Long-term memory contribution as applied to the motion of discrete dynamical systems

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 00.15 the memory effects win over chaos.