## Seminars and Colloquia by Series

Monday, May 12, 2014 - 11:00 , Location: Skiles 006 , Jordi-Lluis Figueras Romero , Department of Mathematics, Uppsala University , Organizer:
We provide several explicit examples of 3D quasiperiodic linear skew-products with simple Lyapunov spectrum, that is with 3 different Lyapunov multipliers, for which the corresponding Oseledets bundles are measurable but not continuous, colliding in a measure zero dense set.
Friday, April 18, 2014 - 11:00 , Location: Skiles 006 , Professor Joe Auslander , University of Maryland , Organizer: Yingfei Yi
Let (X, T) be a flow, that is a continuous left action of the group T on the compact Hausdorff space X. The proximal P and regionally proximal RP relations are dened, respectively (assuming X is a metric space) by P = {(x; y) | if \epsilon > 0 there is a t \in T such that d(tx, ty) < \epsilon} and RP = {(x; y) | if \epsilon > 0 there are x', y' \in X and t \in T such that d(x; x') < \epsilon, d(y; y') < \epsilon and t \in T such that d(tx'; ty') < \epsilon}. We will discuss properties of P and RP, their similarities and differences, and their connections with the distal and equicontinuous structure relations. We will also consider a relation V defined by Veech, which is a subset of RP and in many cases coincides with RP for minimal flows.
Monday, March 24, 2014 - 15:00 , Location: Skiles 005 , Professor Ken Palmer , Providence University, Taiwan , Organizer: Yingfei Yi
Theoretical aspects: If a smooth dynamical system on a compact invariant set is structurally stable, then it has the shadowing property, that is, any pseudo (or approximate) orbit has a true orbit nearby. In fact, the system has the Lipschitz shadowing property, that is, the distance between the pseudo and true orbit is at most a constant multiple of the local error in the pseudo orbit. S. Pilyugin and S. Tikhomirov showed the converse of this statement for discrete dynamical systems, that is, if a discrete dynamical system has the Lipschitz shadowing property, then it is structurally stable. In this talk this result will be reviewed and the analogous result for flows, obtained jointly with S. Pilyugin and S. Tikhomirov, will be described. Numerical aspects: This is joint work with Brian Coomes and Huseyin Kocak. A rigorous numerical method for establishing the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous system of ordinary differential equations is presented. Given a suitable approximate connecting orbit and assuming that a certain associated linear operator is invertible, the existence of a true connecting orbit near the approximate orbit and for a nearby parameter value is proved provided the approximate orbit is sufficiently good''. It turns out that inversion of the operator is equivalent to the solution of a boundary value problem for a nonautonomous inhomogeneous linear difference equation. A numerical procedure is given to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse (the latter determines how good'' the approximating orbit must be).
Monday, March 24, 2014 - 11:00 , Location: Skiles 005 , Professor Michael Li , Univeristy of Alberta , , Organizer: Yingfei Yi
Many complex models from science and engineering can be studied in the framework of coupled systems of differential equations on networks. A network is given by a directed graph. A local system is defined on  each vertex, and directed edges represent couplings among vertex  systems. Questions such as stability in the large, synchronization,  and complexity in terms of dynamic clusters are of interest. A more  recent approach is to investigate the connections between network  topology and dynamical behaviours. I will present some recent results  on the construction of global Lyapunov functions for coupled systems  on networks using a graph theoretic approach, and show how such  a construction can help us to establish global behaviours of compelx  models.
Monday, February 24, 2014 - 11:00 , Location: Skiles 005 , Aynur Bulut , Univ. of Michigan , Organizer: Rafael de la Llave
In this talk we will discuss recent work, obtained in collaboration with Jean Bourgain, on new global well-posedness results along Gibbs measure evolutions for the radial nonlinear wave and Schr\"odinger equations posed on the unit ball in two and three dimensional Euclidean space, with Dirichlet boundary conditions. We consider initial data chosen according to a Gaussian random process associated to the Gibbs measures which arise from the Hamiltonian structure of the equations, and results are obtained almost surely with respect to these probability measures. In particular, this renders the initial value problem supercritical in the sense that there is no suitable local well-posedness theory for the corresponding deterministic problem, and our results therefore rely essentially on the probabilistic structure of the problem. Our analysis is based on the study of convergence properties of solutions. Essential ingredients include probabilistic a priori bounds, delicate estimates on fine frequency interactions, as well as the use of invariance properties of the Gibbs measure to extend the relevant bounds to arbitrarily long time intervals.
Monday, February 17, 2014 - 11:00 , Location: Skiles 006 , Chunhua Shan , School of Mathematics, Georgia Institute of Technology , Organizer:
In 1994, Dumortier, Roussarie and Rousseau launched a program aiming at proving the ﬁniteness part of Hilbert’s 16th problem for the quadratic system. For the program, 121 graphics need to be proved to have ﬁnite cyclicity. In this presentation, I will show that 4 families of HH-graphics with a triple nilpotent singularity of saddle or elliptic type have finite cyclicity. Finishing the proof of the cyclicity of these 4 families of HH-graphics represents one important step towards the proof of the finiteness part of Hilbert’s 16th problem for quadratic systems. This is a joint work with Professor Christiane Rousseau and Professor Huaiping Zhu.
Friday, February 7, 2014 - 15:00 , Location: Skiles 06 , Alex Haro , Univ. of Barcelona , Organizer: Rafael de la Llave
This talk is devoted to quasi-periodic Schrödinger operators beyond theAlmost Mathieu, with more general potentials and interactions. The  linksbetween the spectral properties of these operators and the dynamicalproperties of the associated quasi-periodic linear skew-products rule thegame. In particular, we present a Thouless formula  and some consequencesof Aubry duality. This is a joint work with Joaquim Puig~
Monday, February 3, 2014 - 11:00 , Location: Skiles05 , Marta Canadell , Univ. of Barcelona , Organizer: Rafael de la Llave
We present a KAM-like theorem for the existence of quasi-periodic tori with a prescribed Diophantine rotation for a discrete family of dynamical system. The theorem is stated in an a posteriori format, so it can be used to validate numerical computations. The method of proof provides an efficient algorithm for computing quasi-periodic tori. We also present implementations of the algorithm, illustrating them throught several examples and observing different mechanisms of breakdown of qp invariant tori. This is a joint work with Alex Haro.
Wednesday, January 22, 2014 - 11:00 , Location: Skiles 006 , Renato Calleja , IIMAS UNAM , Organizer: Rafael de la Llave
We present a numerical  study of the dynamics of a state-dependent delay equation with two state dependent delays that are linear in the state. In particular, we study some of the the dynamical behavior driven by the existence of two-parameter families of invariant tori. A formal normal form analysis predicts the existence of torus bifurcations and the appearance of a two parameter family of stable invariant tori. We investigate the dynamics on the torus thought a Poincaré section. We find some boundaries of Arnold tongues and indications of loss of normal hyperbolicity for this stable family. This is joint work with A. R. Humphries and B. Krauskopf.
Friday, January 17, 2014 - 11:05 , Location: Skiles 005 , , University of British Columbia, Vancouver. , Organizer:
(Joint work with A. Avila and S. Jitomirskaya). An analytic, complex, one-frequency cocycle is given by a pair $(\alpha,A)$ where $A(x)$ is an analytic and 1-periodic function that maps from the torus $\mathbb(R) / \mathbb(Z)$ to the complex $d\times d$ matrices and $\alpha \in [0,1]$ is a frequency. The pair is interpreted as the map $(\alpha,A)\,:\, (x,v) \mapsto (x+\alpha), A(x) v$. Associated to the iterates of this map are (averaged) Lyapunov exponents $L_k(\alpha,A)$ and an Osceledets filtration. We prove joint-continuity in $(\alpha,A)$ of the Lyapunov exponents at irrational frequencies $\alpha$, give a criterion for domination and prove that for a dense open subset of cocycles, the Osceledets filtration comes from a dominated splitting which is an analogue to the Bochi-Viana Theorem.