Chirality　is　an　indispensable　geometric　property　in　the　world　that　has　become　invariably　interlocked　with　life.　The　main　goal　of　this　paper　is　to　study　the　nonlinear　dynamic　behavior　and　periodic　vibration　characteristic　of　a　two-coupled-oscillator　model　in　the　optics　of　chiral　molecules.　We　systematically　discuss　the　stability　and　local　dynamic　behavior　of　the　system　with　two　pairs　of　identical　conjugate　complex　eigenvalues.　In　particular,　the　existence　and　number　of　periodic　solutions　are　investigated　by　establishing　the　curvilinear　coordinate　and　constructing　a　Poincare　map　to　improve　the　Melnikov　function.　Then,　we　verify　the　accuracy　of　the　theoretical　analysis　by　numerical　simulations,　and　take　a　comprehensive　look　at　the　nonlinear　response　of　multiple　periodic　motion　under　certain　conditions.　The　results　might　be　of　important　significance　for　the　vibration　control,　safety　stability　and　design　optimization　for　chiral　molecules.