3 - Bifurcation Theory

Abstract

This chapter presents an overview of the bifurcation theory. This theory is applicable to nonlinear differential equations. Given a nonlinear differential equation that depends on a set of parameters, the number of distinct solutions may change as the parameters change. Points where the number of solutions changes are called bifurcation points, while bifurcations occur in all types of equations, the chapter focuses on ordinary differential equations. The chapter discusses Abelson’ computer program in LISP that automatically explores the steady-state orbits of one-parameter families of periodically driven oscillators, the program generates both textual descriptions and schematic diagrams. This program is capable of recognizing the different types of bifurcations, including fold bifurcations, supercritical and subcritical flip bifurcations, supercritical and subcritical Niemark bifurcations, supercritical and subcritical pitchfork bifurcations, and transcritical bifurcations.