The elastic pendulum: Dzhanibekov-like transitions of symmetric and asymmetric periodic responses

The free-vibration response of the conservative planar elastic pendulum falls into one of several categories, depending on the system parameters and the initial conditions. It has been shown in many previous studies that the free-vibration response of the system can be chaotic, quasi-periodic, or periodic. In this work, we present a new category of response: a slow sequential, back-and-forth switching between a nearly periodic response and its mirror-image. This switching resembles the Dzhanibekov effect. We show that the frequency of switching is unrelated to the frequency of the pendulum swings but rather depends on initial conditions. Specifically, we demonstrate Dzhanibekov-like transitions of symmetric and asymmetric nearly periodic responses. For each of these responses, we present Fourier transforms that reveal frequency combs at two scales. One frequency comb is related to the nearly periodic response of the elastic pendulum, and the other frequency comb is related to the much slower Dzhanibekov transitions.