Chaos control in a pendulum system with excitations and phase shift

Melnikov methods are used for suppressing homoclinic and heteroclinic chaos of a pendulum system with a phase shift and excitations. This method is based on the addition of adjustable amplitude and phase-difference of parametric excitation. Theoretically, we give the criteria of suppression of homoclinic and heteroclinic chaos, respectively. Numerical simulations are given to illustrate the effect of the chaos control in this system. Moreover, we calculate the maximum Lyapunov exponents (LEs) in parameter plane, and study how to vary the maximum LE when the parametric frequency varies.