Random dynamical systems from SDEs, Lyapunov exponents and bifurcations

Video: lecture 8   (passwd: rds&et2021) Maximilian Engel

Lecture slides:

1. RDSET2021L8-slides

Stationary and quasi-stationary measures near a bifurcation point

 Video: lecture 7   (passwd: rds&et2021) by Guillermo Olicón Méndez

Reading material:

1. RDSET2021L7-slides

Python notebook:

1. RDSERL7.ipynb
   
   

Intermittency

Video: lecture 6  (passwd: rds&et2021) by Ale Jan Homburg

Reading material:

1. RDSET2021L6-slides
2. A.J. Homburg, H. Peters. Critical intermittency in rational maps.
3. A.J. Homburg, V.F. Rabodonandrianandraina. On-off intermittency and chaotic walks. To appear in Ergodic theory and dynamical systems. [doi]
4. N. Abbasi, M. Gharaei, A.J. Homburg. Iterated function systems of logistic maps: synchronization and intermittency. Nonlinearity 31, 3880-3913. [doi]

Lyapunov exponents

 Video: lecture 5  (passwd: rds&et2021) by Jeroen Lamb

Reading material:

1. RDSET2021L5-slides
2. Lectures on Lyapunov exponents (Marcelo Viana)  [you may have online access through your institution's library]

Suggested reading material

A few (non-exhaustive) suggestions for papers (in addition to the material already provided in the course):

"Chaos game":

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

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)

Random interval maps:

A.J. Homburg, V.F. Rabodonandrianandraina. On-off intermittency and chaotic walks. To appear in Ergodic theory and dynamical systems. [doi]

N. Abbasi, M. Gharaei, A.J. Homburg. Iterated function systems of logistic maps: synchronization and intermittency. Nonlinearity 31 (2018), 3880-3913. [doi]

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

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)

Random circle maps

Video: lecture 4  (passwd: rds&et2021) by Julian Newman

Reading material:

1. RDSET2021L4-slides
2. TCC2017-L3
   
3. TCC2017-L3b
   

The correspondence theorem (between stationary measures and Markov measures)

Video: lecture 3 (passwd: rds&et2021) by Zheng Bian

Reading material: 

1. RDSET2021L3-notes.pdf

Jupyter notebooks (python):

1. RDSdensities_1.ipynb
2. RDSdensities_JL.ipynb
3. RDSstat_meas_Doubling+Logistic.ipynb

External links:

1. Introduction to the transfer operator method (Omri Sarig)