Incollection,
On the phase diagram of globally coupled piecewise linear maps
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
Globally coupled chaotic map is one of the simplest dynamical system
which exhibit high dimensional chaos. Despite its simple structure,
behavior of the system in `thermodynamic limit' is not yet clearly
understood. In this presentation, I will report some numerical
results on globally coupled piecewise linear map systems
eqnarray*
x_i(t+1)=(1-k)f(x_i(t))+kN\sum_j=1^N f_j(t),\cr
f(x)=łeft\
arrayll
a(-2-x),& ( x<-1 ) \\
ax,& ( -1 x 1 )\\
a(2-x),& ( x>1 )
array
\right.\
eqnarray*
with relatively large number of elements. The results of numerical
experimets on systems with large number of elements indicate that
intermittency with power-law distribution of laminer duration length
appear in wide area in parameter space. (Upper part of the figure is
an example of intermittent mean-field sequence ($a=3.0$, $k=0.56$, $N=50$).
Average over 1024 step is plotted for (60000 x 1024) time steps. Lower
part of teh figure is power spectrum of mean-field sequence (for
$a=3.0$, $k=0.54$, $N=64$). Even in the parameter region where
intermittency is observed, the limiting behavior of the system in the
increasing system size $N$ are seemingly have some variety. Depending
on the parameter $a$ and $k$, non-stationary behavior, as well as
ferromagnetic phase like and anti-ferromagnetic phase like ones, are
observed.
