SIAM Georgia Tech Student Conference 2019

Abstracts

String and Escape Rate

• Speaker: Hassan Attarchi (Georgia Tech)
• Abstract:
  • It is well-known that the survival probability depends on the position of a hole in an open dynamical system. Here, we consider doubling map as a strongly chaotic map and a hole which is an element from the partition of $2^n$ subintervals of the same length on $[0, 1]$. Thus, we can use a binary string of length n to determine the hole position and with help of that string we can find all periods in the hole. Then, the faster escape (smaller survival probability) occurs through the hole where the minimal period of the hole assumes its maximal value among those of other holes. Working with string not only provides us some information about the survival probability but also tells us how many holes with same periods exist in our partition. Moreover, if we consider all holes of the same period in a class then we can also approximation the number of these classes of different periods.

Volumes of polytopes associated with chemical reaction networks

• Speaker: Cvetelina Hill (Georgia Tech)
• Abstract:
  • The steady-state degree of a chemical reaction network is the number of complex steady-states, which gives an upper bound for the number of positive real steady-states. In general, the steady-state degree may be dicult to compute. Here, we give an upper bound to the steady-state degree of a reaction network by utilizing the underlying polyhedral geometry associated with the corresponding polynomial system. We focus on three case studies of innite families of networks, each generated by joining smaller networks to create larger ones. For each family, we state results on the steady-state degree or give an upper bound in terms of the mixed volume of the corresponding polynomial system.

Computation and Usage of High-Order Resonant Orbit Manifold Expansions For Mission Design In the Planar Circular Restricted $3$-Body Problem

• Speaker: Bhanu Kumar (Georgia Tech)
• Abstract:
  • In recent years, stable and unstable manifolds of invariant objects (such as libration points and periodic orbits) have been increasingly recognized as an efficient tool for designing transfer trajectories in space missions. However, most methods currently used in mission design rely on using eigenvectors of the linearized dynamics as local approximations of the manifolds. Since such approximations are not accurate except very close to the base invariant object, this requires large amounts of numerical integration to globalize the manifolds and locate intersections.
  • The goal of this talk is to discuss methods of studying hyperbolic resonant periodic orbits in the planar circular restricted $3$-body problem, and transfer trajectories between them, by: 1) determining where to search for resonant periodic orbits; 2) developing efficient and accurate methods for computation and parametrization of their invariant manifolds; and 3) developing an algorithm to compute intersections of the stable and unstable manifolds. We develop and implement algorithms that accomplish these three goals, and apply them to the problem of transferring between resonances in the Jupiter-Europa system

On the Stability of the Kevin-Stuart Cat’s Eyes Flow

• Speaker: Shasha Liao (Georgia Tech)
• Abstract:
  • Discovered by Stuart (1967), the Kevin-Stuart Cat’s Eyes flows are a class of important steady solutions of the $2$D Euler equation. We study the linear stability of them in the domain $\mathbb{T}_{2\pi} \times \mathbb{R}$. Both analytical and numerical methods are implemented. The analytical part relies on the index theorem for general linear Hamiltonian PDEs developed by Lin and Zeng (2017). The numerical part mainly focuses on studying the spectrum of the approximation linear operator using spectral methods. Our preliminary results include the stability under co-periodic perturbations and instability under perturbations with a different period. This is a joint work with Zhiwu Lin.

Response solutions for quasi-periodic forced strongly dissipative systems with any frequency vectors via fixed point theorem

• Speaker: Fenfen Wang (Shandong University and Georgia Tech)
• Abstract:
  • We consider several models of strongly dissipative systems, (including both ordinary differential equations (ODEs) and partial differential equations (PDEs), possibly ill-posed), subject to very strong damping and quasi-periodic external forcing. We study the existence of response solutions (i.e., quasi-periodic solutions with the same frequency as the forcing). Under analyticity assumptions on the nonlinear term, forcing term and other suitable hypothesis, without any arithmetic condition on the forcing frequency $\omega$, we show that the response solutions indeed exist and depend analytically on $\varepsilon$ (where $\varepsilon$ is the inverse of the coefficients multiplying the damping) in a complex domain, which accumulates the origin. Moreover, these response solutions depend continuously on $\varepsilon$ when $\varepsilon\rightarrow 0$. We allow multidimensional systems and we do not require that the unperturbed equations are Hamiltonian.
  • In the proof of our results, we first reformulate the existence of response solutions as a fixed point problem in appropriate Banach spaces and then use the contraction mapping principle to get the solutions.
  • One advantage of the present method is that it gives results for finitely differentiable nonlinearity and forcing. As a matter of fact, we do not even need that the forcing is continuous. We have results when the forcing is in $L^2$ or in $H^1$.

$(-1)$-homogeneous solutions of stationary incompressible Navier-Stokes equations with singular rays

• Speaker: Xukai Yan (Georgia Tech)
• Abstract:
  • In 1944, L.D. Landau first discovered explicit $(-1)$-homogeneous solutions of $3$-d stationary incompressible Navier-Stokes equations (NSE) with precisely one singularity at the origin, which are axisymmetric with no swirl. These solutions are now called Landau solutions. In 2006 V. Sverak proved that with just the $(-1)$-homogeneous assumption Landau solutions are the only solutions with one singularity. Our work focuses on the $(-1)$-homogeneous solutions of $3$-d incompressible stationary NSE with finitely many singularities on the unit sphere.
  • In this talk we will first classify all $(-1)$-homogeneous axisymmetric no-swirl solutions of $3$-d stationary incompressible NSE with one singularity at the south pole on the unit sphere as a two dimensional solution surface. We will then present our results on the existence of a one parameter family of $(-1)$-homogeneous axisymmetric solutions with non-zero swirl and smooth on the unit sphere away from the south pole, emanating from the two dimensional surface of axisymmetric no-swirl solutions. We will also present asymptotic behavior of general $(-1)$-homogeneous axisymmetric solutions in a cone containing the south pole with a singularity at the south pole on the unit sphere. We also constructed families of solutions smooth on the unit sphere away from the north and south poles.
  • This is a joint work with Professor Yanyan Li and Li Li.