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In this work, we have studied the spin dynamics of a synthethicAntiferromagnet (SAFM)$|$Heavy Metal (HM)$|$Ferromagnet (FM) double barriermagnetic tunnel junction (MTJ) in presence of Ruderman-Kittel-Kasuya-Yosideinteraction (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 toinvestigate the AFM dynamics of the proposed system. We found that the systemexhibits a transition from regular to damped oscillations with the increase instrength of STT for systems with weaker iDMI than RKKYI while display sustainedoscillatons for system having same order of iDMI and RKKYI. On the other handthe iDMI dominating system exhibits self-similar but aperiodic patterns inabsence of N\'eel field. In the presence of N\'eel field, the RKKYI dominatingsystems exhibit chaotic oscillations for low STT but display sustainedoscillation under moderate STT. Our results suggest that the decay time ofoscillations can be controlled via SOC. The system can works as an oscillatorfor low SOC but display nonlinear characteristics with the rise in SOC forsystems having weaker iDMI than RKKYI while an opposite characteristic arenoticed for iDMI dominating systems. We found periodic oscillations under lowexternal magnetic field in RKKYI dominating systems while moderate field arenecessary for sustained oscillation in iDMI dominating systems. Moreover, thesystem exhibits saddle-node bifurcation and chaos under moderate N\'eel fieldand SOC with suitable iDMI and RKKYI. In addition, our results indicate thatthe magnon lifetime can be enhanced by increasing the strength of iDMI for bothoptical and acoustic modes.

Effect of interfacial Dzyaloshinskii-Moriya interaction in spin dynamics  of an Antiferromagnet coupled Ferromagnetic double-barrier Magnetic Tunnel  Junction