On mean field fluctuations in globally coupled logistic-type maps

We consider mean field fluctuations in globally coupled map x(N+1)(i) = f(x(n)(i)) + epsilon(1/N) Sigma(j=1)(N) x(n)(j) when the local map f has a flat power-law top, e.g. f(x) = 1 - a\x\(beta) with beta > 1. It is shown that in the thermodynamic limit N --> infinity the amplitude of mean field fluctuations is O(epsilon(1/(beta-1))) for epsilon much less than 1 which is in good agreement with numerical calculations. We also demonstrate the relation between this problem and behaviour of averages as functions of the parameter a in 1D maps. For the latter, we give both theoretical grounds and experimental evidence that the ''mean deviation'' of an average value behaves as a power of the deviation of the parameter, e.g. integral \[x](a + delta a) - [x](a)\da similar to \delta a\(1/beta).