Based　on　temporal　rescaling　and　harmonic　balance,　an　extended　asymptotic　perturbation　method　for　parametrically　excited　two-degree-of-freedom　systems　with　square　and　cubic　nonlinearities　is　proposed　to　study　the　nonlinear　dynamics　under　1:2　internal　resonance.　This　asymptotic　perturbation　method　is　employed　to　transform　the　two-degree-of-freedom　nonlinear　systems　into　a　four-dimensional　nonlinear　averaged　equation　governing　the　amplitudes　and　phases　of　the　approximation　solutions.　Linear　stable　analysis　at　equilibrium　solutions　of　the　averaged　equation　is　done　to　show　bifurcations　of　periodic　motion　and　homoclinic　motions.　Furthermore,　analytical　expressions　of　homoclinic　orbits　and　heteroclinic　cycles　for　the　averaged　equation　without　dampings　are　obtained.　Considering　the　action　of　the　damping,　the　bifurcations　of　limit　cycles　are　also　investigated.　A　concrete　example　is　further　provided　to　discuss　the　correctness　and　accuracy　of　the　extended　asymptotic　perturbation　method　in　the　case　of　small-amplitude　motion　for　the　two-degree-of-freedom　nonlinear　system.