Nonlinear Maps with Memory

The maps for which x N +1 depends expliticly not only on X n but also on x n -1 , x n -2 ,... are considered as the analogues of the integro-differential equations of the non-equilibrium statistical physics. The transformation to the memoryless form is discussed. The asymptotic forms ( n → ∞ ) either are linear or reproduce the behaviour of the corresponding nonlinear maps without memory, or are the linear combination of both cases, depending on the memory properties, especially on the length of the memory and on the convolution properties.