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

Physica D: Nonlinear Phenomena

Volumn 144, Issue 3-4, 2000, Pages 231-258

  Plumecoq, Jérôme ^a   Lefranc, Marc ^a  

^a UNIVERSITÉ DES SCIENCES ET TECHNOLOGIES DE LILLE   (France)

Author keywords

05.45.+b; Generating partitions; Knot theory; Symbolic dynamics; Template analysis

Indexed keywords

ALGORITHMS; COMPUTER SIMULATION; IMAGE ANALYSIS; IMAGE CODING; IMAGE RECONSTRUCTION; INVARIANCE; PRECISION ENGINEERING; TIME SERIES ANALYSIS;

KNOT THEORY; SYMBOLIC DYNAMICS; TEMPLATE ANALYSIS;

CHAOS THEORY;

EID: 0034300276     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(00)00082-8     Document Type: Article

References (81)

1. J. Guckenheimer, P.J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, Berlin, 1983.
2. A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambdrige, 1995.
3. Hao Bai-Lin, Elementary Symbolic Dynamics and Chaos in Dissipative Systems, World Scientific, Singapore, 1989.
4. R. Badii, A. Politi, Complexity: Hierarchical Structures and Scaling in Physics, Cambridge Nonlinear Science Series, Vol. 6, Cambridge University Press, Cambridge, 1997.
5. Hayes S., Grebogi C., Ott E. Communicating with chaos. Phys. Rev. Lett. 70:1993;3031-3034.
6. Hayes S., Grebogi C., Ott E., Mark A. Experimental control of chaos for communication. Phys. Rev. Lett. 73:1994;1781-1784.
7. Grassberger P., Kantz H. Generating partitions for the dissipative Hénon map. Phys. Lett. A. 113:1985;235-238.
8. Lefranc M., Glorieux P., Papoff F., Molesti F., Arimondo E. Combining topological analysis and symbolic dynamics to describe a strange attractor and its crises. Phys. Rev. Lett. 73:1994;1364-1367.
9. Gilmore R. Topological analysis of chaotic dynamical systems. Rev. Mod. Phys. 70:1998;1455-1530.
10. Flepp L., Holzner R., Brun E., Finardi M., Badii R. Model identification by periodic-orbit analysis for NMR-laser chaos. Phys. Rev. Lett. 67:1991;2244-2247.
11. Finardi M., Flepp L., Parisi J., Holzner R., Badii R., Brun E. Topological and metric analysis of heteroclinic crisis in laser chaos. Phys. Rev. Lett. 68:1992;2989-2991.
12. Badii R., Brun E., Finardi M., Flepp L., Holzner R., Parisi J., Reyl C., Simonet J. Progress in the analysis of experimental chaos through periodic orbits. Rev. Mod. Phys. 66:1994;1389-1415.
13. Davidchack R.L., Lai Y.-C., Bollt E.M., Dhamala M. Estimating generating partitions of chaotic systems by unstable periodic orbits. Phys. Rev. E. 61:2000;1353-1356.
14. Sterling D., Dulling H.R., Meiss J. Homoclinic bifurcations in the Hénon map. Physica D. 134:1999;153-184.
15. N.B. Tufillaro, T.A. Abbott, J.P. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA, 1992.
16. H.G. Solari, M.A. Natiello, G.B. Mindlin, Nonlinear Dynamics: A Two-way Trip from Physics to Math, IOP Publishers, London, 1996.
17. Mindlin G.B., Hou X.-J., Solari H.G., Gilmore R., Tufillaro N.B. Characterization of strange attractors by integers. Phys. Rev. Lett. 64:1990;2350-2353.
18. Mindlin G.B., Solari H.G., Natiello M.A., Gilmore R., Hou X.-J. Topological analysis of chaotic time series data from Belousov-Zhabotinski reaction. J. Nonlinear Sci. 1:1991;147-173.
19. Birman J.S., Williams R.F. Knotted periodic orbits in dynamical systems I: Lorenz's equations. Topology. 22:1983;47-82.
20. R.W. Ghrist, P.J. Holmes, M.C. Sullivan, Knots and Links in Three-dimensional Flows, Lecture Notes in Mathematics, Vol. 1654, Springer, Berlin, 1997.
21. J. Plumecoq, M. Lefranc, From template analysis to generating partitions. II: Characterization of the symbolic encodings, Physica D 144 (2000) 259-278.
22. Abarbanel H.D., Brown R., Sidorowich J.J., Tsimring L.S. The analysis of observed chaotic data in physical systems. Rev. Mod. Phys. 65:1993;1331-1392.
23. Holmes P.J. Knotted periodic orbits in suspensions of Smale's horseshoe: period multiplying and cabled knots. Physica D. 21:1986;7-41.
24. P.J. Holmes, in: T. Bedford, J. Swift (Eds.), New Directions in Dynamical Systems, Cambridge University Press, Cambridge, 1988, pp. 150-191.
25. Grassberger P., Kantz H., Moenig U. On the symbolic dynamics of the Hénon map. J. Phys. A. 22:1989;5217-5230.
26. Cvitanović P., Gunaratne G.H., Procaccia I. Topological and metric properties of Hénon-type strange attractors. Phys. Rev. A. 38:1988;1503-1520.
27. D'Alessandro G., Grassberger P., Isola S., Politi A. On the topology of the Hénon map. J. Phys. A. 23:1990;5285-5294.
28. Giovannini F., Politi A. Homoclinic tangencies, generating partitions and curvature of invariant manifolds. J. Phys. A. 24:1991;1837-1848.
29. Jaeger L., Kantz H. Structure of generating partitions for two-dimensional maps. J. Phys. A. 30:1997;L567-L576.
30. Wu Z.-B. Symbolic dynamics analysis of chaotic time-series with a driven frequency. Phys. Rev. E. 53:1996;1446-1452.
31. Jaeger L., Kantz H. Homoclinic tangencies and non-normal Jacobians - effects of noise in nonhyperbolic chaotic systems. Physica D. 105:1997;79-96.
32. R. Gilmore, Catastrophe Theory for Scientists and Engineers, Wiley, New York, 1981 [reprinted by Dover, New York, 1993].
33. Solari H.G., Gilmore R. Relative rotation rates for driven dynamical systems. Phys. Rev. A. 37:1988;3096-3109.
34. Auerbach D., Cvitanović P., Eckmann J.-P., Gunaratne G., Procaccia I. Exploring chaotic motion through periodic orbits. Phys. Rev. Lett. 58:1987;2387-2389.
35. Lathrop D., Kostelich E.J. Characterization of a strange attractor by periodic orbits. Phys. Rev. A. 40:1989;4028-4031.
36. Ott E., Grebogi C., Yorke J. Controlling chaos. Phys. Rev. Lett. 64:1990;1196-1199.
37. L.H. Kaufmann, Knots and Physics, World Scientific, Singapore, 1991.
38. Natiello M.A., Solari H.G. Remarks on braid theory and the characterisation of periodic orbits. J. Knot Theory Ramifications. 3:1994;511.
39. Solari H.G., Natiello M.A., Vázquez M. Braids on the Poincaré section: a laser example. Phys. Rev. E. 54:1996;3185-3195.
40. Hall T. The creation of horseshoes. Nonlinearity. 7:1994;861-924.
41. Birman J., Williams R. Knotted periodic orbits in dynamical systems II: knot holders for fibered knots. Cont. Math. 20:1983;1-60.
42. Holmes P.J., Williams R.F. Knotted periodic orbits in suspensions of Smale's horseshoe: torus knots and bifurcation sequences. Arch. Rational Mech. Anal. 90:1985;15-194.
43. Holmes P.J. Knotted periodic orbits in suspensions of annulus maps. Proc. R. Soc. London A. 411:1987;351-378.
44. Holmes P.J. Knotted period orbits in suspensions of Smale's horseshoe: extended families and bifurcation sequences. Physica D. 40:1989;42-64.
45. Tufillaro N.B., Holzner R., Flepp L., Brun R., Finardi M., Badii R. Template analysis for a chaotic NMR laser. Phys. Rev. A. 44:1991;R4786-R4788.
46. Papoff F., Fioretti A., Arimondo E., Mindlin G.B., Solari H.G., Gilmore R. Structure of chaos in the laser with a saturable absorber. Phys. Rev. Lett. 68:1992;1128-1131.
47. Fioretti A., Molesti F., Zambon B., Arimondo E., Papoff F. Topological analysis of laser with saturable absorber in experiments and models. Int. J. Bifurc. Chaos, Appl. Sci. Eng. 3:1993;559-564.
48. 2 laser with modulated losses Int. J. Bifurc. Chaos, Appl. Sci. Eng. 3:1993;643-649.
49. Braun T., Correira R.R.B., Altmann N. Topological model of homoclinic chaos in a glow discharge. Phys. Rev. E. 51:1995;4165-4168.
50. Letellier C., Sceller L.L., Dutertre P., Gouesbet G., Fei Z., Hudson J.L. Topological characterization and global vector field reconstruction of an experimental electrochemical system. J. Phys. Chem. 99:1995;7016-7027.
51. Tufillaro N.B., Wyckoff P., Brown R., Schreiber T., Molteno T. Topological time series analysis of a string experiment and its synchronized model. Phys. Rev. E. 51:1995;164-174.
52. Letellier C., Gouesbet G., Rulkov N.F. Topological analysis of chaos in equivariant electronic circuits. Int. J. Bifurc. Chaos, Appl. Sci. Eng. 6:1996;2531-2555.
53. Boulant G., Lefranc M., Bielawski S., Derozier D. Horseshoe templates with global torsion in a driven laser. Phys. Rev. E. 55:1997;5082-5091.
54. Boulant G., Bielawski S., Derozier D., Lefranc M. Experimental observation of a chaotic attractor with a reverse horseshoe topological structure. Phys. Rev. E. 55:1997;R3801-R3804.
55. McCallum J.W.L., Gilmore R. A geometric model for the Duffing oscillator. Int. J. Bifurc. Chaos, Appl. Sci. Eng. 3:1993;685-691.
56. Gilmore R., McCallum J.W.L. Structure in the bifurcation diagram of the Duffing oscillator. Phys. Rev. E. 51:1995;935-956.
57. Letellier C., Dutertre P., Gouesbet G. Characterization of the Lorenz system, taking into account the equivariance of the vector field. Phys. Rev. E. 49:1994;3492-3495.
58. Letellier C., Dutertre P., Maheu B. Unstable periodic orbits and templates of the Rössler system: toward a systematic topological characterization. Chaos. 5:1995;271-282.
59. Tufillaro N.B. Braid analysis of a bouncing ball. Phys. Rev. E. 50:1994;4509-4522.
60. Letellier C., Gouesbet G., Soufi F., Buchler J.R., Kolláth Z. Chaos in variable stars: topological analysis of W Vir model pulsations. Chaos. 6:1996;466-476.
61. Boulant G., Lefranc M., Bielawski S., Derozier D. A non-horseshoe template in a chaotic laser model. Int. J. Birfurc. Chaos, Appl. Sci. Eng. 8:1998;965-975.
62. Gilmore R., Vilaseca R., Corbalan R., Roldan E. Topological analysis of chaos in the optically pumped laser. Phys. Rev. E. 55:1997;2479-2487.
63. Roldan E., de Valcárcel G.J., Vilaseca R., Martínez V.J., Gilmore R. The dynamics of optically pumped molecular lasers. On its relation with the Lorenz-Haken laser model. Quant. Semiclassical Opt. 9:1997;1-35.
64. Gilmore R., Pei X., Moss F. Topological analysis of chaos in neural spike train bursts. Chaos. 9:1999;812-8817.
65. Melvin P., Tufillaro N.B. Templates and framed braids. Phys. Rev. A. 44:1991;3419-3422.
66. M. Lefranc, unpublished.
67. Hansen K. Remarks on the symbolic dynamics for the Hénon map. Phys. Lett. A. 165:1992;100-104.
68. Giovannini F., Politi A. Generating partitions in Hénon-type maps. Phys. Lett. A. 161:1992;332-336.
69. Hénon M. A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50:1976;69-77.
70. Mindlin G.B., Lopez-Ruiz R., Solari H.G., Gilmore R. Horseshoe implications. Phys. Rev. E. 48:1993;4297-4304.
71. Hansen K.T., Cvitanović P. Bifurcation structures in maps of Hénon type. Nonlinearity. 11:1998;1233-1261.
72. Hall T. Weak universality in two-dimensional transitions to chaos. Phys. Rev. Lett. 71:1993;58-61.
73. Tredicce J.R., Arecchi F.T., Puccioni G.P., Poggi A., Gadomski W. Dynamics behavior and onset of low-dimensional chaos in a modulated homogeneously broadened single-mode laser: experiments and theory. Phys. Rev. A. 3:1986;2073-2081.
74. 2 laser with modulated parameters: experiments and numerical simulations Phys. Rev. A. 36:1987;4775-4791.
75. Aurenhammer F. Voronoi diagrams - a survey of a fundamental geometric data structure. ACM Comput. Surveys. 23:1991;345-405.
76. F.P. Preparata, M.I. Shamos, Computational Geometry: An Introduction, Springer, Berlin, 1985.
77. A. Okabe, B. Boots, K. Sugihara, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Wiley, Chichester, UK, 1992.
78. Watson D.F. Computing the. n -dimensional Delaunay tesselation with application to Voronoi polytopes Comput. J. 24:1981;167-172.
79. J.-D. Boissonat, M. Teillaud, in: Proceedings of the Second Annual ACM Symposium on Computing Geometry, ACM, New York, 1986, pp. 260-268.
80. Watson D.F. Natural neighboring sorting. Austral. Comput. J. 17:1985;189-193.
81. R. Gilmore, in: B. Bosacchi, J.C. Bezdek, D.B. Fogel (Eds.), Applications of Soft Computing, Proceedings of SPIE, Vol. 3165, SPIE, Bellingham, 1997, pp. 243-257.