential systems.For a deterministic non-autonomous differential system${\rm~d}{\boldsymbol~x}(t)={\boldsymbol~f}({\boldsymbol~x}(t),t){\rm~d}t$, which can explode at a finite time,we introduce polynomial noise and study the perturbed system${\rm~d}{\boldsymbol~x}(t)={\boldsymbol~f}({\boldsymbol~x}(t),t){\rmd}t~+h(t)^{\frac{1}{2}}|{\boldsymbol~x}(t)|^{\beta}A{\boldsymbol~x}(t){\rm~d}B(t)$.We demonstrate that the polynomial noise can not only guarantee the existence and uniqueness of theglobal solution for the perturbed system, but can also make almost every path of the global solutiongrow 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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