Fast-slow systems with chaotic noise.
David Kelly, CIMS
Abstract:
It has long been observed that multi-scale systems, particularly
those in climatology, exhibit behavior typical of stochastic models,
most notably in the unpredictability and statistical variability of
events. This is often in spite of the fact that the underlying
physical model is completely deterministic. One possible
explanation for this stochastic behavior is deterministic chaotic
effects. In fact, it has been well established that the statistical
properties of chaotic systems can be well approximated by stochastic
differential equations.
In this talk, we focus on fast-slow ODEs, where the fast, chaotic
variables are fed into the slow variables to yield a diffusion
approximation. In particular we focus on the case where the fast
process is multi-dimensional and enters the slow equation as
multiplicative “noise”. The tools from rough path theory prove
useful in this difficult setting. From a stochastic modeling
perspective, the limiting slow variables are somewhat surprising.
Even though the noise is approximated by a smooth (chaotic) signal,
one does not obtain a Stratonovich integral in the limiting
equation.