  Derivatives of fractional order with respect to time describe long-term
memory effects. Using nonlinear differential equation with Caputo fractional
derivative of arbitrary order $\alpha>0$, we obtain discrete maps with
power-law memory. These maps are generalizations of well-known universal map.
The memory in these maps means that their present state is determined by all
past states with power-law forms of weights. Discrete map equations are
obtained by using the equivalence of the Cauchy-type problem for fractional
differential equation and the nonlinear Volterra integral equation of the
second kind.
