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In this paper, we consider the impacts of noise on ordinary differential equations. We first prove that the weak noise can change the value of equilibrium and the strong noise can destroy the stability of equilibrium. Then we consider the competition between the nonlinear term and noise term, which shows that noise can induce singularities (finite time blow up of solutions) and that the nonlinear term can prevent the singularities. Besides that, some simulations are given in order to illustrate our results.

IMPACTS OF NOISE ON ORDINARY DIFFERENTIAL EQUATIONS