  Using kicked differential equations of motion with derivatives of noninteger
orders, we obtain generalizations of the dissipative standard map. The main
property of these generalized maps, which are called fractional maps, is
long-term memory.The memory effect in the fractional maps means that their
present state of evolution depends on all past states with special forms of
weights. Already a small deviation of the order of derivative from the integer
value corresponding to the regular dissipative standard map (small memory
effects) leads to the qualitatively new behavior of the corresponding
attractors. The fractional dissipative standard maps are used to demonstrate a
new type of fractional attractors in the wide range of the fractional orders of
derivatives.
