A fractional map, also called a fractional-order map, is a discrete dynamical system with memory whose update weights are associated with a fractional derivative, fractional difference, or fractional differential equation. A general one-dimensional map with long-term memory can be written
 |
where 
is the memory kernel, 
is a function, 
characterizes the memory, and 
is a parameter. Fractional maps
typically use kernels with power-law decay (Edelman 2014).
For example, sampling a periodically kicked fractional differential equation defined using the Caputo fractional derivative gives the unit-step family
 |
for 
,
where 
is the gamma function. As 
, this family reduces to the associated one-step
map. Different choices of Riemann-Liouville
fractional derivatives or Caputo fractional
derivatives and continuous or discrete fractional
formulations lead to different fractional maps, so the term denotes a class
rather than a unique map.
