Alpha-Families of Maps and General Properties of Fractional Dynamical Systems
We introduce the notion of α-Families of Maps depending on a
single parameter α > 0 which is the order of the fractional derivative
in the nonlinear fractional differential equation describing a system
experiencing periodic kicks. The α-Families of Maps represent a
very general form of the multidimensional maps with power law memory
and may be applicable to studying such systems with memory as viscoelastic
materials, electromagnetic fields in dielectric media, Hamiltonian
systems, adaptation in biological systems, human memory, etc.
Fractional maps are invaluable tools in studying the general properties
of the nonlinear fractional dynamical systems. On the examples
of the fractional Logistic and Standard α-Families of Maps we demonstrate
that the nonlinear fractional dynamical systems may have solutions in
the form of periodic sinks, attracting slow diverging trajectories,
attracting accelerator mode trajectories, chaotic attractors, and
cascade of bifurcations type trajectories with some new properties.