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In this work we studied the model proposed by Blume-Emery-Griffiths, with interaction between first and second neighbors, through a Bethe lattice with finite coordination number, without magnetic field. This model was developed in an attempt to explain phenomena in which the Ising model could not be applied, as in the case of bosons, with a main focus on experimentally detected effects on blends such as superfluidity. In this work we chose not to focus on these effects and applications, but to study better the fundamental properties of the model itself and mainly to understand it's characteristics in relation to magnetization. From the hamiltonian of the system and the partial partition functions, we arrive at the recurrence relations for the magnetization at each layer of the network. Through the iteration of this map, we investigate the different phases of the system through parameters such as the wave vector and the mean quadratic magnetization. With these parameters and from the analysis of the different phases, we constructed the phase diagram, which allowed us to better study some general characteristics of this model, such as the appearance of Arnold languages and devil stairs at low temperatures.