Topological analysis of chaotic time series

Proceedings of SPIE - The International Society for Optical Engineering

Volumn 3165, Issue , 1997, Pages 243-257

  Gilmore, Robert ^a  

^a DREXEL UNIVERSITY   (United States)

Author keywords

Branched manifold; Knot holder; Linking numbers; Singularities; Template; Topological analysis

Indexed keywords

CHAOS THEORY; CONTROL SYSTEM ANALYSIS; CONTROL THEORY; ORBITS; SOFT COMPUTING; TIME SERIES;

BRANCHED MANIFOLD; KNOT-HOLDER; LINKING NUMBERS; SINGULARITIES; TEMPLATE; TOPOLOGICAL ANALYSIS;

TIME SERIES ANALYSIS;

EID: 0040125168     PISSN: 0277786X     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1117/12.279595     Document Type: Conference Paper

References (42)

1. R. Gilmore, "Topological analysis of chaotic time series", Drexel University preprint, 1997.
2. G. B. Mindlin, H. G. Solari, M. A. Natiello, R. Gilmore, and X.-J. Hou, "Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction", J. of Nonlinear Sci. 1, pp. 147-173, 1991.
3. G. B. Mindlin and R. Gilmore, "Topological analysis and synthesis of chaotic time series", Physica D58, pp. 229-242, 1992.
4. H. G. Solari and R. Gilmore, "Relative rotation rates in driven dynamical systems", Phys. Rev. A37, pp. 3096-3109, 1988.
5. N. B. Tufillaro, T. A. Abbott, and J. P. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA, 1992.
6. H. G. Solari, M. A. Natiello, and G. B. Mindlin, Nonlinear Physics and its Mathematical Tools, IOP Publishing, Ltd., Cambridge, 1996.
7. G. H. Gunaratne, P. S. Linsay, and M. J. Vinson, "Chaos beyond the onset: A comparison of theory and experiment", Phys. Rev. Lett. 67, pp. 1-4, 1989.
8. H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, "The analysis of observed chaotic data in physical systems", Revs. Mod. Phys. 65, pp. 1331-1392, 1993.
9. N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, "Geometry from a time series", Phys. Rev. Lett. 45, pp. 712-714, 1980.
10. F. Takens, "Detecting strange attractors in turbulence", In: Dynamical Systems and Turbulence, Lecture Notes in Mathematics, Vol. 989, D. A. Rand and L. S. Young (Eds.), Springer-Verlag, Heidelberg, 1981, pp. 366-381.
11. H. Whitney, "Differentiable manifolds", Ann. Math. 37, pp. 645-680, 1936.
12. N. B. Tufillaro, H. G. Solari, and R. Gilmore, "Relative rotation rates: Fingerprints for strange attractors", Phys. Rev. A41, pp. 5717-5720, 1990.
13. J. S. Birman and R. F. Williams, "Knotted periodic orbits in dynamical systems-I: Lorenz's equations", Topology 22, pp. 47-82, 1983.
14. J. S. Birman and R. F. Williams, "Knotted periodic orbits in dynamical system II: Knot holders for fibered knots", Cont. Math. 20, pp. 1-60, 1983.
15. G. B. Mindlin, X.-J. Hou, H. G. Solari, R. Gilmore, and N. B. Tufillaro, "Classification of strange attractors by integers", Phys. Rev. Lett. 64, pp. 2350-2353, 1990.
16. D. Auerbach, P. Cvitanovic, J.-P. Eckmann, G. Gunaratne, and I. Procaccia, "Exploring chaotic motion through periodic orbits", Phys. Rev. Lett. 58, pp. 2387-2389, 1987.
17. R. Gilmore and J. W. L. McCallum, "Superstructure in the bifurcation diagram of the Duffing oscillator", Phys. Rev. E51, pp. 935-956, 1995.
18. J. L. W. McCallum and R. Gilmore, "A geometric model of the driven Duffing oscillator", Int. J. Bifurcation and Chaos 3, pp. 685-691, 1993.
19. F. Argoul, A. Arneodo, P. Richetti, J. C. Roux, and H. L. Swinney, "Chemical chaos: from hints to confirmation", Acc. Chem. Res. 20, pp. 436-442, 1987.
20. J. C. Roux, R. H. Simoyi, and H. L. Swinney, "Observations of a strange attractor", Physica 8D, pp. 257-266, 1983.
21. F. Papoff, A. Fioretti, F. Arimondo, G. B. Mindlin, R. Gilmore, and H. G. Solari, "Structure of chaos in the laser with saturable absorber", Phys. Rev. Lett. 68, pp. 1128-1131, 1992.
22. R. Brown, N. Rulkov, and E. Tracy, "Modeling and synchronizing chaotic systems from time series data", Phys. Rev. E49, pp. 3784-3800, 1994.
23. H. G. Solari, M. A. Natiello, and M Vázquez, "Braids on the Poincaré section: A laser example", Phys. Rev. E54, pp. 3185-3195, 1996.
24. M. Lefranc, P. Glorieux, F. Papoff, F. Molesti, and E. Arimondo, "Combining topological analysis and symbolic dynamics to describe a strange attractor and its crises", Phys. Rev. Lett. 73, pp. 1364-1367, 1994.
25. D. Dangoisse, P. Glorieux, and D. Hennequin, "Laser chaotic attractors in crisis", Phys. Rev. Lett. 57, pp. 2657-2660, 1986.
26. 2 laser with modulated losses", Int. J. Bifurcation & Chaos 3, pp. 643-649, 1993.
27. N. B. Tufillaro, R. Holzner, L. Flepp, R. Brun, M. Finardi, and R. Badii, "Template analysis for a chaotic NMR laser", Phys. Rev. A44, pp. 4786-4789, 1991.
28. G. Boulant, M. Lefranc, S. Bielawski, and D. Derozier, "Horseshoe templates with global torsion in a driven laser", Phys. Rev. E55,5082-5091, 1997.
29. C. Letellier, L. Le Sceller, P. Dutertre, G. Gouesbet, Z. Fei, and J. L. Hudson, "Topological characterization and global vector field reconstruction of an experimental electrochemical system", J. Phys. Chem. 99, pp. 7016-7027, 1995.
30. T. Braun, R. R. B. Correira, and N. Altmann, "Topological model of homoclinic chaos in a glow discharge", Phys. Rev. E51, pp. 4165-4168, 1995.
31. N. B. Tufillaro, P. Wyckoff, R. Brown, T. Scheiber, and T. Molteno, "Topological time series analysis of a string experiment and its synchronized model", Phys. Rev. E51, pp. 164-174, 1995.
32. S. O. Firle, M. A. Natiello, and M. Eiswirth, "Topological dynamics in a catalysis experiment", Phys. Rev. E53, pp. 1257-1260, 1996.
33. E. Roldán, G. J. de Valcárel, R. Vilaseca, R. Corbalán, V. J. Martinez, and R. Gilmore, "The dynamics of optically pumped molecular lasers. On its relation with the Lorenz-Haken model", Quantum and Semiclassical Optics 9, pp. R1-R35, 1997.
34. R. Gilmore, R. Vilaseca, R. Corbalán, and E. Roldán, "Analysis of data from a pumped molecular laser", Phys. Rev. E55, pp. 2479-2487, 1997.
35. H. F. Creveling, J. F. dePaz, J. Y. Baladi, and R. J. Schoenhals, "Stability characteristics of a single-phase free convection loop", J. Fluid Mechanics, 67, pp. 65-84, 1975.
36. J. Singer, Y.-Z. Wang, and H. Bau, "Controlling a chaotic system", Phys. Rev. Lett. 66, pp. 1123-1126, 1991.
37. G. Boulant, M. Lefranc, S. Bielawski, and D. Derozier, "A non-horseshoe template in a chaotic laser model", Int. J. Bifurcation & Chaos (in print).
38. C. Letellier, P. Dutertre, and B. Maheu, "Unstable periodic orbits and templates of the Rössler system: toward a systematic topological characterization", Chaos 5, pp. 271, 1995.
39. G. B. Mindlin, R. Lopez-Ruiz, H. G. Solari, and R. Gilmore, "Horseshoe implications", Phys. Rev. E48, pp. 4297-4304, 1993.
40. R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, Wiley, New York, 1974.
41. R. Gilmore, Catastrophe Theory for Scientists and Engineers, Wiley, New York, 1981.
42. H. Whitney, "On singularities of mappings of Euclidean spaces I, Mappings of the plane onto the plane.", Ann. Math., pp. 374-410, 1955.