In this work, we have studied the spin dynamics of a synthetic antiferromagnet (AFM)/heavy metal/ferromagnet double barrier magnetic tunnel junction in the presence of Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, interfacial Dzyaloshinskii–Moriya (iDM) interaction, Néel field, and Spin–Orbit Coupling (SOC) with different Spin-Transfer Torque (STT). We employ the Landau–Lifshitz–Gilbert–Slonczewski equation to investigate the AFM dynamics of the proposed system. We found that the system exhibits a transition from regular to damped oscillations with the increase in strength of STT for systems with a weaker strength of iDM interaction than RKKY interaction while displaying sustained oscillations for systems having the same order of RKKY and iDM interactions. On the other hand, the systems with sufficiently strong iDM interaction strength exhibit self-similar but aperiodic patterns in the absence of the Néel field. In the presence of the Néel field, the RKKY interaction dominating systems exhibit chaotic oscillations for low STT but display sustained oscillations under moderate STT. Our results suggest that the decay time of oscillations can be controlled via SOC. The system can work as an oscillator for low SOC but displays non-linear characteristics with the rise in SOC for systems having weaker iDM interaction than RKKY interactions. In contrast, opposite characteristics are noticed for iDM interaction dominating systems. We found periodic oscillations under low external magnetic fields in RKKY interaction dominating systems. However, moderate fields are necessary for sustained oscillation in iDM interaction dominating systems. Moreover, the system exhibits saddle-node bifurcations and chaos under moderate Néel field and SOC with suitable RKKY and iDM interactions. In addition, our results indicate that the magnon lifetime can be enhanced by increasing the strength of iDM interaction for both optical and acoustic modes.