ABSTRACT
A recurrence plot is a two-dimensional visualization technique for sequential data. These plots are useful in that they bring out correlations at all scales in a manner that is obvious to the human eye, but their rich geometric structure can make them hard to interpret. In this paper, we suggest that the unstable periodic orbits embedded in a chaotic attractor are a useful basis set for the geometry of a recurrence plot of those data. This provides not only a simple way to locate unstable periodic orbits in chaotic time-series data, but also a potentially effective way to use a recurrence plot to identify a dynamical system.
- 1. http://home.netcom.com/∼eugenek/download.html, Google Scholar
- 2. http://www.cs.colorado.edu/∼lizb/rps.html, Google Scholar
- 3. R. Artuso, E. Aurell, and P. Cvitanovic, “Recycling of strange sets: I. Cycle expansions,” Nonlinearity 3, 325–359 (1990). Google ScholarCrossref
- 4. R. Artuso, E. Aurell, and P. Cvitanovic, “Recycling of strange sets: II. Applications,” Nonlinearity 3, 361–386 (1990). Google ScholarCrossref
- 5. D. Auerbach, P. Cvitanovic, J.-P. Eckmann, G. Gunaratne, and I. Procaccia, “Exploring chaotic motion through periodic orbits,” Phys. Rev. Lett. 58, 2387–2389 (1987). Google ScholarCrossref
- 6. R. Balocchi, A. DiGarbo, C. Michelassi, S. Chillemi, M. Varanini, M. Barbi, J. Legramente, G. Raimondi, and J. Zbilut, “Heart rate and blood pressure response to short-term head-down bed rest: A nonlinear approach,” Methods Inf. Med. 39, 157–159 (2000). Google ScholarCrossref
- 7. O. Bihamand W. Wenzel, “Characterization of unstable periodic orbits in chaotic attractors and repellers,” Phys. Rev. Lett. 63, 819–822 (1989). Google ScholarCrossref
- 8. J. C. Webber, “Rhythmogenesis of deterministic breathing patterns,” in Rhythms in Physiological Systems, edited by H. Haken and H. Koepchen (Springer-Verlag, Berlin, 1991). Google Scholar
- 9. M. Casdagli, “Recurrence plots revisited,” Physica D 108, 12–44 (1997). Google ScholarCrossref
- 10. F. Christiansenand P. Cvitanovic, “Periodic orbit quantization of the anisotropic Kepler problem,” Chaos 2, 61–69 (1992). Google ScholarScitation
- 11. P. Cvitanovic, “Invariant measurement of strange sets in terms of circles,” Phys. Rev. Lett. 61, 2729–2732 (1988). Google ScholarCrossref
- 12. J.-P. Eckmann, S. Kamphorst, and D. Ruelle, “Recurrence plots of dynamical systems,” Europhys. Lett. 4, 973–977 (1987). Google ScholarCrossref
- 13. J.-P. Eckmannand D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617–656 (1985). Google ScholarCrossref
- 14. J. H. Friedman, J. L. Bentley, and R. A. Finkel, “An algorithm for finding best matches in logarithmic expected time,” ACM Trans. Math. Softw. 3, 209–226 (1977). Google ScholarCrossref
- 15. J. Gaoand H. Cai, “On the structures and quantification of recurrence plots,” Phys. Lett. A 270, 75–87 (2000). Google ScholarCrossref
- 16. A. Giuliani, R. Benigni, P. Sirabella, J. Zbilut, and A. Colosimo, “Exploiting the information content of protein sequences using time-series methods: A case study in rubredoxins,” Biophys. J. 78, 136–149 (1998). Google ScholarCrossref
- 17. J. Guckenheimerand B. Meloon, “Computing periodic orbits and their bifurcations with automatic differentiation,” SIAM J. Sci. Comput. (USA) 22, 951–985 (2000). Google ScholarCrossref
- 18. G. H. Gunaratne, P. S. Linsay, and M. J. Vinson, “Chaos beyond onset: A comparison of theory and experiment,” Phys. Rev. Lett. 63, 1 (1989). Google ScholarCrossref
- 19. S. Ikagawa, M. Shinohara, T. Fukunaga, J. Zbilut, and C. Webber, “Nonlinear time-course of lumbar muscle fatigue using recurrence quantifications,” Biol. Cybern. 82, 373–382 (2000). Google ScholarCrossref
- 20. J. Iwanskiand E. Bradley, “Recurrence plots of experimental data: To embed or not to embed?” Chaos 8, 861–871 (1998). Google ScholarScitation
- 21. P. Kaluznyand R. Tarnecki, “Recurrence plots of neuronal spike trains,” Biol. Cybern. 68, 527–534 (1993). Google ScholarCrossref
- 22. M. Koebbe and G. Mayer-Kress, “Use of recurrence plots in the analysis of time-series data,” in Nonlinear Modeling and Forecasting, edited by M. Casdagli and S. Eubank (Addison Wesley, New York, 1992). Google Scholar
- 23. H. Löffelman, Ph.D. thesis, Vienna Technical University, November 1998. See also http://www.cg.tuwien.ac.at/∼helwig/diss/node42.htm, Google Scholar
- 24. G. McGuire, N. Azar, and M. Shelhammer, “Recurrence matrices and the preservation of dynamical properties,” Phys. Lett. A 237, 43–47 (1997). Google ScholarCrossref
- 25. P. So, J. Francis, T. Netoff, B. Gluckman, and S. Schiff, “Periodic orbits: A new language for neuronal dynamics,” Biophys. J. 74, 2776–2785 (1998). Google ScholarCrossref
- 26. P. So, E. Ott, T. Sauer, B. Gluckman, C. Grebogi, and S. Schiff, “Extracting unstable periodic orbits from chaotic time-series data,” Phys. Rev. E 55, 5398–5417 (1997). Google ScholarCrossref
- 27. P. So, E. Ott, S. Schiff, D. Kaplan, T. Sauer, and C. Grebogi, “Detecting unstable periodic orbits in chaotic experimental data,” Phys. Rev. Lett. 76, 4705–4708 (1996). Google ScholarCrossref
- 28. C. Webber, M. Schmidt, and J. Walsh, “Influence of isometric loading on biceps EMG dynamics as assessed by linear and nonlinear tools,” J. Appl. Physiol. 78, 814–822 (1995). Google ScholarCrossref
- 29. C. Webberand J. Zbilut, “Dynamical assessment of physiological systems and states using recurrence plot strategies,” J. Appl. Physiol. 76, 965–973 (1994). Google ScholarCrossref
- 30. J. Zbilut, A. Giuliani, and C. Webber, “Detecting deterministic signals in exceptionally noisy environments using cross-recurrence quantification,” Phys. Lett. A 246, 122–128 (1998). Google ScholarCrossref
- 31. J. Zbilut, A. Giuliani, and C. Webber, “Recurrence quantification analysis and principal components in the detection of short complex signals,” Phys. Lett. A 237, 131–135 (1998). Google ScholarCrossref
- 32. J. Zbilutand C. Webber, “Embeddings and delays as derived from recurrence quantification analysis,” Phys. Lett. A 171, 199–203 (1992). Google ScholarCrossref
Please Note: The number of views represents the full text views from December 2016 to date. Article views prior to December 2016 are not included.
