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| Name |
Title |
Date |
Time |
Room |
| Raphael Krikorian (Université de Cergy-Pontoise) | Exotic rotation domains and Herman rings for quadratic Hénon mapsAbstract: Quadratic Hénon maps are polynomial automorphism of $\mathbb{C}^2$ of the form $h:(x,y)\mapsto (\lambda^{1/2}(x^2+c)-\lambda y,x)$. They have constant Jacobian equal to $\lambda$ and they admit two fixed points. If $\lambda$ is on the unit circle (one says the map $h$ is conservative) these fixed points can be elliptic or hyperbolic. In the elliptic case, a simple application of Siegel Theorem shows (under a Diophantine assumption) that $h$ admits many quasi-periodic orbits with two frequencies in the neighborhood of its fixed points. Surprisingly, in some hyperbolic cases, S. Ushiki observed some years ago what seems to be quasi-periodic orbits (though no Siegel disks exist). I will explain why this is the case. This theoretical framework also predicts (and proves), in the dissipative case ($\lambda$ of module less than 1), the existence of (attractive) Herman rings. These Herman rings, which were not observed before, can be produced in numerical experiments. |
Tuesday, 30 April 2024 |
11:00 |
Huxley 340 |
| Artur Avila (Universität Zürich) | Renormalization, Fractal Geometry and the Newhouse Phenomenon Abstract: As discovered by Poincaré in the end of the 19th century, even small perturbations of very regular dynamical systems may display chaotic features, due to complicated interactions near a homoclinic point. In the 1960's Smale attempted to understand such dynamics in term of a stable model, the horseshoe, but this was too optimistic.Indeed, Newhouse showed that even in only two dimensions, a homoclinic bifurcation gives rise to particular wild dynamics, such as the generic presence of infinitely many attractors. This Newhouse phenomenon is associated to a renormalization mechanism, but also with particular geometric properties of some fractal sets within a Smale horseshoe. When considering two-dimensional complex dynamics those fractal sets become much more beautiful but unfortunately also more difficult to handle. |
Tuesday, 30 April 2024 |
15:00 |
Huxley 340 |
| Peter Giesl (University of Sussex) | Solving differential inequalities with applications to complete Lyapunov functionsAbstract: Meshfree collocation can be used to solve partial differential equations in a Reproducing Kernel Hilbert space by discretising the problem, which leads to a system of linear equations. In this talk, we seek to solve partial differential inequalities which leads to a quadratic programming problem. We discuss the discretised problem and prove the convergence of solutions of the discretised problem to the original one.
Furthermore, we apply the theory to the computation of complete Lyapunov functions for ODEs. These are functions which characterise the dynamics of the ODE; in particular, they serve to find the connected components of the chain-recurrent set, consisting of attractors and repellers, and to determine their stability. This is joint work with Holger Wendland (Bayreuth), Stefan Suhr (Bochum), Sigurdur Hafstein and Carlos Argaez (Iceland). |
Tuesday, 7 May 2024 |
13:00 |
Huxley 140 |
| Emmanuel Fleurentin (George Mason University and UNC Chapel Hill) | Investigating Most Probable Escape Paths over Periodic Boundaries: a Dynamical Systems ApproachAbstract: Noise-induced tipping (N-tipping) emerges when random fluctuations prompt transitions from one (meta)stable state to another, potentially as a rare event. In this talk, we delineate new techniques for determining Most Probable Escapes Paths (MPEPs) in stochastic differential equations over periodic boundaries. We utilize a dynamical system approach to unravel MPEPs for the intermediate noise regime. We discuss the framework for computing the MPEPs by first looking at intersections of stable and unstable manifolds of invariant sets of a Hamiltonian system derived from the Euler-Lagrange equations of the Freidlin-Wentzell (FW) functional. The Maslov index helps identify which critical points of the FW functional are local minimizers and assists in explaining the effects of the interaction of noise and the deterministic flow. The Onsager-Machlup functional, which is treated as a perturbation of the FW functional, will provide a selection mechanism to pick out the MPEP. We will illustrate our approach and compare our theoretical prediction with Monte Carlo simulations in the Inverted Van der Pol system and a carbon cycle model. |
Tuesday, 18 June 2024 |
13:00 |
Huxley 140 |
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