# 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.

# NPDE Lecture 1 Discussion

I am going to start a series of lectures on nonlinear partial differential equations. Advancements made in the theory of nonlinear PDEs is one of the main achievements in XX century mathematics. Let me first make a remark on linear PDEs. In some sense one can say that there is a complete theory of linear PDEs, and perhaps the best source would be the four volumes of “The Analysis of Linear Partial Differential Operators” by Lars Hörmander [3].

The importance of the nonlinear PDEs is associated with the fact that they reflect more precise nonlinear laws of the nature, while linear PDEs were derived from the linearized versions of the nonlinear laws. Despite the great advances in the theory of nonlinear PDEs it is far from being complete. It is remarkable that almost every major nonlinear PDE has its own personality and requires unique approach. However, looking to current state of art in the theory of nonlinear PDEs, one can observe a similarity with the classical linear theory in the following sense: there are some major individual nonlinear PDEs, and analysis and understanding of those PDEs is a key towards the general theory of a broad class of nonlinear PDEs. I am going to concentrate on one of those key nonlinear PDEs called the nonlinear diffusion equation, which is a generalization of the classical heat/diffusion equation.
Continue reading NPDE Lecture 1 Discussion

# Nonlinear Partial Differential Equations

Partial differential equations (PDEs) are central to mathematics, whether pure or applied. They arise in mathematical models of real world problems, where dependent variables vary continuously as functions of several independent variables, usually space and time. PDEs arising in a majority of real world applications are nonlinear. Despite its complexity, in the theory of nonlinear PDEs one can
observe key equations or systems which are essential for the development of the theory for a particular class of equations. One of those key equations is the nonlinear reaction-diffusion-convection equation:

$u_t=(u^m)_{xx}+bu^{\beta}+c(u^{\gamma})_x,$

where $u=u(x,t),x\in \mathbb{R}, t>0, b,c \in \mathbb{R}, m>0, \beta>0, \gamma>0$
It arises, for instance, in the theory of gas flow through a porous medium, heat radiation in plasmas, spatial spread of biological populations, diffusion of particles in plasma etc.

In summer session of 2014 students will pursue research on the open problem of evolution of interfaces
of solutions to the reaction-diffusion-convection equations.

Full solution of this problem for the reaction-diffusion equation is given in the following two papers by using significantly the theory of reaction-diffusion equations in general non-smooth domains developed in my JDE paper.

Introductory lectures on this topic will be posted soon.

# Welcome to Ugur Abdulla’s Mathematical Blog

Welcome to my Mathematical Blog .  The purpose is to share updates in my research,  discussion of mathematical ideas in the fields of my research expertise, discussion of open problems, updates and discussions on my new papers and posted lectures, and in general discussion of any math related topics.  It is open to everyone in the mathematical community worldwide.

In particular, this is the weblog for the Research Experience for Undergraduates (REU) Site on Partial Differential Equations and Dynamical Systems, which I am running for 2014-2016 with the grant from the US National Science Foundation.  For application, grant, and program information, please see the REU Main Page.

Each summer from 2014 to 2016, 9 undergraduate students will participate in an 8 week summer REU Site on Partial Differential Equations and Dynamical Systems held at the Florida Institute of Technology Mathematical Sciences Department.

The REU Site is designed to involve undergraduate students in innovative research in nonlinear partial differential equations, optimal control and inverse problems for systems with distributed parameters, and dynamical systems and chaos theory, while utilizing modern tools of mathematical and numerical analysis. Students will have a great opportunity to pursue hands-on, original research on the frontier of modern mathematics, which will include the evolution of interfaces for nonlinear reaction-diffusion-convection equations, inverse free boundary problems and optimal control of phase transition processes, and the fine classification of minimal periodic orbits of discrete dynamical systems with application in chaos theory.

For more information on the REU faculty and program, see Dr. Abdulla’s REU page. For an extended description and other related links, see the FIT Research Portal. I will post here periodically both before, during, and after the REU to communicate with other students and the community at large about our results and other interesting information.

# 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.

# NPDE Lecture 4 Discussion

In this lecture I am going to talk about nonlinear diffusion equation in general non-smooth domains.
In a series of papers I developed the theory of nonlinear diffusion type equations in general non-smooth domains.
In this talk I am going to explain the results of my papers [1,2].
A particular motivation for this works arises from the problem about the evolution of interfaces in problems for porous medium equation.
Special interest concerns the cases when support of the initial data contains a corner or cusp singularity at some points.
What about the movement of these kind of singularities along the interface? To solve this problem, it is important at the first stage to develop general theory of boundary-value problems in non-cylindrical domains with boundary surfaces which has the same kind of behaviour as the interface. In many cases this may be non-smooth and characteristic. It should be mentioned that in the one-dimensional case Dirichlet and Cauchy-Dirichlet problems for the reaction-diffusion equations in irregular domains were studied in [3,4]. Primarily applying this theory a complete description of the evolution of interfaces was presented in [5,6].

# Optimal Control and Inverse Problems for PDEs

Development and investigation of the nonlinear models of the mathematical physics is one of the most important problems of the modern science. This project will shed light into modeling and control of an important class of nonlinear processes with phase transition, so called free boundary problems, arising in thermophysics and mechanics of continuous media, bioengineering and materials science. Free boundary problems involve the solution of the partial differential equations in domains that are unknown apriori: not only the solutions of the equations but also the domain of definition of the equations must be determined.
The main goal of this project is to gain insight into inverse free boundary problems. These problems arise when not only the solution of the PDEs and the free boundaries are unknown, but also some other characteristics of the media are not available. For example, temperature or heat flux on the fixed boundaries, intensity of the source term or phase transition temperature may not be available. Another important motivation for the analysis of the inverse Stefan problem arises in optimal control of processes with phase transitions.

In summer REU session of 2014 students will pursue research on the optimal control of the free boundary problem for the heat equation motivated by the new variational formulation suggested in a recent paper:

# NPDE Lecture 3 Discussion

In this talk, I am going to prove existence of the weak solution of the Dirichlet problem for the nonlinear diffusion equation under minimal restrictions on the data. Continue reading NPDE Lecture 3 Discussion

# NPDE Lecture 2 Discussion

### Weak Solutions

Let me remind you what I did in a previous lecture. I considered the problem
\begin{align} u_{t}-\text{div}\big( u^{\sigma} \nabla u\big) &=0,~x \in\R^{N},~t>0; \ \sigma > 0,\label{eq:inst-pme-1}\\ \int_{\R^{N}}u(x,t)\,dx &=1,~t > 0\label{eq:inst-pme-2}\\ u(x,0) = \delta(x)\label{eq:inst-pme-3} \end{align}
and constructed instantaneous point-source solution, also called the Zeldovich-Kompaneets-Barenblatt (ZKB) solution.
$$u_*(x,t)=t^{-\frac{N}{2+N\sigma}}\left[\frac{\sigma}{2(2+N\sigma)} \left( \eta_{0}^{2}-\frac{|x|^{2}}{t^{\frac{2}{2+N\sigma}}}\right)_{+} \right]^{\frac{1}{\sigma}}$$
Note that the nonlinear diffusion equation is invariant under translation of time and space coordinate, and the solution to the problem
$\begin{cases} u_{t}=\text{div}\big(u^{\sigma }u\big),~&(x,t) \in \R^{N}\times \R_{+}\\ u(x,0)=u_*(x,T),~&x \in \R^{N} \end{cases}$
is exactly $u_*(x,t+T)$, which has compact support for all $t\geq 0$.

However, several important questions are left open. First of all, ZKB solution is not a classical solution: it is not even differentiable on the boundary of the support. First important question which we need to answer is the following: In what sense is $u_*$ actually a solution of the problem? Hence, we need to define the notion of the weak solution. Second important question we need to answer is the following: May be there is a smooth solution of the same problem with different properties, and ZKB solution is just physically irrelevant mathematical example. Main goal of this lecture is to answer these important questions.
Continue reading NPDE Lecture 2 Discussion