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Today many scientists see nonlinear science as the most important frontier for the fundamental understanding of Nature. Especially, the recent implementation of the memristor has led to the interpretation of phenomena not only in electronic devices but also in biological systems. Many research teams work on projects, which use memristor to simulate the behavior of biological synapses. Based on this research approach, we have studied using computer simulations, the dynamic behavior of two coupled, via a memristor, identical nonlinear circuits, which play the role of “neurons”. The proposed memristor is a flux-controlled memristor where the relation between charge and magnetic flux is a smooth continuous cubic function. Very interesting synchronization phenomena such as inverse π-lag synchronization and complete chaotic synchronization were observed.