Björn Sandstede

A group of 26 undergraduate students (including 8 Brown students) worked in 8 teams on projects on lead propagation, voting and zebrafish models, equation-free modelling, snaking bifurcations, and Koopman theory. The summaries of the research teams give an overview of the results obtained during the summer.

Chukiat worked on an analytical proof of the transition from localized planar spot patterns to nonlocalized target patterns. He focused on the Swift-Hohenberg equation near onset and used geometric blow-up techniques to study the existence of algebraically decaing spots at onset and their transition to target patterns.

Nadia explored how "snaking diagrams" of localized roll patterns in the Swift-Hohenberg equations change when one moves from dimension one to higher dimensions. She used AUTO to compute the change of the Hamiltonian energy of rolls under the change of dimension and used her results to study snaking diagrams analytically. Nadia found that the snaking diagrams narrow in an exponential fashion and determined that the resulting bifurcation curves narrow onto the so-called Maxwell point, where rolls have zero energy.

Zuhal worked on the spectrum on steady velocity fields that satisfy the Euler equations of incompressible inviscid fluids. Zuhal wrote a suite of Matlab routines that allow her to compute and continue stream functions, calculate the associated vorticity spectra using spectral methods, and compare the shape of eigenfunctions with the particle dynamics generated by the velocity field. She used these scripts to test a recent conjecture on the shape of eigenfunctions associated with point eigenvalues.

Dmitry and Sameer explored degree theory, a theory that produces a robust counting of zeros of functions and can be used to prove many interesting fixed-point theorems including famous results by Brouwer and Borsuk-Ulam. Dmitry and Sameer also studied infinite-dimensional extensions of degree theory, including the Leray-Schauder degree and Schauder's fixed-point theorem.

Nadia continued her previous summer research as an independent study towards an honors thesis. The goal is to find a simple proof that certain algorithms for the computation of Floquet exponents give accurate results without generating spurious spectrum. Nadia found a convenient formulation of this problem that allowed her to prove that spurious solutions are impossible. She also explored the use of operator pencils to prove that the algorithm reproduces the multiplicity of eigenvalues.

Wenhao deepened his understanding of Monte-Carlo simulations. He also studied random walks and explored connections and applications of these topics and techniques in financial applications such as option pricing and Black-Scholes models.

In his independent study, William continued the topics outlined in APMA 1360 by learning more about chaotic dynamics in two dimensions. William explored symbolic dynamics via the Baker's map and Smale's horseshoe, and carried out numerical studies of the forced pendulum to see whether the inverted pendulum state can indeed be stabilized by rapid oscillations.

Nadia explored how well spectra of ODE operators with periodic coefficients are approximated by finite differences and spectral methods. For the numerical computations, Nadia used Matlab. She also investigated how her findings could be made rigorous using Galerkin approximations and Fourier series.

Outcomes: (i) a poster that was presented at the 2011 Summer Research Symposium at Brown.

Nics gave a self-contained proof of the regularity of center manifolds that followed an idea proposed by Henry. He also derived formulas for the coefficients of the cubic terms in the normal form of Turing, Hopf, and Turing-Hopf bifurcations and implemented an algorithm in Mathematica that calculates these coefficients for a given reaction-diffusion system.

Outcomes: (i) Mathematica software; (ii) a poster that was presented at the 2011 Summer Research Symposium at Brown; (iii) a manuscript on the regularity of center manifolds.

Do Young continued the work that Kesinee Ninsuwan had started in the previous summer. Do Young investigated a four-component model proposed by Vanag and Epstein for light-sensitive Belousov-Zhabotinsky reactions. Using Matlab and Auto07p, Do Young computed and path-followed radial stationary spots and time-periodic radial oscillon solutions using finite differences and spectral discretization schemes.

Outcomes: (i) Auto07p and Matlab software; (ii) a poster that was presented at the 2011 Summer Research Symposium at Brown; (iii) a paper is in preparation.

This independent study explored discrete dynamical systems and in particular complex dynamics, Mandelbrot sets, and Julia sets. Dmitry and Sameer also studied fractals, fractional dimensions, and fractional calculus, and their applications in dynamical systems.

In this continuation of their summer project, Dan and Nat proceed with the numerical computation of rotational water waves using the codes that they developed during the summer. In addition, they study the mathematical theory of water waves and review material on finite-difference schemes and numerical continuation methods. The picture to the left shows a color plot of the pressure of a planar water wave with constant nonzero vorticity computed by Dan and Nat.

The goal of this study is to get a comprehensive overview of the modelling and the dynamics of chemical reactions. Chongwu reviews the derivation of reaction-diffusion equations as models for chemical reactions and studies their dynamics via bifurcation theory, singular perturbation theory, and numerical computations. One specific goal is to understand Turing bifurcations which result in spatially periodic patterns that emerge from homogeneous rest states.

Jeff looks at several models that describe pancreatic glucose-insulin oscillations. Three different types of oscillations with different characteristic temporal periods have been found in experiments, and various differential equation and delay different equation models have been proposed to study their behavior. Jeff reviews these models, carries out numerical computations of their dynamics, and studies the mathematical theory of delay differential equations and the bifurcations that they can exhibit.

Eve used Matlab and Auto07p to calculate radial stationary spots and time-periodic radial oscillon solutions of a three-component model that was derived by Somogyi and Stucki to describe intracellular calcium oscillations. She used both finite-difference and spectral discretizations and continued oscillons in several parameters to study how oscillons emerge and for which parameter values they are stable. The figure shows a color plot of an oscillon solution as a function of radius and time.

Outcomes: (i) Auto07p and Matlab software; (ii) a poster that was presented at the 2010 Summer Research Symposium at Brown; (iii) a paper is in preparation.

The goal of this summer project was to explore whether Matlab and Auto07p can be used to compute planar rotational water waves. Previous work by Ko and Strauss on water waves with vorticity utilized Trilinos. The underlying model is a quasilinear elliptic partial differential equation on a fixed two-dimensional domain. Dan and Nat discretized the equation with finite differences, and implemented the resulting algebraic or differential equations in Matlab and Auto07p. Both programs gave very good results, and the Auto07p code was also able to accurately calculate and continue stagnation points.

Outcomes: (i) Auto07p and Matlab software; (ii) a poster that was presented at the 2010 Summer Research Symposium at Brown.