A note on exact solutions of the logistic map

CITATION:Maritz MF. A note on exact solutions of the logistic map. Chaos. 2020 Mar;30(3):033136. doi: 10.1063/1.5125097The logistic map, whose iterations lead to period doubling and chaos as the control parameter, is increased and has three cases of the control parameter where exact solutions are known. In this paper, we show that general solutions also exist for other values of the control parameter. These solutions employ a special function, not expressible in terms of known analytical functions. A method of calculating this function numerically is proposed, and some graphs of this function are given, and its properties are discussed. The logistic map is often studied as a model of the period doubling route to chaos as its control parameter is increased. For three particular values of the control parameter, exact solutions are known. In particular, for one of these values, when the solution is chaotic, an exact solution in terms of the square of the sine function is known. In this study, a general solution for the logistic map for all values of the control parameter in a continuous range is proposed. This solution employs a special function, not expressible in terms of known analytical functions. A method of calculating this function numerically, as well as calculating its inverse numerically is proposed and some of its properties are discussed. This study, therefore, contributes to the field of nonlinear dynamics by introducing a novel tool for visualization and for investigation. The technique may be extendable to other nonlinear maps