Seminars and Colloquia by Series

Periodic approximation of Lyapunov exponents for cocycles over hyperbolic systems.

Series
CDSNS Colloquium
Time
Monday, February 18, 2019 - 10:10 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Victoria Sadovskaya – Penn State
We consider a hyperbolic dynamical system (X,f) and a Holder continuous cocycle A over (X,f) with values in GL(d,R), or more generally in the group of invertible bounded linear operators on a Banach space. We discuss approximation of the Lyapunov exponents of A in terms of its periodic data, i.e. its return values along the periodic orbits of f. For a GL(d,R)-valued cocycle A, its Lyapunov exponents with respect to any ergodic f-invariant measure can be approximated by its Lyapunov exponents at periodic orbits of f. In the infinite-dimensional case, the upper and lower Lyapunov exponents of A can be approximated in terms of the norms of the return values of A at periodic points of f. Similar results are obtained in the non-uniformly hyperbolic setting, i.e. for hyperbolic invariant measures. This is joint work with B. Kalinin.

Fluctuation of ergodic sums over periodic orbits

Series
CDSNS Colloquium
Time
Thursday, January 17, 2019 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Manfred Denker – Penn State University –
The fluctuations of ergodic sums by the means of global and local specifications on periodic points will be discussed. Results include a Lindeberg-type central limit theorems in both setups of specification. As an application, it is shown that averaging over randomly chosen periodic orbits converges to the integral with respect to the measure of maximal entropy as the period approaches infinity. The results also suggest to decompose the variances of ergodic sums according to global and local sources.

The Technique to Solve the Variable Coefficients Homological Equations

Series
CDSNS Colloquium
Time
Monday, December 17, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hongyu Cheng – MSRI & Nankai University
In the infinite-dimensional KAM theory, solving the homological equations is the one of the main parts. Generally, the coefficients of the homological equations are constants, by comparing the coefficients of the functions, it is easy to solve these equations. If the coefficients of homological equations depend on the angle variables, we call these equations as the variable coefficients homological equations. In this talk we will talk about how to solve these equations.

Gradient-like dynamics: motion near a manifold of quasi-equilibria

Series
CDSNS Colloquium
Time
Monday, September 24, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter Bates – Michigan State University
This concerns general gradient-like dynamical systems in Banach space with the property that there is a manifold along which solutions move slowly compared to attraction in the transverse direction. Conditions are given on the energy (or, more generally, Lyapunov functional) that ensure solutions starting near the manifold stay near for a long time or even forever. Applications are given with the vector Allen-Cahn and Cahn-Morral equations. This is joint work with Giorgio Fusco and Georgia Karali.

Chaotic Transition States on the Monkey Saddle

Series
CDSNS Colloquium
Time
Monday, April 16, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Thomas Bartsch – Loughborough University

Please Note: Transition State Theory describes how a reactive system crosses an energy barrier that is marked by a saddle point of the potential energy. The transition from the reactant to the product side of the barrier is regulated by a system of invariant manifolds that separate trajectories with qualitatively different behaviour. The situation becomes more complex if there are more than two reaction channels, or possible outcomes of the reaction. Indeed, the monkey saddle potential, with three channels, is known to exhibit chaotic dynamics at any energy. We investigate the boundaries between initial conditions with different outcomes in an attempt to obtain a qualitative and quantitative description of the relevant invariant structures.

TBA

Occupation times

Series
CDSNS Colloquium
Time
Monday, April 2, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Manfred Heinz Denker – Penn State University
Consider a $T$-preserving probability measure $m$ on a dynamical system $T:X\to X$. The occupation time of a measurable function is the sequence $\ell_n(A,x)$ ($A\subset \mathbb R, x\in X$) defined as the number of $j\le n$ for which the partial sums $S_jf(x)\in A$. The talk will discuss conditions which ensure that this sequence, properly normed, converges weakly to some limit distribution. It turns out that this distribution is Mittag-Leffler and in particular the result covers the case when $f\circ T^j$ is a fractal Gaussian noise of Hurst parameter $>3/4$.

Invariant Measures for the derivative nonlinear Schrödinger equation

Series
CDSNS Colloquium
Time
Monday, March 12, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Giuseppe Genovese – University of Zurich
The derivative nonlinear Schrödinger equation (DNLS) is an integrable, mass-critical PDE. The integrals of motion may be written as an infinite sequence of functionals on Sobolev spaces of increasing regularity. I will show how to associate to them a family of invariant Gibbs measures, if the L^2 norm of the solution is sufficiently small (mass-criticality). A joint work with R. Lucà (Basel) and D. Valeri (Beijing).

Oscillatory motions for the restricted three body problem

Series
CDSNS Colloquium
Time
Tuesday, March 6, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Marcel Guardia – Universitat Politècnica de Catalunya
The restricted three body problem models the motion of a body of zero mass under the influence of the Newtonian gravitational force caused by two other bodies, the primaries, which describe Keplerian orbits. In 1922, Chazy conjectured that this model had oscillatory motions, that is, orbits which leave every bounded region but which return infinitely often to some fixed bounded region. Its existence was not proven until 1960 by Sitnikov in a extremely symmetric and carefully chosen configuration. In 1973, Moser related oscillatory motions to the existence of chaotic orbits given by a horseshoe and thus associated to certain transversal homoclinic points. Since then, there has been many atempts to generalize their result to more general settings in the restricted three body problem.In 1980, J. Llibre and C. Sim\'o, using Moser ideas, proved the existence of oscillatory motions for the restricted planar circular three body problem provided that the ratio between the masses of the two primaries was arbitrarily small. In this talk I will explain how to generalize their result to any value of the mass ratio. I will also explain how to generalize the result to the restricted planar elliptic three body problem. This is based on joint works with P. Martin, T. M. Seara. and L. Sabbagh.

Computing rotation numbers from a quasiperiodic trajectory

Series
CDSNS Colloquium
Time
Monday, March 5, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Evelyn Sander – George Mason University
A trajectory is quasiperiodic if the trajectory lies on and is dense in some d-dimensional torus, and there is a choice of coordinates on the torus for which F has the form F(t) = t + rho (mod 1) for all points in the torus, and for some rho in the torus. There is an extensive literature on determining the coordinates of the vector rho, called the rotation numbers of F. However, even in the one-dimensional case there has been no general method for computing the vector rho given only the trajectory (u_n), though there are plenty of special cases. I will present a computational method called the Embedding Continuation Method for computing some components of r from a trajectory. It is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem. There is however a caveat; the coordinates of the rotation vector depend on the choice of coordinates of the torus. I will give a statement of the various sets of possible rotation numbers that rho can yield. I will illustrate these ideas with one- and two-dimensional examples.

Shilnikov bifurcations in the Hopf-zero singularity

Series
CDSNS Colloquium
Time
Monday, February 26, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Tere M. Seara – Departament de Matemàtiques. Universitat Politècnica de Catalunya (UPC)
The so-called Hopf-zero singularity consists in a vector field in $\mathbf{R}^3$ having the origin as a critical point, with a zero eigenvalue and a pair of conjugate purely imaginary eigenvalues. Depending of the sign in the second order Taylor coefficients of the singularity, the dynamics of its unfoldings is not completely understood. If one considers conservative (i.e. one-parameter) unfoldings of such singularity, one can see that the truncation of the normal form at any order possesses two saddle-focus critical points with a one- and a two-dimensional heteroclinic connection. The same happens for non-conservative (i.e. two-parameter) unfoldings when the parameters lie in a certain curve (see for instance [GH]).However, when one considers the whole vector field, one expects these heteroclinic connections to be destroyed. This fact can lead to the birth of a homoclinic connection to one of the critical points, producing thus a Shilnikov bifurcation. For the case of $\mathcal{C}^\infty$ unfoldings, this has been proved before (see [BV]), but for analytic unfoldings it is still an open problem.Our study concerns the splittings of the one and two-dimensional heteroclinic connections (see [BCS] for the one-dimensional case). Of course, these cannot be detected in the truncation of the normal form at any order, and hence they are expected to be exponentially small with respect to one of the perturbation parameters. In [DIKS] it has been seen that a complete understanding of how the heteroclinic connections are broken is the last step to prove the existence of Shilnikov bifurcations for analytic unfoldings of the Hopf-zero singularity. Our results [BCSa, BCSb] and [DIKS] give the existence of Shilnikov bifurcations for analytic unfoldings. [GH] Guckenheimer, J. and Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New York (1983), 376--396. [BV] Broer, H. W. and Vegter, G., Subordinate Sil'nikov bifurcations near some singularities of vector fields having low codimension. Ergodic Theory Dynam. Systems, 4 (1984), 509--525. [BSC] Baldoma;, I., Castejon, O. and Seara, T. M., Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity. Journal of Dynamics and Differential Equations, 25(2) (2013), 335--392. [DIKS] Dumortier, F., Ibanez, S., Kokubu, H. and Simo, C., About the unfolding of a Hopf-zero singularity. Discrete Contin. Dyn. Syst., 33(10) (2013), 4435--€“4471. [BSCa] Baldoma, I., Castejon, O. and Seara, T. M., Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (I). Preprint: https://arxiv.org/abs/1608.01115 [BSCb] Baldoma, I., Castejon, O. and Seara, T. M., Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (II). The generic case. Preprint: https://arxiv.org/abs/1608.01116

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