Random Dynamical Systems
Prof Jeroen S.W. Lamb, Imperial College London
Thu 12:00-14:00 / Fri 12:00-13:00 in Huxley 658 (to be reconfirmed)
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 and ergodic theory. 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 theory is useful, but is not a strict prerequisite as we make an effort to remain self-contained as much as possible.
Topics (tbc):
- Invariant measures and ergodic theory
- Forward and pullback attractors
- Random circle maps
- Random interval maps
- Lyapunov exponents
- Bifurcations
Assessment: Students taking this course for credit will need to write an essay on a specific aspect of the course material. In addition, an oral will be held to examine the student on the project content, also in the broader context of the course. Detailed instructions on the essay and oral will be given in due course.
