On the cyclicity of hyperbolic polycycles

TL;DR

The paper studies the cyclicity of a hyperbolic polycycle $\Gamma^n$ in planar vector fields by linking the cyclicity to the product of hyperbolicity ratios $r_i$ through a combinatorial measure $\Delta(\Gamma^n)$ computed over permutations. It leverages sharp regularity results for the Dulac map due to Marin–Villadelprat to control how perturbations unfold heteroclinic connections, and constructs displacement maps that track bifurcations of limit cycles when breaking the polycycle. The main contribution is a general lower bound $\textit{Cycl}(X,\mathcal{X},\Gamma^n)\ge\Delta(\Gamma^n)$, extended to the smooth and polynomial settings, together with a method to realize these bounds via polynomial perturbations (even of high degree) and an explicit inverse problem demonstrating realizability of prescribed hyperbolicity data. The results illuminate how many limit cycles can emerge from a polycycle under small perturbations and provide concrete polynomial realizations and examples, with implications for the broader theory of polycycle unfolding and cyclicity in planar systems.

Abstract

Let $X$ be a planar smooth vector field with a polycycle $Γ^n$ with $n$ sides and all its corners, that are at most $n$ singularities, being hyperbolic saddles. In this paper we study the cyclicity of $Γ^n$ in terms of the hyperbolicity ratios of these saddles, giving explicit conditions that ensure that it is at least $k,$ for any $k\leqslant n.$ Our result extends old results and also provides a more accurate proof of the known ones because we rely on some recent powerful works that study in more detail the regularity with respect to initial conditions and parameters of the Dulac map of hyperbolic saddles for families of vector fields. We also prove that when $X$ is polynomial there is a polynomial perturbation (in general with degree much higher that the one of $X$) that attains each of the obtained lower bounds for the cyclicities. Finally, we also study some related inverse problems and provide concrete examples of applications in the polynomial world.

Figures (11)

• Figure 1: Illustration of $\Gamma^2$, with $(a)$ distinct and $(b)$ non-distinct hyperbolic saddles.
• Figure 2: Illustration of the bifurcation process. Blue means stable and red means unstable. Colors available in the online version.
• Figure 3: Illustration of $\Gamma^6$ with $(a)$ trivial and $(b)$ non trivial permutation on the indexes of the singularities.
• Figure 4: The Dulac map near a hyperbolic saddle.
• Figure 5: An example of a perturbation of $\Gamma^3$, with $d_1(\mu)<0$, $d_2(\mu)>0$ and $d_3(\mu)<0$. For simplicity, we omitted the dependence on $\mu$ in the expressions of $x_i^{s,u}$ and $L_i^{s,u}$.

Theorems & Definitions (29)

• Theorem 1
• Proposition 1
• Remark 1
• Proposition 2
• Remark 2: Theorem $B$ and Lemma $4.3$ of MarVil2021
• Remark 3
• Remark 4
• Proposition 3
• proof
• Remark 5