Abstract
We consider a particular class of multidimensional nonlinear stochastic differential equations with 0 as a fixed point. The almost sure stability or instability of 0 is determined by the Lyapunov exponent λ for the associated linear system. If parameters in the stochastic differential equation are varied in such a way that λ changes sign from negative to positive then 0 changes from being (almost surely) stable to being (almost surely) unstable and a new stationary probability measure μ appears. There also appears a new Lyapunov exponent \( \tilde \lambda \), say, corresponding to linearizing the original stochastic differential equation along a trajectory with stationary distribution μ. The value of \( \tilde \lambda \) determines stability or instability along trajectories. We show that, under appropriate conditions, the ratio \( \tilde \lambda \)/λ has a limiting value Γ at a bifurcation point, and we give a Khasminskii-Carverhill type formula for Γ. We also provide examples to show that Γ can take both negative and positive values.
Research supported in part by Office of Naval Research contract N00014-96-1-0413.
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Baxendale, P.H. (1999). Stability Along Trajectories at a Stochastic Bifurcation Point. In: Stochastic Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-22655-9_1
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DOI: https://doi.org/10.1007/0-387-22655-9_1
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