Dominance of Milnor attractors in globally coupled dynamical systems with more than [formula presented] degrees of freedom

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

Volumn 66, Issue 5, 2002, Pages 4-

  Kaneko, Kunihiko ^a  

^a UNIVERSITY OF TOKYO   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

BIFURCATION (MATHEMATICS); COMBINATORIAL MATHEMATICS; JOSEPHSON JUNCTION DEVICES; MATHEMATICAL MODELS; NEURAL NETWORKS; PROBABILITY; PROBLEM SOLVING;

COMBINATORIAL COMPLEXITY; DYNAMICAL SYSTEMS; MILNOR ATTRACTORS; PERMUTATIONS;

DEGREES OF FREEDOM (MECHANICS);

ARTICLE;

EID: 41349083976     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.66.055201     Document Type: Article

References (28)

1. K. Kaneko, Phys. Rev. Lett. 78, 2736 (1997);
2. K. KanekoPhysica D 124, 322 (1998).
3. M. Timme, F. Wolf, and T. Geisel, Phys. Rev. Lett. 89, 154105 (2002).
4. J. Milnor, Commun. Math. Phys. 99, 177 (1985);
5. Commun. Math. Phys.J. Milnor102, 517 (1985), Milnor’s definition of attractors includes usual (asymptotically stable) attractors. Following terminology recently used, Milnor attractors refer only to asympototically unstable ones.
6. P. Ashwin, J. Buescu, and I. Stuart, Phys. Lett. A 193, 126 (1994);
7. P. AshwinJ. BuescuI. StuartNonlinearity 9, 703 (1996).
8. J.C. Sommerer and E. Ott, Nature (London) 365, 138 (1993);
9. E. Ott, Phys. Rev. Lett. 71, 4134 (1993);
10. Y-C. Lai and C. Grebogi, Phys. Rev. E 53, 1371 (1996).
11. K. Kaneko, Phys. Rev. Lett. 63, 219 (1989);
12. K. KanekoPhysica D 41, 38 (1990).
13. The basin is riddled 5, as discussed in Ref. 6.
14. In the estimate here, “attractors” with an escape rate of more than 50% are not counted as Milnor attractors in order to avoid possible inclusion of transient states. Even if they are included, however, there is only a small increase in the basin fraction. Note that in spite of the “severe” criterion adopted here, Milnor attractors are dominant. See also Ref. 1.
15. D. Dominguez and H.A. Cerdeira, Phys. Rev. Lett. 71, 3359 (1993).
16. S.C. Venkataramani, B.R. Hunt, and E. Ott, Phys. Rev. E 54, 1346 (1996).
17. K. Kaneko, Physica D 75, 55 (1994).
18. The degree of freedoms we use in the present paper is the number of units that has orbital instability. In the Josephson junction array model, each element with temporal forcing can show chaotic instability, and hence the “number” we mean is just the number of elements N, not (Formula presented)
19. Since the perfect permutational symmetry is not required, the present result will be extended to a system with weak local couplings besides the global one. It cannot be directly applied to a system with local coupling only (such as coupled map lattice or partial differential equation). However, by considering the number of degrees of freedom that are directly coupled within a correlation length in space, the present argument may be applied to such a system.
20. G.A. Miller, The psychology of communication (Basic Books, N.Y., 1975).
21. J.S. Nicolis and I. Tsuda, Bull. Math. Biol. 47, 343 (1985).
22. Still, Milnor attractors, for example, may be relevant to some cognitive processes, say for chaotic search processes 1 (see also Refs. 1718) as may be supported in an experiment on the olfactory bulb 17.
23. W. Freeman and C.A. Skarda, Brain Res. Rev. 10, 147 (1985);
24. W. FreemanC.A. SkardaPhysica D 75, 151 (1994).
25. I. Tsuda, Neural Networks 5, 313 (1992).
26. One possible means to resolve the (Formula presented) bound is the use of module structure instead of all-to-all couplings.
27. S. Sasa and T.S. Komatsu, Phys. Rev. Lett. 82, 912 (1999);
28. N. Nakagawa and K. Kaneko, Phys. Rev. E 64, 055205(R) (2001).