```Information flow and Lyapunov exponent on coupled map lattice

The information flow is studied in many fields, not only network
theory but also meteorology, economics and so on.

In such fields, the system on which information is transported is
often modelled as a dynamical system which consists of the source
part, the receiver part and the other parts. We regard the state of
the source part at a certain time as the input, and the information of
the input is transported to the receiver part. However, the dynamical
system cannot transport information perfectly. Uncertainty is one of
the reasons of the imperfect information flow. Unobserved states,
which are states of the other parts for instance, are uncertainty on
the system. External noise is also included in the uncertainty if it

The dynamical system is characterized by the Lyapunov exponent, and
hence it is interesting to investigate the relation between the
information flow and the Lyapunov exponent. If the Lyapunov exponent
of the system is too small, difference of the inputs damps in the way
of transportation and hence no correlation appears between the inputs
and the receiver states. If the Lyapunov exponent of the system is too
large, the receiver state responds sharply to the uncertainty. We,
therefore, conjecture that the system does not transport information
when the Lyapunov exponent is too small or too large and amount of
information flow takes a peak value when the Lyapunov exponent is one
middle value. In addition, we are also interested in the change of the
information flow in the view point from the Lyapunov exponent when we
add the noise to the system.

We calculate the information flow and the Lyapunov exponent on a
1-dimentional coupled map lattice numerically, and verify the
conjecture. The sign of the Lyapunov exponent is negative when the
amount of the information flow is maximized. The negative sign is
observed in the two types of the coupled map lattices with the
logistic map and with the tent map. We also show that, depending on
the system parameter values, the noise added to a lattice point
increases or decreases the information flow when the Lyapunov exponent
is around the middle value. If both the source part and the receiver
part are coupled with the noisy lattice point, it is no wonder that
the amount of the information flow increses because both the parts
share the common noise. One of remarkable findings is that the amount
of the information may increase even when the noisy lattice point
couples with the receiver part only.

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