Dynamics controlled by additive noise

Nuovo Cimento D 17, 855-861 (1995)

G.D. Lythe

Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 9EW, United Kingdom.


Analysis is presented of a system whose dynamics are dramatically simplified by tiny amounts of additive noise. The dynamics divide naturally into two phases. In the slower phase, trajectories are close to an invariant manifold; this allows small random disturbances to exert a controlling influence. A map is derived which provides an accurate description of the trajectories.

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The four trajectories shown are solutions differing only in the level of added noise, ε.
The greater the noise level, the simpler the solution.
Sensitivity to noise arises because the solutions spend most of their time near y=z=0 with |x| slowly increasing. Here δ=0.3 and μ=0.01. Fixed points are marked with crosses.
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