Assessment

This courses is assessed by project/essay and oral.

Projects/essays (which have a weight of 60% for your mark for the course; the remaining 40% is for the oral examination) are supposed to treat in some depth a specific aspect of "random dynamical systems and ergodic theory", building on the material presented in the lectures. Normally, students focus on one or two specific research paper(s) or  ((a) chapter(s) of) a research monograph and study (a part of) this in detail. During the running of the course I will give some specific suggestions for material that you could use, but you are also welcome to suggest alternative material.

What I am most looking for in your project/essay is depth and demonstration of understanding of the presented material. Of course, a degree of ambition is also appreciated and the difficulty of the chosen material will be considered as well (but the latter should not come at the expense of clarity and depth).

Everyone taking the course for credit will need to discuss the topic/material for their project/essay with the lecturer (jswlamb@ic.ac.uk) during the second half of term. It is anticipated that everyone makes good progress with their project/essay during the second half of the second term.

The official deadline for hand-in of the project/essay will be early June 2027 (precise deadline tba). I would like to emphasize that is the very latest you can still submit. In previous years, many students submitted shortly after the easter break, before the start of the exam period. I make the deadline flexible to facilitate your organisation of the various other tasks and deadlines during this time. It is my aim to do the oral examinations shortly after your hand-in.

Welcome to Random Dynamical Systems and Ergodic Theory

This blog is dedicated to the the course  Random Dynamical Systems and Ergodic Theory, running in the department of Mathematics of Imperial College London, moderated by Prof Jeroen S.W. Lamb in 2027.

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 most likely include (tbc):

Invariant measures and ergodic theory

(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, see also next post.

For the course material in previous years see

https://rdset2026.blogspot.com/

https://rdset2024.blogspot.com/

https://rdset2023.blogspot.com/

https://rdset2022.blogspot.com/

https://rdset2021.blogspot.com/

https://rdset2020.blogspot.com/

https://rdset2019.blogspot.com/