On a Difference Equation with Exponentially Decreasing Nonlinearity

Figure 5

The bifurcation diagram for (2.1), with 𝛼=0.5,𝑝=2, as 𝑞 is growing (a) and for the perturbed model (4.6) with 𝛼=0.5,𝑝=2,and𝜆=0.2 (b). We see that there is a deterministic chaos in (2.1) for 𝑞 large enough, while the perturbed model experiences chaos and then (as 𝑞 grows) the break of chaos through period-halving bifurcations.

(a)

(b)