Random
Dynamical Systems
Prof Jeroen S.W. Lamb,
Imperial College London
Wed 11:10-13:00, from 11/10/17 until 29/11/17
Abstract:
Dynamical systems describe the time-evolution of variables that characterize
the state of a system. In deterministic autonomous dynamical systems, the corresponding
equations of motion are independent of time. In contrast, in random dynamical
systems the equations of motion explicitly depend on a stochastic process or
random variable.
The development of the
field of deterministic dynamical systems – including “chaos” theory - has been
one of the scientific revolutions of the twentieth century, originating with
the pioneering insights of Poincaré, providing a geometric qualitative
understanding of dynamical processes, aiding and complementing analytical and
quantitative viewpoints.
During the last decades
there has been an increasing interest in time-dependent and in particular
random dynamical systems, often – but not necessarily - described by stochastic
differential equations. Despite the obvious scientific importance of the field,
with applications ranging from physics and engineering to bio-medical and social
sciences, a geometric qualitative theory for random dynamical systems is still
in its infancy.
This course provides
an introduction to random dynamical systems. The main aim is to introduce key
concepts and results in the context of relatively simple examples. We will also
highlight open problems.
Some background in
dynamical systems and probability/ergodic theory is useful, but is not a strict
prerequisite as we make an effort to remain self-contained as much as possible.
Topics (tentative):
- Invariant measures and ergodicity
- Forward and pullback attractors
- Random
circle maps
- Random interval maps
- Lyapunov exponents
- Bifurcations
Assessment: Anyone taking this course for credit will be asked to submit a written presentation on some aspect of the course (as pre-agreed with me) which will be discussed during a short oral in January 2018. Please let me know as soon as possible, but in any case before the final lecture, if you would like to take this course for credit.
