Asynchronous updating of coupled maps leads to synchronization

Chaos

Volumn 10, Issue 2, 2000, Pages 350-358

  Mehta, Mitaxi ^a   Sinha, Sudeshna ^a  

^a INSTITUTE OF MATHEMATICAL SCIENCES   (India)

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[No Author keywords available]

Indexed keywords

EID: 0034195483     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.166501     Document Type: Article

References (21)

1. Theory and Applications of Coupled Map Lattices, (Wiley, 1993); J. Crutchfield and K. Kaneko, in Directions in Chaos, edited by H. Bai-Lin (1987, World Scientific, Singapore), and references therein.
2. Theory and Applications of Coupled Map Lattices, (Wiley, 1993); J. Crutchfield and K. Kaneko, in Directions in Chaos, edited by H. Bai-Lin (1987, World Scientific, Singapore), and references therein.
3. J. Hertz, A. Krogh, and R. G. Palmer, Introduction to the Theory of Neural Computation (Addison-Wesley, Reading, 1991).
4. P. Marcq, H. Chate, and P. Manneville, Phys. Rev. Lett. 77. 4003 (1996); Phys. Rev. E 55, 2606 (1997).
5. P. Marcq, H. Chate, and P. Manneville, Phys. Rev. Lett. 77. 4003 (1996); Phys. Rev. E 55, 2606 (1997).
6. J. Rolf, T. Bohr, and M. H. Jensen, Phys. Rev. E 57, R2503 (1998).
7. In Ref. 4 the significance of asynchronous updating was argued on the basis of the connection between CML and physical spatially extended systems (which is not straightforward at all). Heuristically, the closest physical parallel is probably a collection of coupled elements, each evolving chaotically. In order to derive a discrete local map for the sites, one may apply a Poincare section. In the absence of an external periodic forcing, the time interval between consecutive crossings of the section will generically not be a constant and in fact vary from unit to unit in space. The state of the extended system at fixed time intervals can then be roughly mimicked by using asynchronous updates, in which the elements experience slightly different update times.
8. There are cases where synchronous updates can lead to complicated spatial structures in cellular automata [Ref: M. Nowak and R. M. May, Nature (London) 359, 826 (1992)] which disappear under asynchronous updating
9. [B. A. Huberman and N. Glance, Proc. Natl. Acad. Sci. USA 90, 7716 (1993)] and are therefore considered unphysical in the absence of an external clock.
10. In the context of Monte Carlo simulations, the importance of different kinds of updating has been appreciated. For instance, R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987), studied the differences induced by varied asynchronous updating schemes in Monte Carlo simulations of equilibrium systems.
11. This kind of updating is usual in the Monte Carlo context [see R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987). In fact, updating in two parallel clusters (bipartite lattice) is often done in order to speed up simulations on parallel machines.
12. E. D. Lumer and G. Nicolis, Physica D 71, 440 (1994).
13. G. Perez, S. Sinha, and H.A. Cerdeira, unpublished (1997).
14. Note, for coupling form g(x)=f(x), in the range ∈∼0.8-0.85, we observed very long quasistationary transient configurations, that have antiferro spatial order broken by thin lines of kinks or defects, for certain initial conditions. We are further investigating the nature of this transient phenomenon.
15. Some references of this rapidly growing field of interest are: L. Pecora and T. M. Caroll, Phys. Rev. Lett. 64, 821 (1990); Phys. Rev. A 44, 2374 (1991); T. L. Caroll, Phys. Rev. E 50, 2580 (1994); F. Heagy, T. L. Caroll, and L. M. Pecora, 50, 1874 (1994); Auerbach, Phys. Rev. Lett. 72, 1184 (1994).
16. Some references of this rapidly growing field of interest are: L. Pecora and T. M. Caroll, Phys. Rev. Lett. 64, 821 (1990); Phys. Rev. A 44, 2374 (1991); T. L. Caroll, Phys. Rev. E 50, 2580 (1994); F. Heagy, T. L. Caroll, and L. M. Pecora, 50, 1874 (1994); Auerbach, Phys. Rev. Lett. 72, 1184 (1994).
17. Some references of this rapidly growing field of interest are: L. Pecora and T. M. Caroll, Phys. Rev. Lett. 64, 821 (1990); Phys. Rev. A 44, 2374 (1991); T. L. Caroll, Phys. Rev. E 50, 2580 (1994); F. Heagy, T. L. Caroll, and L. M. Pecora, 50, 1874 (1994); Auerbach, Phys. Rev. Lett. 72, 1184 (1994).
18. Some references of this rapidly growing field of interest are: L. Pecora and T. M. Caroll, Phys. Rev. Lett. 64, 821 (1990); Phys. Rev. A 44, 2374 (1991); T. L. Caroll, Phys. Rev. E 50, 2580 (1994); F. Heagy, T. L. Caroll, and L. M. Pecora, 50, 1874 (1994); Auerbach, Phys. Rev. Lett. 72, 1184 (1994).
19. Some references of this rapidly growing field of interest are: L. Pecora and T. M. Caroll, Phys. Rev. Lett. 64, 821 (1990); Phys. Rev. A 44, 2374 (1991); T. L. Caroll, Phys. Rev. E 50, 2580 (1994); F. Heagy, T. L. Caroll, and L. M. Pecora, 50, 1874 (1994); D. Auerbach, Phys. Rev. Lett. 72, 1184 (1994).
20. W. L. Ditto et al.. Nature (London) 376, 465 (1995).
21. Note that very small lattices (of 2-4 maps) can synchronize spatially under simultaneous updates (though usually not to a temporal fixed point), in certain cases, as reported in some references in Ref. 12. Of course, here we have been interested in synchronization of very large systems and typically we deal with very large numbers of maps (N> 100). Such large system sizes also validate our mean-field approach in the analytical treatment of the problem.