Inhibition of chaos in a pendulum equation

Analytical and numerical results concerning the inhibition of chaos in a pendulum equation with parametric and external excitations are given by using Melnikov methods proposed in [Chacón R, Palmero F, Balibrea F. Taming chaos in a driven Josephson junction. Int J Bifurcat Chaos 2001;11(7):1897-909]. We theoretically give parameter-space region and intervals of initial phase-difference, where homoclinic chaos or herteroclinic chaos can be inhibited. Numerical simulations show the consistency and difference with the theoretical analysis and the chaotic behavior can be converted to periodic orbits by adjusting amplitude and phase-difference of parametric excitation. Moreover, we consider the influence of parametric frequency on maximum Lyapunov exponent (LE) for different phase-differences, and give the distribution of maximum Lyapunov exponents in parameter-plane, which indicates the regions of non-chaotic states (non-positive LE) and chaotic states (positive LE).