Fig. 3

Unfolding of the deg. TB singularity, focus case. A The fixed points of the system are found for \(y_{0}=0\), \(x_{0}^{3}+\mu_{2} x_{0}-\mu _{1}=0\). The blue surface represents \(x_{0}\) plotted against the two parameters \(\mu_{1}\), \(\mu_{2}\). In orange are marked the curves of saddle-node bifurcations at which the saddle solution (i.e. the middle branch) collides with the focus in the upper or lower branch and annihilates. B The manifold of the saddle-node bifurcation (in orange) is plotted in the three dimensional unfolding parameters space together with a sphere of radius \(R = 0.4\) centered at the origin of the parameter space \((\mu_{1},\mu_{2},\nu)=(0,0,0)\). The intersection between the surface of saddle-node bifurcations and the spherical surface gives a curve of saddle-node bifurcation. C The bifurcation curves obtained at the intersection between the sphere and all the bifurcation surfaces of the unfolding. In this figure we do not separate between supercritical and subcritical Hopf (both in green) and between saddle node and SNIC (both in orange)