  In this work, we have studied the spin dynamics of a synthethic
Antiferromagnet (SAFM)$|$Heavy Metal (HM)$|$Ferromagnet (FM) double barrier
magnetic tunnel junction (MTJ) in presence of Ruderman-Kittel-Kasuya-Yoside
interaction (RKKYI), interfacial Dzyaloshinskii-Moriya interaction (iDMI),
N\'eel field and Spin-Orbit Coupling (SOC) with different Spin Transfer Torque
(STT). We employ Landau-Lifshitz-Gilbert-Slonczewski (LLGS) 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 weaker iDMI than RKKYI while display sustained
oscillatons for system having same order of iDMI and RKKYI. On the other hand
the iDMI dominating system exhibits self-similar but aperiodic patterns in
absence of N\'eel field. In the presence of N\'eel field, the RKKYI dominating
systems exhibit chaotic oscillations for low STT but display sustained
oscillation under moderate STT. Our results suggest that the decay time of
oscillations can be controlled via SOC. The system can works as an oscillator
for low SOC but display nonlinear characteristics with the rise in SOC for
systems having weaker iDMI than RKKYI while an opposite characteristic are
noticed for iDMI dominating systems. We found periodic oscillations under low
external magnetic field in RKKYI dominating systems while moderate field are
necessary for sustained oscillation in iDMI dominating systems. Moreover, the
system exhibits saddle-node bifurcation and chaos under moderate N\'eel field
and SOC with suitable iDMI and RKKYI. In addition, our results indicate that
the magnon lifetime can be enhanced by increasing the strength of iDMI for both
optical and acoustic modes.
