title: 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 the 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.
authors.affiliation: Institute of Radio Astronomy, Ukrainian National Academy of Sciences, 4 Chervonopraporna St., Kharkov 61002, Ukraine.
authors.forename: A A
authors.initials: AA
authors.lastname: Stanislavsky
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D008568:Memory; D008962:Models, Theoretical; D009038:Motion; D017711:Nonlinear Dynamics; D012815:Signal Processing, Computer-Assisted; D013997:Time Factors