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

Abstract: 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.