Upper bounds in phase synchronous weak coherent chaotic attractors

Physica D: Nonlinear Phenomena

Volumn 216, Issue 2, 2006, Pages 260-268

  Baptista, M S ^a   Pereira, T ^a   Kurths, J ^a  

^a UNIVERSITY OF POTSDAM   (Germany)

Author keywords

Chaotic phase synchronization; Phase of chaotic attractors

Indexed keywords

EQUATIONS OF MOTION; SYNCHRONIZATION;

CHAOTIC PHASE SYNCHRONIZATION; COHERENT MOTION; FUNNEL ATTRACTOR; PHASE OF CHAOTIC ATTRACTORS; PHASE SYNCHRONOUS STATES; RULKOV MAP;

CHAOS THEORY;

EID: 33748116348     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2006.02.007     Document Type: Article

References (29)

1. Pikovsky A., Rosenblum M., and Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences (2001), Cambridge University Press
2. Boccaletti S., Kurths J., Osipov G., Valladares D., and Zhou C. Phys. Rep. 366 (2002) 1
3. Kurths J., Boccaletti S., Grebogi C., et al. Chaos 13 (2003) 126
4. Rosenblum M.G., Pikovsky A.S., and Kurths J. Phys. Rev. Lett. 76 (1996) 1804
5. Rosa Jr. E., Ott E., and Hess M.H. Phys. Rev. Lett. 80 (1998) 1642
6. Park E.-H., Zaks M.A., and Kurths J. Phys. Rev. E 60 (1999) 6627
7. Parlitz U., Junge L., Lauterborn W., and Kocarev L. Phys. Rev. E 54 (1996) 2115
8. Kiss I.Z., and Hudson J.L. Phys. Rev. E 64 (2001) 046215
9. Baptista M.S., Pereira T., Sartorelli J.C., Caldas I.L., and Rosa Jr. E. Phys. Rev. E 67 (2003) 056212
10. Ticos C.M., Rosa Jr. E., Pardo W.B., et al. Phys. Rev. Lett. 85 (2000) 2929
11. Fell J., Klaver P., Elger C.E., and Fernandez G. Rev. Neurosc. 13 (2002) 299
12. Mormann F., Kreuz T., Andrzejak R.G., David P., Lehnertz K., and Elger C.E. Epilepsy Res. 53 (2003) 173
13. Osipov V.G., Bambi H., Zhou C., Ivanchenko V.M., and Kurths J. Phys. Rev. Lett. 91 (2003) 024101
14. Josić K., and Beck M. Chaos 13 (2003) 247
15. Josić K., and Mar D.J. Phys. Rev. E 64 (2001) 056234
16. r is considered to be a rational. However, as shown in Ref. [12], PS, as defined by the boundness of the phase difference, was found in two chaotic systems for a finite but very large time interval, as r approaches an irrational. Therefore, although in this work we consider r to be rational, we should make the remark that for the special situation as the one presented in Ref. [12], Eq. (20) can only be satisfied for a finite but large time if r is considered to be irrational.
17. Baptista M.S., Boccaletti S., Josić K., and Leyva I. Phys. Rev. E 69 (2004) 056228
18. j.
19. Rulkov N.F. Phys. Rev. E 65 (2002) 041922
20. A trivial case, where | over(over(A),→)' | can be analytically computed, is in a planar circular motion. The position vector is [ x, y ] = [ cos ( w t ), sin ( w t ) ], and the unitary tangent vector has the coordinates [ - sin ( w t ), cos ( w t ) ]. The derivative of the tangent vector has the coordinates over(over(A,→),' ) = [ - w cos ( w t ), - w sin w t ], and therefore, | over(over(A, →),' ) | = w, and φ ( t ) = w t.
21. 2 ( t ) | = 0. But, this assumption is too strong to detect PS, once that the trajectories of phase synchronous systems may be uncorrelated. So, the difference between events might differ by an unit, in a generic case.
22. Jalan S., and Amritkar R.E. Phys. Rev. Lett. 90 (2003) 014101
23. Pinto R.D., Gonçalves W.M., Sartorelli J.C., Caldas I.L., and Baptista M.S. Phys. Rev. E 58 (1998) 4009
24. Strong S.P., Koberle R., van Steveninck R.R.D., et al. Phys. Rev. Lett. 80 (1998) 197
25. j = - 1.2, in the next event.
26. Baptista M.S., Pereira T., Sartorelli J.C., Caldas I.L., and Kurths J. Non-transitive maps in phase synchronization. Physica D 212 (2005) 216
27. Another interesting way of detecting PS in terms of the recurrence of trajectories is by means of the Recurrence Plots as proceed in Ref. [23].
28. Romano M.C., Thiel M., Kurths J., Kiss I.Z., and Hudson J. Europhys. Lett. 71 (2005) 466
29. Thiel M., Romano M.C., and Kurths J. Phys. Lett. A 330 (2004) 343-349