Investigating Cases of Jump Phenomenon in a Nonlinear Oscillatory System

Abstract

A two degree-of-freedom (DOF) nonlinear oscillatory system is presented which exhibits jump phenomena where the period of oscillation jumps to an integer multiple of its original period when the state changes by a small amount due to damping. The jump phenomenon is investigated within the framework of Catastrophe theory where the abrupt change in the response characteristic of the system is explained by the interaction between its invariant manifolds. The authors recently presented an approach based on the concept of the Instantaneous Center Manifold (ICM), where, the center manifold of a conservative nonlinear system was replaced by an equivalent set of ICMs as two-dimensional invariant manifolds of the system that contain all of its periodic orbits. This study attempts to explain the sudden change in the response of a slightly damped oscillator as its response decays, based on the ICMs of its underlying undamped system. Such abrupt changes in the response of the system are explained based on the orientability properties of the local ICMs of the undamped system. This study also presents a bifurcation analysis of periodic orbits of the undamped system to illustrate the evolution of different types of periodic orbits, orientability of their local ICMs and their effect on the transient response of the damped system. These effects are expounded in cases of jump phenomena observed in numerical simulation of the free response of the damped system.