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We study the Galam majority rule dynamics with contrarian behavior and anoscillating external propaganda, in a population of agents that can adopt oneof two possible opinions. In an iteration step, a random agent interacts withother three random agents and takes the majority opinion among the agents withprobability $p(t)$ (majority behavior) or the opposite opinion with probability$1-p(t)$ (contrarian behavior). The probability of following the majority rule$p(t)$ varies with the temperature $T$ and is coupled to a time-dependentoscillating field that mimics a mass media propaganda, in a way that agents aremore likely to adopt the majority opinion when it is aligned with the sign ofthe field. We investigate the dynamics of this model on a complete graph andfind various regimes as $T$ is varied. A transition temperature $T_c$ separatesa bimodal oscillatory regime for $T<T_c$ where the population's mean opinion$m$ oscillates around a positive or a negative value, from a unimodaloscillatory regime for $T>T_c$ in which $m$ oscillates around zero. Theseregimes are characterized by the distribution of residence times that exhibitsa unique peak for a resonance temperature $T^*$, where the response of thesystem is maximum. An insight into these results is given by a mean-fieldapproach, which also shows that $T^*$ and $T_c$ are closely related.

Contrarian Majority rule model with external oscillating propaganda and individual inertias