On mean field fluctuations in globally coupled maps
Sergey V. Ershov and Alexei B. Potapov
Keldysh Institute for Applied Mathemetics, Moscow 125047, Russia
We have studied collective behaviour in globally coupled maps
where f is the local map, and 
is the state at the site i at
time n. Such models were originally introduced by K.Kaneko [1]. He also
found that in case of the logistic tent map 
the mean field 
and other statistical characteristics never converge to stationary values but reveal rather irregular
fluctuations around them. A similar behaviour was found by H. Chaté and P.
Manneville [2] in multidimensional map lattices.
To study this behaviour we developed further the theory of self-consistent
Frobenius Perron operator, originally introduced by K.Kaneko [1] and have
shown that spatial correlations in a simultaneous ensemble do decay as 
so in this limit the self-consistent master equation
becomes exact. On the contrary, the correlations obtained by the mixed
ensemble-time averaging do not decay due to the mean field fluctuations.
In case of the tent local map f(x)=1-a|x| we prove that the mean field
does fluctuate and that these fluctuations are due to the discontinuities in
the local map's invariant distribution. An estimate of their characteristic
amplitude 
was derived. Then, since the
self-consistent Frobenius-Perron operator that governs the evolution of
simultaneous distribution is itself a dynamical system (and nonlinear one),
one can study its attractor. The persistent mean field fluctuations indicate
that the latter is of a complex nature. Moreover, we show that it is
chaotic,
so small initial deviations of distribution do not decay (as under the
action of low-dimensional mixing maps) but grow exponentially. The largest
Lyapunov exponent, measuring the growth rate, is shown to have asymptotic 
. Therefore, the time series 
is chaotic
as well; though its irregular component is shown to be only small, and its
behaviour is nearly quasiperiodic. Our theoretical predictions have been
confirmed by numerical simulations.
In case of the local map with a flat top 
the nature of
fluctuations is mainly the same; though now the estimate of their amplitude
is 
. We also study the
relation between this problem and behaviour of averages as functions of the
parameter in 1D maps and derive the estimate for the ``mean deviation'' of
an average value: 
.