The authors investigate bifurcations and existence of chaos in the following three-well duffing system with two periodic forcings \[ \ddot{x}+x(x^2-a^2)(x^2-1)+\overline{\delta}\dot{x}+\overline{\gamma}_1\cos{(\omega_1 t)}+\overline{\gamma}_2\cos{(\omega_2 t)}, \] where \(a,\overline{\gamma}_j\) and \(\omega_j(j=1,2)\) are real parameters. Physically, \(\overline{\delta}\) can be regarded as dissipation or damping \(\overline{\gamma}_j\) and \(\omega_j\) as the amplitudes and the frequences of the forcings, i.e. \(\overline{\delta},\;\overline{\gamma}_j,\;\omega_j \geq 0,\;0<a<1\). The fixed points and phase portraits for the unperturbed system are described. The conditions of chaos existence under periodic perturbation resulting from the homoclinic and heteroclinic bifurcations are performed at the usage of Melnikov’s method. The condition of the existence of chaotic motion under quasi-periodic perturbation for averaged system are derived by using second-order averaging method and Melnikov’s method. Numerical simulations illustrating the theoretical analysis are given.