# 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 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: