Assessment

For those who want to take this course for credit, my suggestion is to prepare a 5+ page document discussing in detail (aspects of ):
- synchronisation of random circle homeomorphisms
- Oseledets theorem
- synchronisation of random diffeomorphisms of the interval (in the Gharaei & Homburg setting)

Please send me this before 8 January. Then I will arrange a brief discussion of your document on skype or in person.

I am also open to alternative suggestions for assessment, let me know if you would like to propose such.

Random interval diffeomorphisms

The final topic of this course concerns random interval diffeomorphisms. We follow M. Gharaei, A.J. Homburg. Random interval diffeomorphisms. Discrete Contin. Dyn. Syst. Ser. S(10), 241-272.

Lyapunov exponents

We discuss this topic in detail in the one- an two-dimensional setting.  For the two-dimensional setting, we follow lecture notes of Marcelo Viana (attached in the right hand side margin).

Synchronisation in random circle maps

Julian Newman speaks in more detail about synchronisation in random circle maps on 1 and 8 November. Accompanying lecture notes can be found in the right hand side margin.

Iterated Function System of contractions

I added a few notes from lectures by Shlomo Sternberg showing that an IFT of contractions on X yields a contraction on the space of non-empty compact subsets of X equipped with the Hausdorff metric. Pages 3-17 are relevant in support of the material on page 4 of my lecture notes.

I also added the relevant passage from Barnsley's book "Fractals Everywhere" proving that the associated Markov operator on probability measures is a contraction in the Hutchinson metric (also known as ($L_1$-) Wasserstein metric).

About this course

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.