On a Difference Equation with Exponentially Decreasing Nonlinearity

Figure 4

The bifurcation diagram for (2.1), with 𝛼=0.5,𝑞=2, as 𝑝 is growing (a), and for the perturbed model (4.6) with 𝛼=0.5,𝑞=2,and𝜆=0.2 (b). For 𝑝<𝑝_𝑐=0.75𝑒³≈15.06 all trajectories of (2.1) converge to the equilibrium point 𝑥. As 𝑝 increases beyond 𝑝_𝑐, there is a series of period-doubling bifurcations leading to a deterministic chaos. We see that the two bifurcation diagrams are similar, with bifurcations for larger 𝑝 in (4.6) compared to (2.1); chaotic behavior is observed in the perturbed equation for any 𝑝, as well as in the original one.

(a)

(b)