The only limit cycle that appears in damped harmonic oscillators affected by state-dependent impulses

Abstract

This study is devoted to planar hybrid dynamical systems with state-dependent impulses to investigate phenomena related to sudden changes in the velocity of an object. In particular, the effect of the state-dependent impulses on the appearance of limit cycles is addressed, which has rarely been studied to date. When a toddler is placed on a swing and pushed gently from behind, the discontinuous changes in the direction and speed of movement of both the toddler and the swing can be described using such a hybrid dynamical system. The main theorem presented in this study reveals that, regardless of the amount of state-dependent impulses, they yield only one stable limit cycle within the region that is identified by the considered system. Moreover, state-dependent impulses lead to destabilization of the equilibrium point. In the proof of the main theorem, a Poincaré mapping is derived by examining the behavior of positive semi-orbits, and the characteristics of the sequence that is obtained from the Poincaré mapping are investigated in detail. Several simulations are performed to demonstrate the main theorem.