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A thorough investigation of the nonlinear dynamics of networks of memristor oscillators is a key step towards the design of systems based upon them, such as neuromorphic circuits and dense nonvolatile memories. A wide gamut of complex dynamic behaviors, including chaos, is observed even in a simple network of memristor oscillators, proposed here as a good candidate for the realization of oscillatory associative and dynamic memories. A detailed study of number and stability of all periodic and nonperiodic oscillations appearing in the network may not leave aside a preliminary deep understanding of the local and global behavior of the basic oscillator. Depending on two bifurcation parameters, controlling memristor nonlinearity, the oscillator exhibits different dynamic behaviors, analyzed here through application of state-of-the-art techniques from the theory of nonlinear dynamics to the oscillator model.