RDS discussion group Fri 10-11 in H642

Until further notice, there will be an RDS discussion group fri 10-11 in H642.

Some project suggestions and relevant literature

The "Chaos game":

Michael Barnsley, Fractals Everywhere (1993)  [Chapter X]

John H Elton, An ergodic theorem for iterated maps. Erg. Theory Dyn. Systems 7 (1987),  481-488. (link)

Pablo G. Barrientos, Fatemeh H. Ghane, Dominique Malicet and Aliasghar Sarizadeh, On the chaos game of iterated function systems. Topological Methods in Nonlinear Analysis 49 (2017),  105-132. (link)

Random circle maps and synchronisation:

Dominique Malicet, Random Walks on Homeo(S^1). Commun. Math. Phys.  356 (2017), 1083-1116. (link)

Julian Newman, Necessary and sufficient conditions for stable synchronization in random dynamical systems. Erg. Theory Dyn. Systems 38 (2018), 1857-1875. (link)

Yves Le Jan, Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. Ann. Inst. Henri Poincaré Probab. Stat. 23 (1987), 111-120. (link)

Some general theory:

Ludwig Arnold, Random Dynamical Systems (1998)
Yuri Kifer, Ergodic Theory of Random Transformations (1986)
Hans Crauel and Franco Flandoli, Attractors for random dynamical systems, Prob. Theory Rel. Fields 100 (1994), 365-393. (link)
Marcelo Viana, Lectures on Lyapunov Exponents (2014) (link)

Lectures time and place - change of room

Please note that the lectures will be given until further notice on Thursdays, 12:00-14:00 in Room Huxley 642. There are no lectures on Fridays.

About this course

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.