A successive integration technique for the solution of the Duffing oscillator equation with damping and excitation

In this paper, a successive integration technique is suggested for solving the Duffing oscillator equation with damping and excitation. In the technique, one performs integrations from some initial values over a wider range. Some of them may even be divergent; however, some of them will give a convergent result such that the periodic condition of motion is satisfied. In fact, the convergent result represents a stable periodic solution for the motion. If a convergent result or a stable periodic solution is obtained, the stability test is passed. A harmonic balance method in conjunction with the successive integration technique (abbreviated as HBMSIT) is also suggested. In the method, the initial values are obtained from the harmonic balance method. Therefore, the HBMSIT belongs to the successive integration techniques. Many examples with calculated results are presented.