Last week of term

There will be no lecture on wed 18/3. Individual discussions on student projects will continue during the week, either in person or via skype.

No lecture on wed 11/3 and individual appointments for projects on thu 12/3

Project students will receive e-mails to confirm details about their appointments.

Thu 5/3: Guest lecture by Prof Ale Jan Homburg (Amsterdam)

During the class, Ale Jan will be talking about on-off intermittency in random dynamical systems.

Wed 4/3: individual project sessions

I will hold individual discussions about your projects. I will all students individually to schedule a time. If you would not have received an e-mail from me by tuesday 4pm, please alert me.

Guest lecture 27/2 by Maximilian Engel on shear-induced chaos

Maximilian Engel (TUM) will discuss the paper

Maximilian Engel, Jeroen S. W. Lamb, and Martin Rasmussen, Bifurcation analysis of a stochastically driven limit cycle, Communications in Mathematical Physics 365, 3 (2019), 935−942.

Wed 26/2 session: projects

Dear all, I will dedicate most of wed’s session on the projects. Please come prepared with some ideas and directions. Apart from discussion some matters of general use, i also hope to identify any opportunity to form smaller subgroups for further discussion. 

Interesting development: turbulent times

Just to make you aware of a couple of preprints, that appeared late last year, proving for the first time some dynamical behaviour observed in turbulence in the context of stochastic Navier-Stokes equations, see this featured article in Quantamagazine. Adding noise helps mastering this problem, but in addition I would like to emphasize the importance of the dynamical systems approach in proving the desired phenomena; in particular, ergodicity (of the one-point motion) is not enough. 

Some suggestions for literature for the project

Please discuss possible directions for project topics for this course. Projects normally centre around one (or more) paper(s) which are studied in depth on the basis of which an essay/report will be prepared.
The topic of projects should concern some aspect of random dynamical systems theory and need to be agreed with me before you start working on it, ideally, to avoid any misunderstandings.

A few (non-exhaustive) suggestions for papers (in addition to the material already provided in the right-hand side margin):

"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)

Topological bifurcations of random dynamical systems with bounded noise:

Ale Jan Homburg and Todd Young  Bifurcations of random differential equations with bounded noise on surfaces. Topological Methods in Nonlinear Analysis 35 (2010), 77-98. [doi]

Hicham Zmarrou, Ale Jan Homburg. Bifurcations of stationary measures of random diffeomorphisms. Ergod. Th. and Dynam. Sys 27 (2007) 1651-1692. [doi]

Jeroen S. W. Lamb, Martin Rasmussen, and Christian S. Rodrigues, Topological bifurcations of minimal invariant sets for set-valued dynamical systems, Proceedings of the American Mathematical Society 143 (2015), 3927−3937. Article Preprint

Chaotic dynamics of SDEs:

Maximilian Engel, Jeroen S. W. Lamb, and Martin Rasmussen, Bifurcation analysis of a stochastically driven limit cycle, Communications in Mathematical Physics 365, 3 (2019), 935−942. Article Preprint

Thai Son Doan, Maximilian Engel, Jeroen S. W. Lamb, and Martin Rasmussen, Hopf bifurcation with additive noise, Nonlinearity 31, 10 (2018), 4567−4601. Article Preprint

General texts:

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)

Lecture on topological bifurcations in 2D diffeomorphisms with bounded noise (20/2)

Lecture will be given by Kalle Timperi and Wei Hao Tey.

Lecture on random interval diffeomorphisms (19/2)

Victoria Klein will talk about the paper

M. Gharaei, A.J. Homburg. Random interval diffeomorphisms. Discrete Contin. Dyn. Syst. Ser. S 10 (2017), 241-272.

Stationary and quasistationary measures for 1D homeomorphisms with bounded noise (12-13/2)

Guillermo Olicón Méndez give lectures. Relevant notes can be found in the right-hind side margin.

Synchronisation for circle maps 5-6/2

We follow notes by Julian Newman (which you can find in the right-hand side margin).

Guest lectures 29-30/1 about stochastic stability

Prof Tiago Pereira (USP) has been so kind to lecture about stochastic stability. Recommended reading for this material is the lecture notes by Marcelo Viana (IMPA) on Stochastic Dynamics of Deterministic Systems, in particular chapter 2 of this text.

Reading material 22/1: IFS + Ergodic Theory

During the class on wed 22/1 we will discuss aspects from last year's project by Jan Stanczuk (see download link in the rhs margin). Please read this in preparation. The material is closely related to the lectures so far.