  In this paper the author presents the results of the preliminary
investigation of fractional dynamical systems based on the results of numerical
simulations of fractional maps. Fractional maps are equivalent to fractional
differential equations describing systems experiencing periodic kicks. Their
properties depend on the value of two parameters: the non-linearity parameter,
which arises from the corresponding regular dynamical systems; and the memory
parameter which is the order of the fractional derivative in the corresponding
non-linear fractional differential equations. The examples of the fractional
Standard and Logistic maps demonstrate that phase space of non-linear
fractional dynamical systems may contain periodic sinks, attracting slow
diverging trajectories, attracting accelerator mode trajectories, chaotic
attractors, and cascade of bifurcations type trajectories whose properties are
different from properties of attractors in regular dynamical systems. The
author argues that discovered properties should be evident in the natural
(biological, psychological, physical, etc.) and engineering systems with
power-law memory.
