Basic Introduction to Bifurcations in 1-D

Abstract

This chapter introduces a simple example, that of a mass attached to a spring where the mass slides along a horizontal bar. We show that this system has several equilibrium solutions and exhibits one of the usual bifurcations, the pitchfork bifurcation. We will paint an intuitive and simple picture explaining the existence of this bifurcation and introduce the universal amplitude equation, as well as some general notions such as symmetry breaking, adiabatic elimination, and the analogy with systems in thermal equilibrium. We conclude with an exact solution to the amplitude equation, confirming classical stability studies.