Locating order-chaos-order transition in elastic pendulum

An undamped elastic pendulum being a nonintegrable Hamiltonian system always has some chaotic trajectories observable on choosing appropriate initial conditions. This is true even if the pendulum is in libration with small amplitude; in this situation, the pendulum may be seen as a nearly integrable system. Since the measure of the set of the local chaotic trajectories in the phase space may be very small, the trajectories are hard to locate. However, the emergence of widespread chaos when the elastic pendulum is at autoparametric resonance is well-documented. The transition from the local and the widespread chaos is typically established through the Chirikov overlap criterion that approximates the phase portrait around a resonance using a one degree-of-freedom pendulum Hamiltonian. We argue in this paper that the aforementioned transition in the elastic pendulum is due to interaction between two resonances of same kind and their coexistence can be analytically located using perturbation methods, like the method of averaging, whereas the technique of the pendulum Hamiltonian is inapplicable. Furthermore, in the course of validating the result numerically, we also showcase the order-chaos-order transition in the elastic pendulum using the fast Lyapunov indicator.