Implications of varying communication speeds in “globally” coupled maps

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

Volumn 57, Issue 4, 1998, Pages 4041-4045

  Sinha, Sudeshna ^a  

^a INSTITUTE OF MATHEMATICAL SCIENCES   (India)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4243339617     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.57.4041     Document Type: Article

References (37)

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18. There is scope and relevance for introducing a more general definition of the “effective mean field” where the influence of the delayed states is weighted by some function of the delay.
19. Note that in our study we have updated the sites in a certain fixed order (say, starting from lattice site i=1, going on to i=N) and at each step we use the current values of x(i) to compute the effective mean field via Eq. (2). On the other hand, if one computed the effective mean field synchronously at the beginning of each time step and updated each site simultaneously, the resulting numbers could sometimes be a little different, but the emergent qualitative picture would be identical. So the results quoted here hold, regardless of the updating algorithm.
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32. One should also note that the above transient behavior is different from the long transience at around ε=0.5, which marks the transition to a spatiotemporally coherent phase. Unlike our situation, the phenomena at ε∼0.5 is akin to the long transience present at bifurcation points due to marginal stability and is characterized by low-frequency noise in the power spectra of both the individual elements and the mean field.
33. See Fig. 55 for an example of a single site spectrum (which is the topmost curve).
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36. The observations in this paper complement a recent study on GCM [G. Perez, S. Sinha, and H. A. Cerdeira, Phys. Rev. E 54, 6936 (1996)] where mean fields were either delayed by a certain amount D or averaged over P contiguous iterations. Our scenario, however, is not equivalent to either delayed or averaged mean field, but involves a bit of both. On the one hand, it has a summation over sites at different time steps (akin to averaging) and, on the other hand, it also has a sum over a subset of sites delayed by a certain amount (akin to a delayed mean field). Analyzing the magnitude of influence of these two competing trends (with respect to increasing D and P on varying subsets of the lattice) may shed some light on the underlying phenomena.PLEEE8
37. Note here that there are two prominent frequencies in the system, given by the most pronounced peak at S→N/2 and at S→1 (see Fig. 66). At intermediate S lying in between these two limits, we observe less pronounced peaks [often with finer frequencies resolved around the broadest peak(s)] at roughly these two positions (as is evident from the case of S=50 in Fig. 66).