Authors’ summary: In their recent work the authors found a correspondence between the logistic map and an infinite-dimensional linear recursion. This correspondence was exploited to obtain a rather general method for the mapping of a polynomial recursion to a linear (but infinite-dimensional) one. However, studying the literature, they found that their work is a rediscovery of a known approach called Carleman linearization.
The authors study a general method to map a nonlinear analytical recursion onto a linear one. The solution of the recursion is represented as a product of matrices whose elements depend only on the form of the recursion and not on initial conditions. First they consider the method for polynomial recursions of arbitrary degree and then the method is generalized to analytical recursions. Some properties of these matrices, such as the existence of an inverse matrix and diagonalization, are also studied. See also Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 837-838 (1997; Zbl 0905.39007), Physica A 218, 457-460 (1995) and J. Math. Phys. 37, 5828-5836 (1996; Zbl 0859.65135).