Scaling laws for noise-induced temporal riddling in chaotic systems

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

Volumn 56, Issue 4, 1997, Pages 3897-3908

  Lai, Ying Cheng ^a  

^a UNIVERSITY OF KANSAS   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BIFURCATION (MATHEMATICS); DIFFUSION; ELECTRON ENERGY LEVELS; LYAPUNOV METHODS; MATHEMATICAL MODELS; MATHEMATICAL TRANSFORMATIONS; NONLINEAR OPTICS; PROBABILITY DENSITY FUNCTION; SPURIOUS SIGNAL NOISE;

CHAOTIC SYSTEMS; DIFFUSION APPROXIMATION; NOISE INDUCED TEMPORAL RIDDLING; SCALING LAWS; UNCERTAINTY EXPONENT;

CHAOS THEORY;

EID: 0031245212     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.56.3897     Document Type: Article

References (41)

1. M. Field and M. Golubitsky, Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature (Oxford University Press, Oxford, 1992).
2. A. Pikovsky and P. Grassberger, J. Phys. A 24, 4587 (1991); JPHAC5
3. A. Pikovsky, M. G. Rosenblum, and J. Kurths, Europhys. Lett. 34, 165 (1996); EULEEJ
4. M. Ding and W. Yang, Phys. Rev. E 54, 2489 (1996).PLEEE8
5. P. Ashwin, J. Buescu, and I. N. Stewart, Phys. Lett. A 193, 126 (1994); PYLAAG
6. Nonlinearity 9, 703 (1996); NONLE5P. Ashwin, P. J. Aston, and M. Nicol, University of Surrey, Technical Report in Mathematics and Statistics, 1996 (unpublished);
7. P. Ashwin and E. Stone, Centre for Interdisciplinary Nonlinear Mathematics Technical Report, 1997 (unpublished);
8. S. C. Venkataramani, B. Hunt, E. Ott, D. J. Gauthier, and J. C. Bienfang, Phys. Rev. Lett. 77, 5361 (1996).PRLTAO
9. J. F. Heagy, T. L. Carroll, and L. M. Pecora, Phys. Rev. Lett. 73, 3528 (1994).PRLTAO
10. J. C. Alexander, J. A. Yorke, Z. You, and I. Kan, Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 795 (1992); IJBEE4
11. I. Kan, Bull. Am. Math. Soc. 31, 68 (1994); BAMOAD
12. Y.-C. Lai and C. Grebogi, Phys. Rev. E 52, R3313 (1995); PLEEE8
13. H. Nakajima and Y. Ueda, Physica D 99, 35 (1996).PDNPDT
14. E. Ott, J. C. Sommerer, J. C. Alexander, I. Kan, and J. A. Yorke, Phys. Rev. Lett. 71, 4134 (1993); PRLTAO
15. E. Ott, J. C. Alexander, I. Kan, J. C. Sommerer, and J. A. Yorke, Physica D 76, 384 (1994).PDNPDT
16. E. Ott and J. C. Sommerer, Phys. Lett. A 188, 39 (1994).PYLAAG
17. Y.-C. Lai, C. Grebogi, J. A. Yorke, and S. C. Venkataramani, Phys. Rev. Lett. 77, 55 (1996).PRLTAO
18. Y.-C. Lai and C. Grebogi, Phys. Rev. Lett. 77, 5047 (1996).PRLTAO
19. S. C. Venkataramani, B. Hunt, and E. Ott, Phys. Rev. E 54, 1346 (1996).PLEEE8
20. In reality there can be situations where this is not true. For systems whose dynamics in the invariant subspace is also influenced by the blowout-bifurcation parameter (Formula presented) our consideration in this paper applies if there is a chaotic attractor in the invariant subspace. As (Formula presented) is changed, the attractor itself can undergo bifurcations. Thus the transverse Lyapunov exponent (Formula presented) is not necessarily a smooth or even a continuous function of (Formula presented) in the vicinity of the blowout bifurcation, as is true for the plots of the Lyapunov exponents of typical chaotic systems versus some parameter. However, in general, we expect (Formula presented) to have a smooth envelope. Our results in this paper should hold with respect to the smooth envelope of the function (Formula presented) On the other hand, situations of normal parameters arise naturally in synchronization of identical coupled chaotic oscillators.
21. C. Grebogi, S. W. McDonald, E. Ott, and J. A. Yorke, Phys. Lett. 99A, 415 (1983); PYLAAG
22. S. W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, Physica D 17, 125 (1985).PDNPDT
23. W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1966).
24. Explicit integration in Eq. (29) yields (Formula presented)where (Formula presented) and (Formula presented) are given by Eq. (15), (Formula presented) (Formula presented) (Formula presented) and (Formula presented) Note that (Formula presented) (Formula presented) in (Formula presented) are not poles. Therefore the only singularity in (Formula presented) is the square-root branch singularity contained in (Formula presented) and (Formula presented)
25. Superpersistent chaotic transients also occur in a variety of systems. See, for example, C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 50, 935 (1983); PRLTAO
26. C. Grebogi, E. Ott, and J. A. Yorke, Erg. Th. Dyna. Sys. 5, 341 (1985);
27. J. P. Crutchfield and K. Kaneko, Phys. Rev. Lett. 60, 2715 (1988); PRLTAO
28. A. Hastings and K. Higgins, Science 263, 1133 (1994); SCIEAS
29. Y.-C. Lai and R. L. Winslow, Phys. Rev. Lett. 74, 5208 (1995); PRLTAO
30. R. Braun and F. Feudel, Phys. Rev. E 53, 6562 (1996).PLEEE8
31. E. A. Spiegel, Ann. (N.Y.) Acad. Sci. 617, 305 (1981); ANYAA9
32. A. S. Pikovsky, Z. Phys. B 55, 149 (1984); ZPCMDN
33. H. Fujisaka and T. Yamada, Prog. Theor. Phys. 74, 919 (1985); PTPKAV
34. 75, 1087 (1986);
35. A. S. Pikovsky, Phys. Lett. A 165, 33 (1992); PYLAAG
36. N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70, 279 (1993); PRLTAO
37. Y.-C. Lai, Phys. Rev. E 53, R4267 (1996); PLEEE8
38. 54, 321 (1996).
39. The effect of noise on on-off intermittency has been investigated. See, for example, J. F. Heagy, N. Platt, and S. M. Hammel, Phys. Rev. E 49, 1140 (1994); PLEEE8
40. S. C. Venkataramani, T. M. Antonsen, Jr., E. Ott, and J. C. Sommerer, Physica D 96, 66 (1996).PDNPDT
41. Y. Nagai and Y.-C. Lai, Phys. Rev. E 55, R1251 (1997). PLEEE8