  In this paper, we study the existence of bifurcation of a van der Pol-Duffing
oscillator with quintic terms and its quasi-periodic solutions by means of
qualitative and bifurcation theories. Firstly, we analyze the autonomous system
and find that it has two kinds of local bifurcations and a global bifurcation:
pitchfork bifurcation, Hopf bifurcation, homoclinic bifurcation. It is worth
noting that the disappearance of the homoclinic orbit is synchronized with the
emergence of a large limit cycle. Then, by discussing the stability of
equilibria at infinity and the orientation of the trajectory, the existence and
stability of limit circles of the autonomous system are analyzed by combining
the Poincar\'{e}-Bendixson theorem and the index theory. The global phase
portrait and the numerical simulation of the autonomous system in different
parameter values are given. Finally, the existence of periodic and
quasi-periodic solutions to periodic forced system is proved by a KAM theorem.
