Computer Science
Scientific paper
Dec 1998
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998phdt........13c&link_type=abstract
Thesis (PHD). THE UNIVERSITY OF TEXAS AT AUSTIN , Source DAI-B 59/06, p. 2801, Dec 1998, 127 pages.
Computer Science
1
Chaos, Sitnikov Problem, Poincare Map, Bifurcations
Scientific paper
This study demonstrates the use of the Poincare map to determine the evolution of areas in the surface of section under a single iteration, rather than scrutinizing a few initial conditions for many iterations as is usually done. This new approach provides details on the underlying structure of what are simply 'chaotic seas' in the traditional surface-of-section plot. Furthermore, since many iterations of the map are not required, the present approach substantially avoids issues of numerical reliability. The isosceles three-body problem has been reduced to a two-dimensional area-preserving Poincare map f depending on two parameters: the mass ratio, and the total angular momentum. The region of allowable motion is divided into subregions according to a symbolic dynamics representation. In this new context one considers how the images of the subregions intersect with their preimages. This leads to an elegant geometric description of the dynamics in the chaotic regions, and to a nearly complete global description of the system. This description immediately provides the existence of various types of motion such as capture-escape, permanent capture, ejection-collision, etc., and their corresponding measures in the map domain. Moreover, this method can be used to show that certain dynamical behaviors are not permitted by the equations of motion. The problem is investigated in three stages. First, the planar (zero angular momentum) case with three equal masses is studied in considerable detail. In this case there exist two important invariant subsets under f. The first is a Cantor set with zero measure. The existence of this chaotic set follows from the fact that f is similar to the Smale horseshoe map in part of the domain. The second subset is an invariant KAM region with positive measure surrounding an elliptic fixed point-the 'main' periodic orbit. The second stage explores the bifurcations in the mass ratio for the planar case. Here the entire chaotic invariant set is simultaneously destroyed at a particular mass ratio, leading to an interesting global bifurcation. The invariant KAM region vanishes at an inverse period doubling bifurcation. parameter space is explored, emphasizing dynamical variations under changing angular momentum. Regions with qualitatively similar behavior are identified, paying special attention to the sequential destruction of the subregions defined in the planar case, and to the stability of the main periodic orbit.
No associations
LandOfFree
The Isosceles Three-Body Problem: a Global Geometric Analysis does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Isosceles Three-Body Problem: a Global Geometric Analysis, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Isosceles Three-Body Problem: a Global Geometric Analysis will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-764325