  We consider the evolution of logistic maps under long-term memory. The memory
effects are characterized by one parameter \alpha. If it equals to zero, any
memory is absent. This leads to the ordinary discrete dynamical systems. For
\alpha = 1 the memory becomes full, and each subsequent state of the
corresponding discrete system accumulates all past states with the same weight
just as the ordinary integral of first order does in the continuous space. The
case with 0 < \alpha < 1 has the long-term memory effects. The characteristic
features are also observed for the fractional integral depending on time, and
the parameter \alpha is equivalent to the order index of fractional integral.
We study the evolution of the bifurcation diagram among \alpha = 0 and \alpha =
0.15 . The main result of this work is that the long-term memory effects make
difficulties for developing the chaos motion in such logistic maps. The
parameter \alpha\ resembles a governing parameter for the bifurcation diagram.
For \alpha\ > 0.15 the memory effects win over chaos.
