From template analysis to generating partitions I: Periodic orbits, knots and symbolic encodings

被引:43
作者
Plumecoq, J [1 ]
Lefranc, M [1 ]
机构
[1] Univ Lille 1, Ctr Etud & Rech Lasers & Applicat, Lab Phys Lasers Atomes Mol, CNRS,UMR 8523, F-59655 Villeneuve Dascq, France
来源
PHYSICA D | 2000年 / 144卷 / 3-4期
关键词
generating partitions; symbolic dynamics; template analysis; knot theory;
D O I
10.1016/S0167-2789(00)00082-8
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We present a detailed algorithm to construct symbolic encodings for chaotic attractors of three-dimensional flows. It is based on a topological analysis of unstable periodic orbits embedded in the attractor and follows the approach proposed by Lefranc et al. [Phys. Rev. Lett. 73 (1994) 1364]. For each orbit, the symbolic names that are consistent with its knot-theoretic invariants and with the topological structure of the attractor are first obtained using template analysis. This information and the locations of the periodic orbits in the section plane are then used to construct a generating partition by means of triangulations. We provide numerical evidence of the validity of this method by applying it successfully to sets of more than 1500 periodic orbits extracted from numerical simulations, and obtain partitions whose border is localized with a precision of 0.01%. A distinctive advantage of this approach is that the solution is progressively refined using higher-period orbits, which makes it robust to noise, and suitable for analyzing experimental time series. Furthermore, the resulting encodings are by construction consistent in the corresponding limits with those rigorously known for both one-dimensional and hyperbolic maps. (C) 2000 Elsevier Science B.V. All rights reserved.
引用
收藏
页码:231 / 258
页数:28
相关论文
共 81 条
[1]   THE ANALYSIS OF OBSERVED CHAOTIC DATA IN PHYSICAL SYSTEMS [J].
ABARBANEL, HDI ;
BROWN, R ;
SIDOROWICH, JJ ;
TSIMRING, LS .
REVIEWS OF MODERN PHYSICS, 1993, 65 (04) :1331-1392
[2]  
[Anonymous], CAMBRIDGE NONLINEAR
[3]  
[Anonymous], 1997, LECT NOTES MATH
[4]   EXPLORING CHAOTIC MOTION THROUGH PERIODIC-ORBITS [J].
AUERBACH, D ;
CVITANOVIC, P ;
ECKMANN, JP ;
GUNARATNE, G ;
PROCACCIA, I .
PHYSICAL REVIEW LETTERS, 1987, 58 (23) :2387-2389
[5]  
AURENHAMMER F, 1991, COMPUT SURV, V23, P345, DOI 10.1145/116873.116880
[6]   PROGRESS IN THE ANALYSIS OF EXPERIMENTAL CHAOS THROUGH PERIODIC-ORBITS [J].
BADII, R ;
BRUN, E ;
FINARDI, M ;
FLEPP, L ;
HOLZNER, R ;
PARISI, J ;
REYL, C ;
SIMONET, J .
REVIEWS OF MODERN PHYSICS, 1994, 66 (04) :1389-1415
[7]  
Birman J., 1983, Contemp. Math., V20, P1, DOI DOI 10.1090/CONM/020/718132
[8]   KNOTTED PERIODIC-ORBITS IN DYNAMICAL-SYSTEMS .1. LORENZ EQUATIONS [J].
BIRMAN, JS ;
WILLIAMS, RF .
TOPOLOGY, 1983, 22 (01) :47-82
[9]  
Boissonnat J. D., 1986, P 2 ANN ACM S COMP G, P260
[10]   A nonhorseshoe template in a chaotic laser model [J].
Boulant, G ;
Lefranc, M ;
Bielawski, S ;
Derozier, D .
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 1998, 8 (05) :965-975