Remarks on transitions order-chaos induced by the shape of the periodic excitation in a parametric pendulum

We study a pendulum parametrically excited by nonharmonic perturbations. Instead of using a circular harmonic function to perturb the pendulum, we use a Jacobi elliptic function as a perturbation, which encompasses it as a limit. Melnikov analysis provides the general condition for the onset of homoclinic bifurcations, adding now the elliptic modulus as a new parameter. Using the elliptic modulus, which is responsible for the shape of the perturbation, as a control parameter, new transitions order-chaos may occur.