Type
Text
Type
Dissertation
Advisor
Martens, Marco | Sutherland, Scott | Winckler, Bjorn | Bonifant, Araceli.
Date
2015-12-01
Keywords
dynamics, route to chaos, two-dimensional maps | Mathematics
Department
Department of Mathematics.
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/76390
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
It has long been conjectured that the two-dimensional dissipative maps, like their one-dimensional counterparts, experience the period-doubling cascade to chaos. In [CEK], Collet, Eckmann and Koch proved this conjecture for highly dissipative families. In this dissertation, we introduce the notion of nested systems that generalizes Henon maps, and construct two operators acting on the space of nested systems based on the idea of the return maps. We use the two operators to show that if a dissipative nested system satisfies some apriori bounds, then they can be replaced with simpler but dynamically equivalent nested systems. We prove that the total number of applications of the operators is bounded by the number of periodic points. When the procedure of the applications stops, we obtain ``little Henon'' maps whose dynamics are well-understood. We then show that if the first nested system of a family contains finitely many periodic points, then the family only experiences either saddle-node or periodic-doubling bifurcations. We conclude that if there are sufficiently many periodic points, then moderate dissipative nested systems can be transformed into highly dissipative ones so that the result of [CEK] applies. | 101 pages
Recommended Citation
Chi, Ying, "On the route to chaos for two-dimensional modestly area-contracting analytic maps" (2015). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2313.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2313