Controlling chaos in a pendulum equation with ultra-subharmonic resonances

Analytical and numerical results concerning control of chaos in a pendulum equation with parametric and external excitations are given by using Melnikov methods. We give the necessary conditions of chaos control with ultra-subharmonic resonances (i.e. X=x ¼ p=q; q > 1; p; q are prime), where homoclinic chaos or heteroclinic chaos can be inhibited. Numerical simulations show that chaotic behavior can be converted to period-nq (n 2 Zþ) orbits by adjusting amplitude and phase-difference of parametric excitation, and the distribution of maximum Lyapunov exponents in parameter-plane ðW; bÞ gives the regions in which chaos can be controlled.