In this paper, the chaotic motion of the driven and damped double Sine-Gordon equation is analyzed. We detect the homoclinic and heteroclinic chaos by Melnikov method. The corresponding Melnikov functions are derived. A numerical method to estimate the Melnikov integral is given and its effectiveness is illustrated through an example. Numerical simulations of homoclinic and heteroclinic chaos are precisely demonstrated through several examples. Further, we employ a state feedback control method to suppress both chaos simultaneously. Finally, numerical simulations are utilized to prove the validity of control methods.