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Josephson junctions due to their memory and non-linearity properties have considerably impacted the exact sciences and technological fields in particular. In this study, we investigate the existence criteria of Smale's horseshoe chaos in a fractal junction by considering the non-harmonic constant of the super current of the junction. After having analytically studied the conditions of existence of this chaos by means of Melnikov's theorem and verifying its predictions by drawing the basins of attraction, we have analyzed the influence of certain control parameters on the dynamics of the system. The control and synchronization of the system have been carried out for its practical use in electronics in integrated circuits , for example, and in telecommunications for the storage of information and the securing of data. The motivation of this study is to bring out the possible and complete dynamics of a fractal junction by considering the non-harmonic constant of the super current of the junction in order to expand the old research work carried out and to promote a more detailed knowledge of the behaviors of this system.