Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

New Feigenbaum constants for four-dimensional volume-preserving symmetric maps

New Feigenbaum constants for four-dimensional volume-preserving symmetric maps We study period doubling in a symmetric four-dimensional volume-preserving quadratic map, i.e., two symmetrically coupled two-dimensional area-preserving Hénon maps. We must vary two parameters and thus obtain two Feigenbaum constants, δ 1 and δ 2 . It is a very important point that for each region of stability (belonging to some period- q orbit) in this parameter plane we find two regions of stability for the period- 2q orbit, four regions for the period-4 q orbit, and so on. Hence we have an infinite number of stability regions and infinities of bifurcation ‘‘paths’’ through these regions. Almost all self-similar bifurcation paths fall into one of three possible ‘‘universality classes,’’ i.e., each class is characterized by its own two Feigenbaum constants, δ 1 and δ 2 . We find δ 2 =+4.000. . ., -2.000. . . , -4.404. . . , respectively, for the three classes. These δ 2 values are also recovered here from some approximate (numerical) renormalization scheme. The δ 1 is, in all cases, the same as in two-dimensional area-preserving maps, δ 1 =8.721. . . . The δ 2 =-15.1. . ., reported in an earlier paper J. M. Mao, I. Satija, and B. Hu, Phys. Rev. A 32 , 1927 (1985), applies to only two exceptional paths. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review A American Physical Society (APS)

New Feigenbaum constants for four-dimensional volume-preserving symmetric maps

Physical Review A , Volume 35 (4) – Feb 15, 1987
9 pages

Loading next page...
 
/lp/american-physical-society-aps/new-feigenbaum-constants-for-four-dimensional-volume-preserving-cDZPUIT2mq

References

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
American Physical Society (APS)
Copyright
Copyright © 1987 The American Physical Society
ISSN
1094-1622
DOI
10.1103/PhysRevA.35.1847
Publisher site
See Article on Publisher Site

Abstract

We study period doubling in a symmetric four-dimensional volume-preserving quadratic map, i.e., two symmetrically coupled two-dimensional area-preserving Hénon maps. We must vary two parameters and thus obtain two Feigenbaum constants, δ 1 and δ 2 . It is a very important point that for each region of stability (belonging to some period- q orbit) in this parameter plane we find two regions of stability for the period- 2q orbit, four regions for the period-4 q orbit, and so on. Hence we have an infinite number of stability regions and infinities of bifurcation ‘‘paths’’ through these regions. Almost all self-similar bifurcation paths fall into one of three possible ‘‘universality classes,’’ i.e., each class is characterized by its own two Feigenbaum constants, δ 1 and δ 2 . We find δ 2 =+4.000. . ., -2.000. . . , -4.404. . . , respectively, for the three classes. These δ 2 values are also recovered here from some approximate (numerical) renormalization scheme. The δ 1 is, in all cases, the same as in two-dimensional area-preserving maps, δ 1 =8.721. . . . The δ 2 =-15.1. . ., reported in an earlier paper J. M. Mao, I. Satija, and B. Hu, Phys. Rev. A 32 , 1927 (1985), applies to only two exceptional paths.

Journal

Physical Review AAmerican Physical Society (APS)

Published: Feb 15, 1987

There are no references for this article.