*4.2. Fundamental and Market Makers*

As already discussed, stability regions, see again Figure 3, the presence of market makers tends to regularize prices enlarging the stability region. Outside of this region dynamics scenarios appear very similar to those in the already studied case with only the fundamental demand component. We focus on *dm* ∈ [0, 1] because values outside this interval own low interest from an economic perspective, indeed such a case complies with a reverse effect of market maker that rather absorbing amplifies the excess demand effect. Even if such dynamics could be interesting from the mathematical point of view, its absence of interest from the economical point of view suggests skipping this study. The simulation outcomes do not add any insight to those already discussed in the fundamental demand map, thus no figures about this case are reported.

**Figure 4.** The (**top-left**) shows the price evolution and (**top-right**) the phase space of fundamental map dynamics with *a* = 1, *b* = 1.9 and *c* = 0.4, while (**bottom-left**) and (**bottom-right**) show the price evolution and the phase space of fundamental map dynamics with *a* = 1, *b* = 0.7 and *c* = 1.4 note that the value of *c* at the border would be <sup>1</sup> *<sup>b</sup>* = 1.4286.

**Figure 5.** The (**top-left**) shows the price evolution and (**top-right**) the phase space of the fundamental map dynamics with *a* = 1, *b* = 1.9 and *c* = <sup>1</sup> *<sup>b</sup>* ; while (**bottom-left**) and (**bottom-right**) show the price evolution and the phase space of fundamental map dynamics with *a* = 0.8, *b* = 1 and *c* = 1.

**Figure 6.** The (**top-left**) shows the price evolution and (**top-right**) the phase space of the fundamental map dynamics with *a* = 1, *b* = 0.7 and *c* = 1.5, note that <sup>1</sup> *<sup>b</sup>* = 1.4286; while (**bottom-left**) and (**bottomright**) show the price evolution and the phase space of fundamental map dynamics with *a* = 1, *b* = 0.9 and *c* = 3.4.

t

**Figure 7.** The (**top-left**) shows the price evolution and (**top-right**) the phase space of the fundamental map dynamics with *a* = 1, *b* = 0.9 and *c* = −0.5; while (**middle-left**) and (**middle-right**) show the price evolution and the phase space of fundamental map dynamics with *a* = 1, *b* = 1.9 and *c* = −0.4, while in the (**bottom**) it holds *a* = 1, *b* = 0.9 and *c* = −0.4.

xt

**Figure 8.** Bifurcation diagrams with respect *c* of the fundamental map. (**left**) shows the diagram with *a* = 1, *b* = 1.9 and *c* ∈ [−0.6, 1.6]. (**middle**) shows the diagram with *a* = 1, *b* = 1 and *c* ∈ [−0.6, 1.6]. (**right**) shows the diagram with *a* = 0.9, *b* = 0.9 and *c* ∈ [−0.6, 1.6]. Note that having *a* and *b* lower than 1 enlarge the area in which a lower/negative *c* generates chaos.

#### *4.3. Chartist Demand*

It is of interest to see how the system behaves when only chartist demand is active. When *a* = 1 and *b* = 1, no matter the chartist contribution, the system converges to a fixed point determined by the initial conditions. When *a* < 1 and *b* = 1 the system shows a rough periodicity for small value of *e*, i.e., for small contribution of the chartist investment style, and a chaotic behavior for high values of *e*, see bifurcation diagrams in Figure 9. Lower the *b* later the chaos appears, since a low *b* reduces the amplitude of the chartist demand and hence its impact on prices, see Figure 9 in the center. Finally, if *a* is greater than 1, but still in that range in which prices do not explode, the contribution of the chartist leads directly to chaos, see Figure 9 (rhs).

**Figure 9.** Bifurcation diagrams for the chartist map with respect to the parameter *e*. (**left**) shows the diagram with *a* = 0.9, *b* = 1 and *e* ∈ [0, 7.6]. (**middle**) shows the diagram with *a* = 0.9, *b* = 0.8 and *e* ∈ [0, 7.6]. (**right**) shows the diagram with *a* = 1.1, *b* = 0.8 and *e* ∈ [0, 7.6]. Note that having the lower the *b* the later the chaos appears.
