Exponentially Small Splitting: A Direct Approach
DateTuesday, February 5, 2019 - 12:30pm
AbstractThis talk is on a study of the splitting distance of the stable and unstable manifold of a Dung equation subject to a time-periodic non-Hamiltonian perturbation. We introduce a simple recursion to write high order splitting distances as collections of certain well-structured multiple integrals, which we name as high order Melnikov integrals. Let ! be the forcing frequency. We also develop an analytic scheme to evaluate high order Melnikov integrals to extract an exponentially small factor out of splitting distances of all order for large !. We reveal that exponentially small splitting is not a phenomenon tied exclusively to Hamiltonian perturbations. It is rather induced by a certain symmetry embedded in the kernel functions of high order Melnikov integrals. This symmetry is beheld by many non-Hamiltonian equation
On the Homoclinic Tangles of Henri Poincar
DateTuesday, February 12, 2019 - 12:30pm
AbstractThis talk is on the dynamics of the time-periodic second order equations.
We assert that Smale’s horseshoe, SRB measure of Benedick-Carleson and Young, and
Newhouse sinks are all participating elements of the homoclinic tangles of this equation.
We further assert that homoclinic tangles of different structures are arranged in a fixed
pattern and this pattern is repeated indefinitely in parameter space.
Modeling and Analysis of patterns in multi-constituent systems with long range interaction
DateTuesday, February 19, 2019 - 12:30pm
AbstractSkin pigmentation, animal coats and block copolymers can be considered as multi-constituent inhibitory systems. Exquisitely structured patterns arise as orderly outcomes of the self-organization principle. Analytically, via the sharp interface model, patterns can be studied as nonlocal geometric variational problems. The free energy functional consists of an interface energy and a long range Coulomb-type interaction energy. The admissible class is a collection of Caccioppoli sets with fixed volumes. To overcome the difficulty that the admissible class is not a Hilbert space, we introduce internal variables. Solving the energy functional for stationary sets is recast as a variational problem on a Hilbert space. We prove the existence of a core-shell assembly and the existence of disc assemblies in ternary systems and also a triple-bubble-like stationary solution in a quaternary system. Numerically, via the diffuse interface model, one open question related to the polarity direction of double bubble assemblies is answered. Moreover, it is shown that the average size of bubbles in a single bubble assembly depends on the sum of the minority constituent volumes and the long range interaction coefficients. One further identifies the ranges for volume fractions and the long range interaction coefficients for double bubble assemblies.
Discrete Conformal and Harmonic Maps for Surface Analysis
DateTuesday, February 26, 2019 - 12:30pm
AbstractConformal and harmonic maps have many nice properties, many due to the fact that they are solutions of elliptic PDE. We will briefly describe some applications and potential applications of conformal and harmonic mappings of domains and surfaces. We will also describe some techniques of discretizing these mappings in a way that preserves some geometric structure. The eventual goal is to have a robust theory of discrete mappings that maintain nice geometric properties to enable efficient computation and robustness of approximation. We will draw some connections between discrete geometry (e.g., Delaunay triangulations) and numerical analysis (e.g., finite element methods).