2025/10/03 by Huzak, Renato, Kristiansen, Kristian Uldall · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.02770
In this paper, we study degenerate entry-exit problems associated with planar slow-fast systems having an invariant line \(x,y) : y=0\ with a turning point at x=0. The degeneracy stems from the fact that the slow flow has a saddle-node of even order 2n, n∈ \mathbb N, at the turning point, i.e. x' = -x2n(1+o(1)) for ε=0. We are motivated by the appearance of such turning point problems (for n=1) in the graphics (I21) and (I41), through a nilpotent saddle-node singularity at infinity, in the Dumortier-Roussarie-Rousseau program (for solving the finiteness part of Hilbert's 16th problem for quadratic polynomial systems). Our results show, under additional hypothesis, that in the case n=1 there is a well-defined entry-exit relation for ε→ 0. The associated Dulac map is smooth w.r.t. (ε,εlog ε-1). On the other hand for the cases n≥ 2, we show that the entry-exit relation requires additional control parameters. Our approach follows the one used by De Maesschalck, P. and Schecter, S. (JDE 2016) for a different type of degenerate entry-exit problem. In particular, we apply blow-up after having first performed a singular coordinate transformation of y. The degeneracy at x=0 requires an additional blow-up. We finally apply the result for n=1 to a normal form for the unfolding of the \NEWrelevant graphics in the Dumortier-Roussarie-Rousseau program. Here we also demonstrate that the singular transformation of y due to De Maesschalck, P. and Schecter, S. (JDE 2016) has practical significance in numerical computations.