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

TL;DR

The paper analyzes entry-exit transitions for planar slow-fast systems with a degenerate turning point along $y=0$, where the slow flow has a saddle-node of even order $2n$. By combining a singular coordinate transform with a blow-up strategy, it derives a precise Dulac map for the case $n=1$, proving smooth dependence on $(ε,ε\log ε^{-1})$ and relating the transition to slow-divergence integrals and principal-value terms. For $n\ge 2$, the exit map is generically obstructed unless unfolding parameters are introduced, and a spherical blow-up reveals how the leading-order mismatch is governed by the integral $\int_{-∞}^{∞} \frac{v}{P_λ(v)} dv$. The results are then connected to the Dumortier–Roussarie–Rousseau program graphics (I2^1, I4^1) and illustrated with an application to the DDR-unfolding, including numerical validation that highlights the practical significance of the $y$-transformation in computation.

Abstract

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 $2n$, $n\in \mathbb N$, at the turning point, i.e. $x' = -x^{2n}(1+o(1))$ for $ε=0$. We are motivated by the appearance of such turning point problems (for $n=1$) in the graphics $(I_2^1)$ and $(I_4^1)$, 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 $ε\rightarrow 0$. The associated Dulac map is smooth w.r.t. $(ε,ε\log ε^{-1})$. On the other hand for the cases $n\ge 2$, 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 \NEW{relevant} 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.

Figures (6)

• Figure 1: A limit periodic set after desingularization of the graphic $(I_2^1)$ through a nilpotent saddle-node at infinity in the Dumortier-Roussarie-Rousseau program.
• Figure 2: Illustration of the base point $(x_{\text{in}/\text{out}}^b,0)$ of $(x_{\text{in}/\text{out}},\delta)\in \Sigma_{\text{in}/\text{out}}$.
• Figure 3: Cylindrical blowup of the degenerate line $(x,z,\epsilon)=(x,0,0)$ by (\ref{['eq:cylbu']}).
• Figure 4: Sketch of (\ref{['eq:z2xb']}) for $n=1$ (red) and $n\ge 2$ (blue).
• Figure 5: Illustration of the spherical blow-up (\ref{['eq:bu-spherical']}) for $n\ge 2$.

Theorems & Definitions (38)

• Remark 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.3
• Theorem 2.4
• Remark 2.5
• Remark 2.6
• Theorem 2.7