Bifurcation and Stability at Finite and Infinite Degrees of Freedom

Abstract

Conventionally problems of finite and infinite degrees of freedom (DOF) are separated in mechanics. The main reason is that different types of mathematical tools are used to study them. For finite DOF systems algebraic equations or systems of ordinary differential equations are used, while at infinite DOF cases vector and tensor fields and sets of partial differential equations should be used. However, the idea and main steps of stability analysis are the same. In both types of systems stability investigation can be done by calculating the spectrum of a linear operator. This operator is an algebraic operator (matrix) for finite DOF and a differential operator for infinite DOF. In nonlinear stability analysis a bifurcation equation should be derived. For finite DOF the general way is to use center manifold reduction while at infinite DOF Lyapunov–Schmit reduction should be performed. The paper aims to find unity in the dynamics of finite and infinite DOF systems. We show how the steps of stability investigation relate to each other in finite and infinite DOF cases. The presentation will explain how the linear operator can be defined and studied for continua, or how Lyapunov–Schmidt reduction can be used for studying oscillations of finite DOF systems.