# 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.

## Abstract

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.

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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