Stochastic suppression and stabilization of delay differential systems

In this paper, we investigate stochastic suppression and stabilization for nonlinear delay differential system. (x) over dot(t) = f(x(t), x(t-delta(t)), t), where delta(t) is the variable delay and f satisfies the one-sided polynomial growth condition. Since f may defy the linear growth condition or the one-sided linear growth condition, this system may explode in a finite time. To stabilize this system by Brownian noises, we stochastically perturb this system into the nonlinear stochastic differential system dx(t) = f (x(t), x(t-delta(t)), t) dt+qx(t) dw(1)(t)+sigma vertical bar x(t)vertical bar(beta)x(t)dw(2)(t) by introducing two independent Brownian motions w(1)(t) and w(2)(t). This paper shows that the Brownian motion w2(t) may suppress the potential explosion of the solution of this stochastic system for appropriate choice of beta under the condition sigma not equal 0. Moreover, for sufficiently large q, the Brownian motion w(1)(t) may exponentially stabilize this system