On a Difference Equation with Exponentially Decreasing Nonlinearity

Figure 3

The unique equilibrium point is stable if the equation has no two-cycle, or the curve 𝑦=𝑓(𝑓(𝑥)) intersects 𝑦=𝑥 only once. Here 𝑓(𝑥) is defined in (3.1), 𝛼=0.5, 𝑞=2, where 𝑝=10 (a) and 𝑝=50 (b). For 𝑝=10 there is one intersection point (stability) while for 𝑝=50 there are three intersection points (the two-cycle is stable rather than the positive equilibrium).

(a)

(b)