Fractional dissipative standard map.

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.