Appleby, J. and Mao, X. and Rodkina, A. (2008) *Stabilization and destabilization of nonlinear differential equations by noise.* IEEE Transactions on Automatic Control, 53 (3). pp. 683-691. ISSN 0018-9286

## Abstract

This paper considers the stabilisation and destabilisa- tion by a Brownian noise perturbation which preserves the equilibrium of the ordinary dierential equation x0(t) = f(x(t)). In an extension of earlier work, we lift the restriction that f obeys a global linear bound, and show that when f is locally Lipschitz, a function g can always be found so that the noise perturbation g(X(t)) dB(t) either stabilises an unstable equilibrium, or destabilises a stable equilibrium. When the equilibrium of the deterministic equation is non{hyperbolic, we show that a non{hyperbolic perturbation suffices to change the stability properties of the solution. .

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