# Florida Tech REU-2014, Final Presentations

National Science Foundation funded Research Experience for Undergraduates Site on Partial Differential Equations and Dynamical Systems has been successfully established at Florida Institute of Technology. In the first summer session in 2014 10 undergraduate students representing University of Chicago, City University of New York, University of Michigan, Williams College, University of Illinois at Urbana-Champaign, University of Auburn, University of Pennsylvania, California State University at Long Beach, University of Redlands and Florida Institute of Technology pursued research under my supervision.

Students worked in there groups working on Nonlinear Partial Differential Equations, Optimal Control and Inverse Problems and Dynamical Systems and Chaos Theory. All 10 students are selected as presenters in YMC-2014 in Ohio State University, August 22-24, 2014. Here is the link to abstracts of all three presentations.

Analysis of Interfaces for the Nonlinear Diffusion Equation with Linear Convection by Taylor P Schluter & Lauren T Lanier & Kev Johnson

On the Structure of Minimal $4(2k+1)$-orbits of Continuous Endomorphisms and Universality in Chaos by
Rashad U Abdulla & Batul Kanawati & Anders Ruden

On Some Inverse Free Boundary Problems for Second Order Parabolic PDEs by
Nicholas Crispi & Daniel Kassler & Paige Williams & Bruno Poggi & Scott Pelton-Stroud

Attached youtube video features final presentations by students at the end of the summer session of 2014.

# Dynamical Systems and Chaos Theory

In a joint work with my undergraduate research students Miss Almas Abdulla (MIT, Class of 2014) and Rashad Abdulla (UPenn, Class of 2017) we revealed a new law of the distribution of periodic orbits in chaotic regime for the one parameter family of discrete dynamical systems. We presented a new constructive proof of the result on the structure of minimal $2(2k+1)$-orbits of the continuous endomorphisms $f:I \to I$, where $I$ is a nondegenerate interval on the real line. It is proved that there are four types of digraphs with accuracy up to inverse graphs. It is demonstrated that the first appearance, as the parameter increased, of the $2(2k+1)$-periodic window within the chaotic regime in the bifurcation diagram of the one-parameter family of logistic type unimodal continuous endomorphisms is always a minimal $2(2k+1)$-orbit with Type I digraph.

In summer session of 2014 students will pursue research on the fine classification of the periodic orbits of the continuous endomorphisms and analysis of the structure of the periodic windows within the chaotic regime of the bifurcation diagram for the unimodal continuous endomorphisms. This research requires creative combination of theoretical and numerical analysis.