|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'504'585
Articles rated: 2609
24 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Period Doubling in Area-Preserving Maps: An Associated One Dimensional Problem | | Denis Gaidashev
; Hans Koch
; | | Date: |
16 Nov 2008 | | Abstract: | It has been observed that the famous Feigenbaum-Coullet-Tresser period
doubling universality has a counterpart for area-preserving maps of
$field{R}^2$. A renormalization approach has been used in a computer-assisted
proof of existence of an area-preserving map with orbits of all binary periods
by J.-P. Eckmann, H. Koch and P. Wittwer (1982 and 1984). As it is the case
with all non-trivial universality problems in non-dissipative systems in
dimensions more than one, no analytic proof of this period doubling
universality exists to date.
We argue that the period doubling renormalization fixed point for
area-preserving maps is almost one dimensional, in the sense that it is close
to the following Henon-like map: $$H^*(x,u)=(phi(x)-u,x-phi(phi(x)-u)),$$
where $phi$ solves $$phi(x)={2 over lambda} phi(phi(lambda x))-x.$$
We then give a ’’proof’’ of existence of solutions of small analytic
perturbations of this one dimensional problem, and describe some of the
properties of this solution.
The ’’proof’’ consists of an analytic argument for factorized inverse
branches of $phi$ together with verification of several inequalities and
inclusions of subsets of $field{C}$ numerically.
Finally, we suggest an analytic approach to the full period doubling problem
for area-preserving maps based on its proximity to the one dimensional. In this
respect, the paper is an exploration of a possible analytic machinery for a
non-trivial renormalization problem in a conservative two-dimensional system. | | Source: | arXiv, 0811.2588 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER