Detecting direction of coupling in interacting oscillators

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 64, Issue 4, 2001, Pages 4-

  Rosenblum, Michael G ^a   Pikovsky, Arkady S ^a  

^a UNIVERSITY OF POTSDAM   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85035249671     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.64.045202     Document Type: Article

References (28)

1. H. D. I. Abarbanel, Analysis of Observed Chaotic Data (Springer-Verlag, New York, 1996);
2. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, England, 1997).
3. P. Tass et al, Phys. Rev. Lett. 81, 3291 (1998).
4. M. Le van Quyen et al, Brain Res. 792, 24 (1998);
5. J. Arnhold et al, Physica D 134, 419 (1999).
6. E. Rodriguez et al, Nature (London) 397, 430 (1999);
7. J.-P. Lachaux et al, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2429 (2000).
8. S. J. Schiff et al, Phys. Rev. E 54, 6708 (1996).
9. F. Drepper, Phys. Rev. E 62, 6376 (2000).
10. R. Quian Quiroga et al, Phys. Rev. E 61, 5142 (2000);
11. Phys. Rev. EA. Schmitz, 62, 7508 (2000).
12. M. Rosenblum, A. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996);
13. A. Pikovsky et al, Physica D 104, 219 (1997);
14. A. Pikovsky, M. Rosenblum, and J. Kurths, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2291 (2000).
15. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, England, 2001).
16. M. Rosenblum, A. Pikovsky, and J. Kurths, IEEE Trans. CAS-I 44, 874 (1997);M. G. Rosenblum et al., in Neuro-informatics, edited by F. Moss and S. Gielen, Vol. 4 of Handbook of Biological Physics, (Elsevier, Amsterdam, 2000), pp. 279–321.
17. For narrow-band signals the (instantaneous) phase can be obtained, e.g., by means of the Hilbert transform; alternatively, one can use wavelet transform with the complex Gabor wavelet 4. For signals containing well-expressed marker events one can use a method similar to the Poncaré map technique. For details, see Ref. 9.
18. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984);
19. K. K. Likharev, Dynamics of Josephson Junctions and Circuits (Gordon and Breach, Philadelphia, 1991);
20. R. E. Best, Phase-Locked Loops (McGraw-Hill, New York, 1984).
21. A. S. Pikovsky, Sov. J. Commun. Technol. Electron. 31, 81 (1986).
22. R. Friedrich et al, Phys. Lett. A 271, 217 (2000).
23. Note that in practice one can encounter more complex cases of interaction, e.g., when two oscillators are not coupled directly but are forced by a common drive; in this case (Formula presented)
24. The equations are solved by the Euler technique with the integration step (Formula presented) every 20th point was used for the subsequent analysis.
25. W. Wang et al, Chaos 10, 248 (2000).
26. S. Nakata et al, Physica D 115, 313 (1998).
27. A. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).
28. T. Schreiber, Phys. Rev. Lett. 85, 461 (2000).