On the construction of bifurcation curves related to limit cycles of multiplicity three for planar vector fields

We consider planar vector fields f(x,y,λ)f(x,y,λ) depending on a three-dimensional parameter vector λλ. We assume f(0,0,λ)≡0f(0,0,λ)≡0 and that there exists a parameter value λ=λ0λ=λ0 connected with the Andronov–Hopf bifurcation of a limit cycle of multiplicity three from the origin. We describe an algorithm to continue the corresponding local Andronov–Hopf bifurcation curve in the parameter space which is based on the continuation of a periodic orbit to some augmented vector field and the construction of a Poincaré function to another augmented system.