Dynamics studies time evolution of systems. The areas of application range from the solar system, global weather on earth, to the future and past of the cosmos and everything between. Although there is an enormous variety of systems, one often observes common features. The role of mathematics in dynamics can be found not only in fundamental contributions to geometrical and probabilistic aspects of dynamics but also in concrete algorithms for predictions. We are still very far from a global understanding of dynamics. Generally speaking we only understand the dynamics of real world systems by numerical simulations. Refined theories are only available in specific and relatively simple systems. The degree of complexity one can expect increases with the number of observables involved and the dimension of the system. Among the most refined areas of dynamics are those where the underling space is an interval or the real line. This is what is called "one-dimensional dynamics". Although one-dimensional dynamics is the simplest of the simplest, the one-dimensional theory is extremely rich. The central tool in one-dimensional dynamics is renormalization. It connects the combinatorial structure of the system with its geometrical and probabilistic behavior. One step up, is the theory of two-dimensional dynamics where the underlying space is the Euclidean plane or a piece of it. This area of research is still wide open. The main goal of this project is to introduce renormalization into two-dimensional dynamics and to explore its consequences for the development of a theory for two-dimensional dynamics.

Henon maps play a crucial role in understanding transitions observed in general smooth dynamics. The most famous example is the transition to chaos in dissipative dynamics. Renormalization plays the central role in understanding these transitions. This has been fully developed for general one-dimensional dynamics and for the transition to chaos in two dimensional dissipative dynamics. This success of renormalization inspired this project. The main goal of the research is to extend the period doubling renormalization theory. In particular, it will introduce Generalized Renormalization which is able to describe systems with general combinatorial structure and the corresponding transitions observed in Henon families. Secondly, it will continue the development of a theory for the dynamics of Henon maps at transition to chaos. Renormalization theory gives us invaluable insights into universality features of dynamical and parameter-regions. It should become an intrinsic part a comprehensive picture of two-dimensional dynamics.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
1600554
Program Officer
Marian Bocea
Project Start
Project End
Budget Start
2016-09-01
Budget End
2019-08-31
Support Year
Fiscal Year
2016
Total Cost
$180,001
Indirect Cost
Name
State University New York Stony Brook
Department
Type
DUNS #
City
Stony Brook
State
NY
Country
United States
Zip Code
11794