Universal Amplitude Equation in the Neighborhood of a Hopf Bifurcation

Abstract

In this chapter we show that all systems manifesting a Hopf bifurcation can be described by the same universal equation (that is, with an equation independent of the system itself). We first derive the universal equation using a concrete example, and then from symmetries reveal and explain its universal character. We introduce a few more general concepts, such as that of the limit cycle associated with the Hopf bifurcation, and will show that a time oscillating solution does not always lead to a limit cycle.