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Imperfect bifurcations in opinion dynamics under external fields

We investigate, through a kinetic-exchange model, the impact that an external field, like advertising and propaganda, has on opinion dynamics. We address the situations where two opposite alternatives can be selected but the possibility of indecision also exists. In this model, individuals influence each other through pairwise interactions, which can be of agreement or disagreement, and there are also external fields that can skew decision making. Two parameters are used to model the interactions with the field: one measures the sensitivity of the individuals to be influenced, another quantifies in which direction. We study this model in a fully connected social network scenario, by means of numerical simulations of the kinetic exchange dynamics and analytical results derived from the mean-field rate equations. We show how the external bias gives rise to imperfect bifurcations, and cusp catastrophes, allowing abrupt changes and hysteresis depending on the level of disagreement in interpersonal interactions and on the strength of the external influence.