Monday, September 14, 2015 - 11:00 , Location: Skiles 005 , Willam T. Gignac , Georgia Tech (Math) , Organizer: Rafael de la Llave
Let f be a rational self-map of the complex projective plane. A central problem when analyzing the dynamics of f is to understand the sequence of degrees deg(f^n) of the iterates of f. Knowing the growth rate and structure of this sequence in many cases enables one to construct invariant currents/measures for dynamical system as well as bound its topological entropy. Unfortunately, the structure of this sequence remains mysterious for general rational maps. Over the last ten years, however, an approach to the problem through studying dynamics on spaces of valuations has proved fruitful. In this talk, I aim to discuss the link between dynamics on valuation spaces and problems of degree/order growth in complex dynamics, and discuss some of the positive results that have come from its exploration.
Friday, September 11, 2015 - 15:00 , Location: Skiles 006 , Yannick Sire , John Hopkins University , Organizer: Rafael de la Llave
I will describe several recent results with N. Nadirashvili where we construct extremal metrics for eigenvalues on riemannian surfaces. This involves the study of a Schrodinger operator. As an application, one gets isoperimetric inequalities on the 2-sphere for the third eigenvalue of the Laplace Beltrami operator.
Construction of quasi-periodic solutions of State-dependent delay differential equations by the parameterization methodWednesday, September 9, 2015 - 11:00 , Location: Skiles 006 , Xiaolong He , Georgia Tech (Math)/Hunan University , Organizer: Rafael de la Llave
We investigate the existence of quasi-periodic solutions for state-dependent delay differential equationsusing the parameterization method, which is different from the usual way-working on the solution manifold. Under the assumption of finite-time differentiability of functions and exponential dichotomy, the existence and smoothness of quasi-periodic solutions are investigated by using contraction arguments We also develop a KAM theory to seek analytic quasi-periodic solutions. In contrast with the finite differentonable theory, this requires adjusting parameters. We prove that the set of parameters which guarantee the existence of analytic quasi-periodic solutions is of positive measure. All of these results are given in an a-posterior form. Namely, given a approximate solution satisfying some non-degeneracy conditions, there is a true solution nearby.
Monday, August 31, 2015 - 11:00 , Location: Skiles 005 , Jiayin Jin , Georgia Inst. of Technology , Organizer: Rafael de la Llave
We construct invariant manifolds of interior multi-spike states for the nonlinear Cahn-Hilliard equation and then investigate the dynamics on it. An equation for the motion of the spikes is derived. It turns out that the dynamics of interior spikes has a global character and each spike interacts with all the others and with the boundary. Moreover, we show that the speed of the interior spikes is super slow, which indicates the long time existence of dynamical multi-spike solutions in both positive and negative time. This result is obtained through the application of a companion abstract result concerning the existence of truly invariant manifolds with boundary when one has only approximately invariant manifolds.
Friday, August 21, 2015 - 15:00 , Location: Skiles 005 , Cinzia Elia , Università degli Studi di Bari , Organizer: Rafael de la Llave
In this talk we examine the typical behavior of a trajectory of a piecewise smooth system in the neighborhood of a co-dimension 2 discontinuity manifold $\Sigma$. It is well known that (in the class of Filippov vector fields, and under commonly occurring conditions) one may anticipate sliding motion on $\Sigma$. However, this motion itself is not in general uniquely defined, and recent contributions in the literature have been trying to resolve this ambiguity either by justifying a particular selection of a Filippov vector field or by substituting the original discontinuous problem with a regularized one. However, in this talk, our concern is different: we look at what we should expect of a typical solution of the given discontinuous system in a neighborhood of $\Sigma$. Our ultimate goal is to detect properties that are satisfied by a sufficiently wide class of discontinuous systems and that (we believe) should be preserved by any technique employed to define a sliding solution on $\Sigma$.
Monday, August 17, 2015 - 23:00 , Location: Skiles 005 , Shangjiang Guo , College of Mathematics and Econometrics, Hunan University , Organizer: Rafael de la Llave
In this talk, the existence, stability, and multiplicity of spatially nonhomogeneous steady-state solution and periodic solutions for a reaction–diffusion model with nonlocal delay effect and Dirichlet boundary condition are investigated by using Lyapunov–Schmidt reduction. Moreover, we illustrate our general results by applications to models with a single delay and one-dimensional spatial domain.
Monday, August 10, 2015 - 11:00 , Location: Skiles 005 , Shangjiang Guo , College of Mathematics and Econometrics, Hunan University , Organizer: Rafael de la Llave
We study the existence and branching patterns of wave trains in a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov-Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and $p:q$ resonance, respectively. We show the impact of the direction $\theta$ of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of travelling waves of the original two-dimensional lattice in the direction $\theta$ of propagation satisfying $\tan\theta$ is rational
Computer assisted proof for coexistence of stationary hexagons and rolls in a spatial pattern formation problem.Wednesday, April 29, 2015 - 11:00 , Location: Skiles 005 , Jason Mireles-James , University of Florida Atlantic , Organizer: Rafael de la Llave
I will discuss a two dimensional spatial pattern formation problem proposed by Doelman, Sandstede, Scheel, and Schneider in 2003 as a phenomenological model of convective fluid flow . In the same work the authors just mentioned use geometric singular perturbation theory to show that the coexistence of certain spatial patterns is equivalent to the existence of some heteroclinic orbits between equilibrium solutions in a four dimensional vector field. More recently Andrea Deschenes, Jean-Philippe Lessard, Jan Bouwe van den Berg and the speaker have shown, via a computer assisted argument, that these heteroclinic orbits exist. Taken together these arguments provide mathematical proof of the existence of some non-trivial patterns in the original planar PDE. I will present some of the ingredients of this computer assisted proof.
Thursday, April 23, 2015 - 13:30 , Location: Skiles 005 , Stuart S. Antman , University of Maryland , Organizer:
This is the 3rd Jorge Ize Memorial lecture, at IIMAS, Mexico City. We will join a videoconference of the event.
The equations governing the motion of a system consisting of a deformable body attached to a rigid body are the partial differential equations for the deformable body subject to boundary conditions that are the equations of motion for the rigid body. (For the ostensibly elementary problem of a mass point on a light spring, the dynamics of the spring itself is typically ignored: The spring is reckoned merely as a feedback device to transmit force to the mass point.) If the inertia of a deformable body is small with respect to that of a rigid body to which it is attached, then the governing equations admit an asymptotic expansion in a small inertia parameter. Even for the simple problem of the spring considered as a continuum, the asymptotics is tricky: The leading term of the regular expansion is not the usual equation for a mass on a massless spring, but is a curious evolution equation with memory. Under very special physical circumstances, an elementary but not obvious process shows that the solution of this equation has an attractor governed by a second-order ordinary differential equation. (This survey of background material is based upon joint work with Michael Wiegner, J. Patrick Wilber, and Shui Cheung Yip.) This lecture describes the rigorous asymptotics and the dimensions of attractors for the motion in space of light nonlinearly viscoelastic rods carrying heavy rigid bodies and subjected to interesting loads. (The motion of the rod is governed by an 18th-order quasilinear parabolic-hyperbolic system.) The justification of the full expansion and the determination of the dimensions of attractors, which gives meaning to these curious equations, employ some simple techniques, which are briefly described (together with some complicated techniques, which are not described). These results come from work with Suleyman Ulusoy.
Monday, April 13, 2015 - 11:00 , Location: Skiles 005 , Alex Haro , Univ. of Barcelona , Organizer: Rafael de la Llave
We present a method to find KAM tori with fixed frequency in degenerate cases, in which the Birkhoff normal form is singular. The method provides a natural classification of KAM tori which is based on Singularity Theory. The method also leads to effective algorithms of computation, and we present some numerical results up to the verge of breakdown. This is a joint work with Alejandra Gonzalez and Rafael de la Llave.