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42. For instance, when and f = 4, there exist M1 and m1 with while M2 and m2 have not emerged yet.
43. In order to show that when a1 and a2 are large enough, we can proceed as follows. First of all, let us observe that when a1 and a2 are sufficiently large, then and thus Let us now consider separately the cases and If we rewrite as follows: observing that and that for while for Hence, since and when and a1 is large enough we have that as desired. A very similar argument can be employed for the case in which suitably rewriting
44. We stress that, when drawing a bifurcation diagram, we chose the starting points by looking closely at the modifications in the shape of the graph of the first iterates of the map and also at the time series of the state variable, on varying the value of the parameter under consideration. Hence, we do believe our choices being well pondered and thus preventing loss of information, such as the existence of further branches not considered in the paper, because their presence is excluded by the shape of and of its iterates. Moreover, because of the way we selected the initial conditions for our bifurcation diagrams, our choices about the number of decimals cannot have a deep influence on the obtained results. Nonetheless, in order to make the discussion more consistent, we tried to homogenize the number of utilized decimals along the paper, setting it equal to three, especially in the descriptions of the bifurcation diagrams. We remark that when the number of decimals is less than three is because those are exact numbers and thus we do not need to round or truncate them.
45. We stress that there are two cases in which we end up with a unique attractor: the first one is when, like in the present framework, there are no (more) external attractors and the internal ones merge through a homoclinic bifurcation; the second possibility, which we observe in Figure 18, is instead that the internal attractor disappears and all orbits are attracted by the external one.

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In the present paper, we consider a nonlinear financial market model in which, in order to decrease the complexity of the dynamics and to achieve price stabilization, we introduce a price variation limiter mechanism, which in each period bounds the price variation so that the current price is forced to belong to a certain interval determined by the price realization in the previous period. More precisely, we introduce such mechanism into a financial market model in which the price dynamics are described by a sigmoidal price adjustment mechanism characterized by the presence of two asymptotes that bound the price variation and thus the dynamics. We show that the presence of our asymptotes prevents divergence and negativity issues. Moreover, we prove that the basins of attraction are complicated only under suitable conditions on the parameters and that chaos arises just when the price limiters are loose enough. On the other hand, for some suitable parameter configurations, we detect multistability phenomena characterized by the presence of up to three coexisting attractors.


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