Bifurcations, Center Manifolds, and Periodic Solutions

Abstract

Nonlinear time-delay differential equations are well known to have arisen in models in physiology, biology, and population dynamics. These delay differential equations (DDEs) usually have parameters in their formulation. How the nature of the solutions change as the parameters vary is crucial to understanding the underlying physical processes. When the DDE is reduced, at an equilibrium point, to leading linear terms and the remaining nonlinear terms, the eigenvalues of the leading coef- ficients indicate the nature of the solutions in the neighborhood of the equilibrium point. If there are any eigenvalues with zero real parts, periodic solutions can arise. One way in which this can happen is through a bifurcation process called a Hopf bifurcation in which a parameter passes through a critical value and the solutions change from equilibrium solutions to periodic solutions. This chapter describes a method of decomposing the DDE into a form that isolates the study of the periodic solutions arising from a Hopf bifurcation to the study of a reduced size differential equation on a surface, called a center manifold. The method will be illustrated by Hopf bifurcation that arises in machine tool dynamics, which leads to a machining instability called regenerative chatter.