Adaptive decision dynamics: Bifurcations, multistability and chaos

Chaotic social interaction via endogenous reactivity

“…Moreover, as observed in Section 1, also the decisional mechanisms leading to their formulations differ. In fact, in the present framework, the reactivity in (2.2) vanishes when the distance between the two opinions is large enough, while the reactivity considered in [14] is decreasing with the distance between the two opinions and tends to zero in the limit, but it never vanishes.…”

“…When comparing the dynamical system generated by the reactivity in (2.2) with the one analysed in [14], some crucial differences emerge. Indeed, although the striking similarity between the shape of the graph of the maps generating the two dynamical systems, we stress that map f in (2.7) permits an analytical treatment, we are going to perform in the next sections, while the map considered in [14] is studied mainly numerically, because of the complexity of the computations involved.…”

“…This process is modelled by an adaptive adjustment mechanism. In particular, similarly to what was assumed in [5,14], the weight with which an agent takes into account the difference between his/her own opinion and the opinion of the other agent is related to the distance between the two opinions, that is, the smaller that distance, the larger the weight given to the comparison of the opinions. However, differently from [5,14], we assume that if the distance between the two opinions is larger than a given threshold, then there is no interaction and each agent does not change his/her own opinion anymore.…”

“…Indeed, although the striking similarity between the shape of the graph of the maps generating the two dynamical systems, we stress that map f in (2.7) permits an analytical treatment, we are going to perform in the next sections, while the map considered in [14] is studied mainly numerically, because of the complexity of the computations involved. Moreover, as observed in Section 1, also the decisional mechanisms leading to their formulations differ.…”

“…A third approach would consist in working with the first three iterates of map f in (2.7), in order to find suitable intervals where to apply Theorem 1 in [10] to prove the existence of chaos in the sense of Li -Yorke, as described in conditions (T1) and (T2) in that result. This method has recently been used to rigorously prove the presence of chaotic dynamics in [14] (see [14], Proposition 4.2). Due to the similarity between the geometry of the present framework and the one in [14] we strongly believe that technique should work in the present framework as well.…”