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No Access Submitted: 15 September 2009 Accepted: 06 December 2009 Published Online: 03 February 2010

An automated algorithm for the generation of dynamically reconstructed trajectories

Chaos 20, 013107 (2010); https://doi.org/10.1063/1.3279680
C. Komalapriya1,2, a), M. C. Romano3,4, M. Thiel3, N. Marwan2, J. Kurths3,5,2, I. Z. Kiss6, and J. L. Hudson7
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ABSTRACT
The lack of long enough data sets is a major problem in the study of many real world systems. As it has been recently shown [C. Komalapriya, M. Thiel, M. C. Romano, N. Marwan, U. Schwarz, and J. Kurths, Phys. Rev. E 78, 066217 (2008)], this problem can be overcome in the case of ergodic systems if an ensemble of short trajectories is available, from which dynamically reconstructed trajectories can be generated. However, this method has some disadvantages which hinder its applicability, such as the need for estimation of optimal parameters. Here, we propose a substantially improved algorithm that overcomes the problems encountered by the former one, allowing its automatic application. Furthermore, we show that the new algorithm not only reproduces the short term but also the long term dynamics of the system under study, in contrast to the former algorithm. To exemplify the potential of the new algorithm, we apply it to experimental data from electrochemical oscillators and also to analyze the well-known problem of transient chaotic trajectories.

ACKNOWLEDGMENTS

C.K. thanks the Virtual Institute-Pole Equator Pole (PEP), NATO projects, and the Microgravity Application Program/Biotechnology from ESA (Contract No. 14592) for their financial support. M.C.R. would like to acknowledge the Scottish Universities Life Science Alliance (SULSA) for financial support. M.T. would like to acknowledge the RCUK academic fellowship from EPSRC. N.M. would like to thank the DFG graduate school 1364 (Shaping Earth's Surface in a Variable Environment) for the financial support. J.K. would like to thank SFB 555 (C4) for financial support. J.L.H. acknowledges the support from the National Science Foundation.

REFERENCES
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  1. 1. E. Ullner, A. Koseska, J. Kurths, E. Volkov, H. Kantz, and J. G.-Ojalvo, Phys. Rev. E 78, 031904 (2008). https://doi.org/10.1103/PhysRevE.78.031904, Google ScholarCrossref
  2. 2. V. S. Pande, I. Baker, J. Chapman, S. P. Elmer, S. Khaliq, S. M. Larson, Y. M. Rhee, M. R. Shirts, C. D. Snow, E. J. Sorin, and B. Zagrovic, Biopolymers 68, 91 (2003). https://doi.org/10.1002/bip.10219, Google ScholarCrossref
  3. 3. J. A. Yorke and E. D. Yorke, J. Stat. Phys. 21, 263 (1979). https://doi.org/10.1007/BF01011469, Google ScholarCrossref
  4. 4. C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 48, 1507 (1982). https://doi.org/10.1103/PhysRevLett.48.1507, Google ScholarCrossref
  5. 5. C. Grebogi, E. Ott, and J. A. Yorke, Physica D 7, 181 (1983). https://doi.org/10.1016/0167-2789(83)90126-4, Google ScholarCrossref
  6. 6. H. Kantz and P. Grassberger, Physica D 17, 75 (1985). https://doi.org/10.1016/0167-2789(85)90135-6, Google ScholarCrossref
  7. 7. T. Tél, Directions in Chaos (World Scientific, Singapore, 1990), Vol. 3. Google Scholar
  8. 8. C. Komalapriya, M. Thiel, M. C. Romano, N. Marwan, U. Schwarz, and J. Kurths, Phys. Rev. E 78, 066217 (2008). https://doi.org/10.1103/PhysRevE.78.066217, Google ScholarCrossref
  9. 9. H. Poincaré, Acta Math. 13, 1 (1890). Google ScholarCrossref
  10. 10. E. Ott, Chaos in Dynamical Systems, 2nd ed. (Cambridge University Press, Cambridge, 2002). Google ScholarCrossref
  11. 11. G. Robinson and M. Thiel, Chaos 19, 023104 (2009). https://doi.org/10.1063/1.3117151, Google ScholarScitation
  12. 12. P. Grassberger and I. Procaccia, Phys. Rev. A 28, 2591 (1983). https://doi.org/10.1103/PhysRevA.28.2591, Google ScholarCrossref
  13. 13. M. Dhamala, Y. -C. Lai, and E. J. Kostelich, Phys. Rev. E 64, 056207 (2001). https://doi.org/10.1103/PhysRevE.64.056207, Google ScholarCrossref
  14. 14. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, 2nd ed. (Cambridge University Press, Cambridge, 2004). Google Scholar
  15. 15. N. Marwan, M. C. Romano, M. Thiel, and J. Kurths, Phys. Rep., Phys. Lett. 438, 237 (2007). Google ScholarCrossref
  16. 16. A. M. Fraser and H. L. Swinney, Phys. Rev. A 33, 1134 (1986). https://doi.org/10.1103/PhysRevA.33.1134, Google ScholarCrossref
  17. 17. M. Thiel, M. C. Romano, P. L. Read, and J. Kurths, Chaos 14, 234 (2004). https://doi.org/10.1063/1.1667633, Google ScholarScitation
  18. 18. O. E. Rössler, Phys. Lett. A 57, 397 (1976). https://doi.org/10.1016/0375-9601(76)90101-8, Google ScholarCrossref
  19. 19. M. Hénon, Commun. Math. Phys. 50, 69 (1976). https://doi.org/10.1007/BF01608556, Google ScholarCrossref
  20. 20. Encyclopedia of Nonlinear Sciences (Taylor & Francis, London, 2005). Google Scholar
  21. 21. E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963). https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2, Google ScholarCrossref
  22. 22. R. Gilmore, Rev. Mod. Phys. 70, 1455 (1998). https://doi.org/10.1103/RevModPhys.70.1455, Google ScholarCrossref
  23. 23. Nonhyperbolic dynamical systems are generally characterized by the existence of multiple main time scales (Refs. 17 and 40). In the chaotic Rössler system, there exist two main time scales that characterize the dynamics. While the first time scale describes the short term dynamics of the attractor and is linked to the amplitude fluctuations of the system, the second one reflects the long term dynamics of the system and is related to the system’s phase diffusion or relaxation behavior. The presence of two time scale dynamics can be identified either by analyzing the decay of autocorrelation function’s envelope (Ref. 40) or the diagonal line distributions of a RP (Ref. 17). Google Scholar
  24. 24. M. Thiel, M. C. Romano, J. Kurths, R. Meucci, E. Allaria, and F. T. Arecchi, Physica D 171, 138 (2002). https://doi.org/10.1016/S0167-2789(02)00586-9, Google ScholarCrossref
  25. 25. S. Schinkel, O. Dimigen, and N. Marwan, Eur. Phys. J. Spec. Top. 164, 45 (2008). https://doi.org/10.1140/epjst/e2008-00833-5, Google ScholarCrossref
  26. 26. I. Z. Kiss and J. L. Hudson, Phys. Chem. Chem. Phys. 4, 2638 (2002). https://doi.org/10.1039/b200716a, Google ScholarCrossref
  27. 27. I. Z. Kiss and J. L. Hudson, Phys. Rev. E 64, 046215 (2001). https://doi.org/10.1103/PhysRevE.64.046215, Google ScholarCrossref
  28. 28. I. Z. Kiss, W. Wang, and J. L. Hudson, Phys. Chem. Chem. Phys. 2, 3847 (2000). https://doi.org/10.1039/b003812l, Google ScholarCrossref
  29. 29. F. Takens, Dynamical Systems and Turbulence (Springer-Verlag, Berlin, 1981), Vol. 898, pp. 366–381. https://doi.org/10.1007/BFb0091924, Google ScholarCrossref
  30. 30. J. P. Crutchfield and K. Kaneko, Phys. Rev. Lett. 60, 2715 (1988). https://doi.org/10.1103/PhysRevLett.60.2715, Google ScholarCrossref
  31. 31. B. Eckhardt and A. Mersmann, Phys. Rev. E 60, 509 (1999). https://doi.org/10.1103/PhysRevE.60.509, Google ScholarCrossref
  32. 32. E. Ott and T. Tél, Chaos 3, 417 (1993). https://doi.org/10.1063/1.165949, Google ScholarScitation
  33. 33. Z. Toroczkai, G. Károlyi, Á. Péntek, T. Tél, and C. Grebogi, Phys. Rev. Lett. 80, 500 (1998). https://doi.org/10.1103/PhysRevLett.80.500, Google ScholarCrossref
  34. 34. M. Dhamala and Y. -C. Lai, Phys. Rev. E 59, 1646 (1999). https://doi.org/10.1103/PhysRevE.59.1646, Google ScholarCrossref
  35. 35. H. E. Nusse and J. A. Yorke, Physica D 36, 137 (1989). https://doi.org/10.1016/0167-2789(89)90253-4, Google ScholarCrossref
  36. 36. I. M. Jánosi, L. Flepp, and T. Tél, Phys. Rev. Lett. 73, 529 (1994). https://doi.org/10.1103/PhysRevLett.73.529, Google ScholarCrossref
  37. 37. P. Moresco and S. P. Dawson, Physica D 126, 38 (1999). https://doi.org/10.1016/S0167-2789(98)00234-6, Google ScholarCrossref
  38. 38. D. Sweet, H. E. Nusse, and J. A. Yorke, Phys. Rev. Lett. 86, 2261 (2001). https://doi.org/10.1103/PhysRevLett.86.2261, Google ScholarCrossref
  39. 39. M. Dhamala, Y. -C. Lai, and E. J. Kostelich, Phys. Rev. E 61, 6485 (2000). https://doi.org/10.1103/PhysRevE.61.6485, Google ScholarCrossref
  40. 40. V. S. Anishchenko, T. E. Vadivasova, G. A. Okrokvertskhov, and G. I. Strelkova, Phys. Usp. 48, 151 (2005). https://doi.org/10.1070/PU2005v048n02ABEH002070, Google ScholarCrossref
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