Abstract: We start to
consider the Lorenz equation where the
bifurcation parameter (the Rayleigh number) is
randomly perturbed. If the perturbation is
influenced by a Gaussian white noise, then we
arrive at a Stratonovich stochastic
differential equation (SDE) of the Lorenz
type; the properties of solution, such as the
nonoccurrence of an explosion, the recurrence
relative to a bounded ellipsoid and the
symmetry under a coordinate transformation,
are investigated. Let e
be a small parameter such that 0
< e
<< 1. Then we consider the SDE on time scales of
order where s is the
Prandtl number appearing in the Lorenz
equation. If the noise-driven bifurcation
parameter increases by e–2
from 1 to 1
+ e–2
with a small parameter 0
< e
<< 1, then by change of spatial variables we arrive
at a new Stratonovich SDE of the Duffing type
with a small parameter e; the solution is regarded as the response to
the stochastic Duffing oscillator. Our main
aim is to derive and identify a limit process
as e®
0. The derived process is the solution of a
Stratonovich SDE and is also regarded as the
response to the stochastic Duffing oscillator.
Lastly, we investigate the asymptotic behavior
of the top Lyapunov exponent for the
linearization of Stratonovich SDE of the
Duffing type with small noise intensity.
Keywords and phrases: stochastic differential equation, Lorenz equation, Duffing equation, explosion time, scale-time change, Lyapunov exponent.
