Georgia Institute of Technology College of Sciences

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

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.

We present a methodology to rigorously validate a given approximation of a quasi-periodic Lagrangian torus of a symplectic map. The approach consists in verifying the hypotheses of a-posteriori KAM theory based of the parameterization method (following Rafael de la Llave and collaborators). A crucial point of our imprementation is an analytic Lemma that allows us to control the norm of periodic functions using their discrete Fourier transform. An outstanding consequence of this approach it that the computational cost of the validation is assymptotically equivalent of the cost of the numerical computation of invariant tori using the parametererization method. We pretend to describe some technical aspects of our implementation. This is a work in progress joint with Jordi-Lluis Figueras and Alejandro Luque.

The Standard Map is a discrete time area-preserving dynamical system and is one of the simplest of such systems to exhibit chaotic dynamics. Traditional studies of the Standard Map have employed symmetric forcing functions that do not induce a net flux. Although the dynamics of these maps is rich there are many systems which cannot be modeled with these restrictions. In this talk we will explore the dynamics of the Standard Map when the forcing is asymmetric and induces a positive flux on the system. We will introduce new numerical methods to study these dynamics and give an overview of how transport in the system changes under these new forces.