The Dynamic Analysis seminar is organized jointly with the analysis group at the UvA.

The seminar (usually) takes place every other Wednesday, 16:00-17:00.

Talks are intended for an audience with a background in analysis and dynamical systems, including PhD students.

For more information, please contact the organizers: Oliver Fabert (VU), Rob van der Vorst (VU), and Ale Jan Homburg (UvA).

A database of earlier years' seminars can be found here.

And a route description can be found here.

## Upcoming talks in 2018

25 April: **Konstantinos Efstathiou** (Groningen), 16:00-17:00, **Room F3.20 (KdV, UvA)**

*Title:* Monodromy and Circle Actions*Abstract:* Standard Hamiltonian monodromy was introduced by Duistermaat as an obstruction to the existence of global action-angle coordinates in integrable Hamiltonian systems. It refers to the monodromy of torus bundles that typically exist in such systems. Fractional Hamiltonian monodromy, introduced by Nekhoroshev, Sadovskií, and Zhilinskií, generalizes standard monodromy by considering not only torus bundles but also more general fibrations with singular fibers. In this talk I present results concerning both standard and fractional monodromy that were recently obtained in collaboration with Nikolay Martynchuk. It turns out that, in integrable Hamiltonian systems with a Hamiltonian circle action, both standard and fractional monodromy can be solely determined through a careful study of the fixed points of the circle action and their weights. A basic ingredient of this approach is the definition of generalized parallel transport of homology cycles. These results will be demonstrated in several examples of integrable Hamiltonian systems.

09 May: **Kees Vuik** (TU Delft), 16:00-17:00, **Room F3.20 (KdV, UvA)**

23 May: **Stefanie Sonner** (Nijmegen), 16:00-17:00, Room WN-623

## Previous talks in 2018

28 March: **Jason Frank** (Utrecht), 16:00-17:00, Room WN-**P640
**

*Title:* Tangent-space splittings for data assimilation*Abstract:* Data assimilation methods are used for marrying instrumental observations of a physical system to numerical prediction models. There are many flavors, depending on whether one takes a probabilistic/statistical, control theoretic, or dynamical systems point of view. Furthermore there are variational methods that consider a whole time window and sequential methods that proceed step-by-step. In this talk I will consider the relationship between the observation operator and the decomposition of the model tangent space in terms of Lyapunov exponents/vectors. The main conclusion is that the observations should constrain the unstable tangent space. Using this point of view we construct two methods, one variational and one sequential and discuss their convergence behavior. Along the way I will mention some other structural considerations in data assimilation.

14 March: **Heinz Han****ß****mann** (Utrecht), 16:00-17:00, Room WN-S623

*Title:* Bifurcations and Monodromy of the Axially Symmetric 1:1:-2 Resonance*Abstract:* We consider integrable Hamiltonian systems in three degrees of freedom near an elliptic equilibrium in 1:1:-2 resonance. The integrability originates from averaging along the periodic motion of the quadratic part and an imposed rotational symmetry about the vertical axis. Introducing a detuning parameter we find a rich bifurcation diagram, containing three parabolas of Hamiltonian Hopf bifurcations that join at the origin. We describe the monodromy of the resulting ramified 3-torus bundle as variation of the detuning parameter lets the system pass through 1:1:-2 resonance.

28 February: **Alef Sterk** (Groningen), 16:00-17:00, **Room F3.20 (KdV, UvA)**

*Title:* Extreme value laws for dynamical systems*Abstract:* Extreme value theory for chaotic, deterministic dynamical systems is a rapidly expanding area of research. Given a dynamical system and a real-valued observable defined on its state space, extreme value theory studies the limit probabilistic laws for asymptotically large values attained by the observable along orbits of the system. Under suitable mixing conditions the extreme value laws are the same as those for stochastic processes of i.i.d. random variables. In this talk I will discuss the classical results for i.i.d. processes, some recently obtained results for dynamical systems, and promising directions for future research.

14 February: **Fabian Ziltener** (Utrecht), 16:00-17:00, Room WN-S623

*Title:***Coisotropic submanifolds of symplectic manifolds and leafwise fixed points for C^0-small Hamiltonian flows**

*Abstract:* Consider a symplectic manifold (M,\omega), a closed coisotropic submanifold N of M, and a Hamiltonian diffeomorphism \phi on M. A leafwise fixed point for \phi is a point x\in N that under \phi is mapped to its isotropic leaf. These points generalize fixed points and Lagrangian intersection points. In classical mechanics leafwise fixed points correspond to trajectories that are changed only by a time-shift, when an autonomous mechanical system is perturbed in a time-dependent way. J. Moser posed the following problem: Find conditions under which leafwise fixed points exist. A special case of this problem is V.I. Arnold's conjecture about fixed points of Hamiltonian diffeomorphisms. In my talk I will provide the following solution to Moser's problem. Namely, leafwise fixed points exist, provided that the Hamiltonian diffeomorphism is the time-1-map of a Hamiltonian flow whose restriction to N stays C^0-close to the inclusion N\to M.