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)
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)
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