Seminars and Colloquia by Series

Invariant Measures for the derivative nonlinear Schrödinger equation

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CDSNS Colloquium
Time
Monday, March 12, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Giuseppe GenoveseUniversity 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

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CDSNS Colloquium
Time
Tuesday, March 6, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Marcel GuardiaUniversitat 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

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CDSNS Colloquium
Time
Monday, March 5, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Evelyn SanderGeorge 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

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CDSNS Colloquium
Time
Monday, February 26, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Tere M. SearaDepartament 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

Self-Excited Vibrations for Higher Dimensional Damped Wave Equations

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CDSNS Colloquium
Time
Monday, February 19, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Nemanja KosovalicUniversity of Southern Alabama
Using techniques from local bifurcation theory, we prove the existence of various types of temporally periodic solutions for damped wave equations, in higher dimensions. The emphasis is on understanding the role of external bifurcation parameters and symmetry, in generating the periodic motion. The work presented is joint with Brian Pigott

Discrete stochastic Hamilton Jacobi equation

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CDSNS Colloquium
Time
Monday, February 5, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Renato IturriagaCIMAT
We present a discrete setting for the viscous Hamilton Jacobi equation, and prove convergence to the continuous case.

KAM for quasi-linear and fully nonlinear PDEs

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CDSNS Colloquium
Time
Monday, February 5, 2018 - 10:10 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Riccardo MontaltoUniversity of Zurich
In this talk I will present some results concerning the existence and the stability of quasi-periodic solutions for quasi-linear and fully nonlinear PDEs. In particular, I will focus on the Water waves equation. The proof is based on a Nash-moser iterative scheme and on the reduction to constant coefficients of the linearized PDE at any approximate solution. Due to the non-local nature of the water waves equation, such a reduction procedure is achieved by using techniques from Harmonic Analysis and microlocal analysis, like Fourier integral operators and Pseudo differential operators.

Multiplicity results for some classes of non-local elliptic equations

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CDSNS Colloquium
Time
Monday, January 29, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Xifeng SuBeijing Normal University
We will consider the nonlinear elliptic PDEs driven by the fractional Laplacian with superlinear or asymptotically linear terms or combined nonlinearities. An L^infinity regularity result is given using the De Giorgi-Stampacchia iteration method. By the Mountain Pass Theorem and other nonlinear analysis methods, the local and global existence and multiplicity of non-trivial solutions for these equations are established. This is joint work with Yuanhong Wei.

A posteriori KAM theorems for systems with first integrals in involution

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CDSNS Colloquium
Time
Monday, January 22, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Alex HaroUniversity of Barcelona
Some relevant Hamiltonian systems in Celestial Mechanics have first integrals in involution. A classic technique to study such systems, known as symplectic reduction, is based in reducing the number of degrees of freedom by using the first integrals. In this talk we present two a posteriori KAM theorems for Hamiltonian systems with first integrals in involution, including the isoenergetic case, without using symplectic reduction. The approach leads to efficient numerical methods and validating techniques.This is a joint work with Alejandro Luque.

Symbolic computations of homoclinic chaos

Series
CDSNS Colloquium
Time
Monday, December 11, 2017 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Andrey ShilnikovGeorgia State University
Over recent years, a great deal of analytical studies and modeling simulations have been brought together to identify the key signatures that allow dynamically similar nonlinear systems from diverse origins to be united into a single class. Among these key structures are bifurcations of homoclinic and heteroclinic connections of saddle equilibria and periodic orbits. Such homoclinic structures are the primary cause for high sensitivity and instability of deterministic chaos in various systems. Development of effective, intelligent and yet simple algorithms and tools is an imperative task for studies of complex dynamics in generic nonlinear systems. The core of our approach is the reduction of the time evolution of a characteristic observable in a system to its symbolic representation to conjugate or differentiate between similar behaviors. Of our particular consideration are the Lorenz-like systems and systems with spiral chaos due to the Shilnikov saddle-focus. The proposed approach and tools will let one detect homoclinic and heteroclinic orbits, and carry out state of the art studies homoclinic bifurcations in parameterized systems of diverse origins.

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