Fig. 1. Extended plane of the parameters e and a. The region M represent the set of economically feasible values of the parameters. The steady state O is stable inside the region LJK.

Fig. 2. For a=1.2, b=0.5, c=0.4, d=0.1 and e=−0.9, just after the Neimark–Hopf bifurcation, a trajectory is numerically generated starting from an initial condition close to the fixed point O: (a) the trajectory is represented in the phase plane (the early iterations, representing the transient, have not been plotted). The white region represents the basin of attraction of Γ, the grey region represents the basin of infinity, i.e., the set of points generating diverging trajectories; (b) a portion of the same trajectory plotted in (a) is represented versus time.

Fig. 3. For a=1.2, b=0.5, c=0.4, d=0.1 and e=0.55 the generic trajectory starting from points of the white region converges to an annular chaotic attractor, as shown in (a), whereas the points of the grey region generate diverging trajectories. In (b) a portion of the trajectory shown in (a) is represented versus time.

Fig. 4. For a=1, b=0.5, c=0.4, d=2 and e=2 the maximum number of preimages is 3. (a) Critical curves of rank-0. (b) Critical curves of rank-1, which separate regions Z_k whose points have k distinct rank-1 preimages. (c) Riemann foliation corresponding to the critical curves shown in (b). Different sheets are associated with different inverses, and the critical curves LC represent folds which join different sheets.

Fig. 5. For a=1, b=0.5, c=0.4, d=0.1 and e=0.1 the maximum number of preimages is 5. (a) Critical curves of rank-1, which separates regions Z_k whose points have k distinct rank-1 preimages. (b) Riemann foliation corresponding to the critical curves shown in (a).

Fig. 6. For a=1, b=0.5, c=0.4, d=0.1 and e=−0.9 the maximum number of preimages is 9. (a) Critical curves of rank-1, which separates regions Z_k whose points have k distinct rank-1 preimages. (b) Riemann foliation corresponding to the critical curves shown in (a).

Fig. 7. (a) Just after the Neimark–Hopf bifurcation, for a=1.2, b=0.5, c=0.4, d=0.1 and e=−0.6, ΓZ₅ and Γ∩LC₋₁=. The five disjoint preimages of Γ are denoted by , k=1,…,5. (b) For a=1.2, b=0.5, c=0.4, d=0.1 and e=0, Γ intersects LC₋₁⁽³⁾ and LC₋₁⁽⁴⁾ and convolutions appear along Γ. (c) For e=0.4 (the other parameters are the same as in (a) and (b)) the convolutions become more evident. (d) For e=0.5 the invariant curve Γ no longer exist and it is substituted by a more complex attracting set, characterized by the presence of loops.

Fig. 8. For a=7.45, b=0.5, c=0.4, d=0.1 and e=−5.6 an attracting cycle of period 6 exists together with an attracting closed invariant curve Γ, on which quasi-periodic motion is numerically observed. Some of the periodic points, denoted by the numbers 1, 3 and 5, are inside Γ and the others are outside Γ.

Fig. 9. (a) For a=1.2, b=0.5, c=0.4, d=0.1 and e=1.1 the generic bounded trajectory is attracted towards a chaotic area. (b) For a=1.2, b=0.5, c=0.4, d=0.1 and e=1.2 the bounded trajectories converge to one of the two symmetric stable cycles of period 7 whose periodic points are respectively denoted by and .

Fig. 10. For the same set of parameters used to obtain Fig. 9(a), the two portions of LC₋₁ included inside the chaotic area, indicated by the arrows, are iterated in order to obtain the boundary of the chaotic area. (a) After four iterations the images of the two segments of LC₋₁, denoted by LC, LC₁, LC₂, LC₃, give the outer boundary of the chaotic area. (b) After seven iterations the whole boundary of the chaotic area is obtained.

Fig. 11. Changes of the structure of the basin of infinity (grey region) due to changes in the parameter e, the other parameters being a=1, b=0.5, c=0.4 and d=0.1. (a) For e=−0.6 the set , whose points generate bounded trajectories, is a simply connected set, represented by the white region. The critical curves LC₋₁ and LC are also represented (compare with Fig. 7(a)). (b) For e=−0.5, after the contacts between LC and the boundary of , nonconnected portion (holes) of appear, nested inside the white region. (c) At e=−0.38 a contact between LC⁽¹⁾ and LC⁽²⁾ and the two cusps of causes a reunion of the holes with the immediate basin of infinity, so that becomes again a simply connected set. (d) At e=0.3 a contact of the other two branches of LC causes the creation of other families of holes of .

Fig. 12. For a=1.12, b=0.5, c=0.4, d=0.5 and e=1.07 there are three coexisting bounded attractors: a cycle C₁₀ of period 10 and two cycles of period seven. The respective basins of attraction are represented by different colors: white for the basin of C₁₀, red and green for the basins of the two symmetric cycles of period 7. The periodic points are represented by black dots. The grey region represents the basin of infinity.