Stabilisation and destabilisation of nonlinear differential equations by noise

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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 (https://doi.org/10.1109/TAC.2008.919255)

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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. .

ORCID iDs

Appleby, John A. D., Mao, Xuerong ORCID logoORCID: https://orcid.org/0000-0002-6768-9864 and Rodkina, Alexandra;
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