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Stabilisation and destabilisation of nonlinear differential equations by noise

Appleby, John A. D. and Mao, Xuerong and Rodkina, Alexandra (2008) Stabilisation and destabilisation of nonlinear differential equations by noise. IEEE Transactions on Automatic Control, 53 (3). pp. 683-691. ISSN 0018-9286

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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. .