A new discrete-time logistic equation having no overshoot and no chaos

This paper proposes a new discretization method for logistic differential equations, for which the popular forward difference model causes chaos and overshoot unless the discrete-time period is chosen to be sufficiently small. While the exact discrete-time model gives exact values at discrete-time instants, it is not really a logistic equation anymore and requires an up-date of a complex gain at each discrete-time instant. In contrast, the proposed discrete-time model has no overshoot for any discretization period T, is more accurate than the forward difference model for large T in general, and requires no update of the gain unlike the exact model.