On entry-exit formulas for degenerate turning point problems in planar slow-fast systems

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 ∈ 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 (I ¹₂ ) and (I ¹₄ ), 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, ϵ log 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 and Schecter (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 relevant graphics in the Dumortier-Roussarie-Rousseau program. Here we also demonstrate that the singular transformation of y due to De Maesschalck and Schecter (JDE 2016) has practical significance in numerical computations.