Regular and slow-fast codimension 4 saddle-node bifurcations (Q340360)

In the present paper the cyclicity of unfoldings of a nilpotent saddle-node singularity with singularity order 4 is analyzed. This amounts to study the family of vector fields given by NEWLINE\[NEWLINE \begin{aligned} \dot{x} & = y \\ \dot{y} & = -xy +\epsilon (b_0+b_1x+b_2x^2+b_3x^3+x^4+x^5G(x,\lambda))+\epsilon y^2 H(x,y,\lambda), \end{aligned} NEWLINE\]NEWLINE where \(G\) and \(H\) are smooth, \(b=(b_0,b_1,b_2,b_3)\) is close to \(0\) and \(\epsilon\geq 0\) is either small (singular perturbation case) or \(\epsilon=1\) (the regular perturbation case).NEWLINENEWLINEThe main result states that the cyclicity is two, i.e. for any choice of \(b\) small and \(\varepsilon\) there can be at most two limit cycles inside a small neighborhood of \((x,y)=(0,0)\). The proof based on geometric singularity theory uses family and phase-directional blow-ups to reduce the problem to the study of slow-fast systems of a similar type but with a lower singularity order so that known results for these cases can be used to get bounds on the local cyclicity. Combining theses local results from slow-fast codimension 1 and 2 Hopf bifurcations, slow-fast Bogdanov-Takens bifurcations and slow-fast codimension 3 saddle and elliptic bifurcations then lead to a proof of the main result.