| Summary: | We study the Galam majority rule dynamics with contrarian behavior and an oscillating external propaganda in a population of agents that can adopt one of two possible opinions. In an iteration step, a random agent interacts with three other random agents and takes the majority opinion among the agents with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> (majority behavior) or the opposite opinion with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>−</mo><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> (contrarian behavior). The probability of following the majority rule <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> varies with the temperature <i>T</i> and is coupled to a time-dependent oscillating field that mimics a mass media propaganda, in a way that agents are more likely to adopt the majority opinion when it is aligned with the sign of the field. We investigate the dynamics of this model on a complete graph and find various regimes as <i>T</i> is varied. A transition temperature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>c</mi></msub></semantics></math></inline-formula> separates a bimodal oscillatory regime for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo><</mo><msub><mi>T</mi><mi>c</mi></msub></mrow></semantics></math></inline-formula>, where the population’s mean opinion <i>m</i> oscillates around a positive or a negative value from a unimodal oscillatory regime for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>></mo><msub><mi>T</mi><mi>c</mi></msub></mrow></semantics></math></inline-formula> in which <i>m</i> oscillates around zero. These regimes are characterized by the distribution of residence times that exhibit a unique peak for a resonance temperature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>T</mi><mo>*</mo></msup></semantics></math></inline-formula>, where the response of the system is maximum. An insight into these results is given by a mean-field approach, which also shows that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>T</mi><mo>*</mo></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>c</mi></msub></semantics></math></inline-formula> are closely related.
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