Linear and Nonlinear Oscillators

Abstract

The linear superposition principle which is valid for linear differential equations is no longer valid for nonlinear ones. A physical consequence is that when the given dynamical system admits oscillatory motion, the associated frequency of oscillation is in general amplitude-dependent in the case of nonlinear systems, while it is not so in the case of linear systems. Particularly, this can have dramatic consequences in the case of forced and damped nonlinear oscillators, leading to nonlinear resonances and jump (hysteresis) phenomenon for low strengths of nonlinearity parameters. Such behaviours can be analysed using various approximation methods. However, as the control parameter varies further, the nonlinear systems can enter into more and more complex motions through different routes, where detailed numerical analysis and possible analog simulations using electronic circuits can be of much help to analyse them. In this chapter, we will introduce some basic features associated with nonlinear oscillations and postpone the discussions on more complex motions to later chapters. However, before discussing the nature of such nonlinear oscillations, we shall first discuss briefly the salient features associated with a damped and driven linear oscillator in order to compare its properties with nonlinear oscillators.