65 research outputs found
A Girsanov approach to slow parameterizing manifolds in the presence of noise
We consider a three-dimensional slow-fast system with quadratic nonlinearity
and additive noise. The associated deterministic system of this stochastic
differential equation (SDE) exhibits a periodic orbit and a slow manifold. The
deterministic slow manifold can be viewed as an approximate parameterization of
the fast variable of the SDE in terms of the slow variables. In other words the
fast variable of the slow-fast system is approximately "slaved" to the slow
variables via the slow manifold. We exploit this fact to obtain a two
dimensional reduced model for the original stochastic system, which results in
the Hopf-normal form with additive noise. Both, the original as well as the
reduced system admit ergodic invariant measures describing their respective
long-time behaviour. We will show that for a suitable metric on a subset of the
space of all probability measures on phase space, the discrepancy between the
marginals along the radial component of both invariant measures can be upper
bounded by a constant and a quantity describing the quality of the
parameterization. An important technical tool we use to arrive at this result
is Girsanov's theorem, which allows us to modify the SDEs in question in a way
that preserves transition probabilities. This approach is then also applied to
reduced systems obtained through stochastic parameterizing manifolds, which can
be viewed as generalized notions of deterministic slow manifolds.Comment: 54 pages, 6 figure
Noise-induced chaos: a conditioned random dynamics perspective
We consider transitions to chaos in random dynamical systems induced by an
increase of noise amplitude. We show how the emergence of chaos (indicated by a
positive Lyapunov exponent) in a logistic map with bounded additive noise can
be analysed in the framework of conditioned random dynamics through expected
escape times and conditioned Lyapunov exponents for a compartmental model
representing the competition between contracting and expanding behaviour. We
find that the noise-induced transition to chaos is caused by a rapid decay of
the expected escape time from the contracting compartment, while all other
order parameters remain approximately constant.Comment: 6 pages, 6 figure
- …