Hopf-Bifurcation

The ultimate purpose of this work is to analyse (non-linear) first order ordinary differential equations. In particular it will be shown under which assumptions periodic solutions occur in a small neighbourhood of an equilibrium. The basic idea is that if the matrix of the linearised differential equation at the equilibrium has two purely imaginary eigenvalues, the solutions close to it are either spiralling or periodic. Once we ascertained the conditions under which they are in fact periodic, we will also parametrise their orbits, their period and their parameter.