Growth estimate for the number of crossing limit cycles in planar piecewise polynomial vector fields

TL;DR

The paper addresses the problem of bounding and understanding the growth of crossing limit cycles in planar piecewise polynomial vector fields, defining the piecewise Hilbert number $\mathcal{H}_c(n)$ and its Hamiltonian variant $\widehat{\mathcal{H}}_c(n)$. It adapts methods from Hilbert’s problem in the smooth setting, notably pseudo-Hopf bifurcation and Christopher–Lloyd-type constructions, to the discontinuous piecewise context. The authors prove a quadratic lower bound $\mathcal{H}_c(n) \ge n^{2}/4$ (in the liminf sense) and establish strict monotonicity $\mathcal{H}_c(n+1) \ge 1+\mathcal{H}_c(n)$ if $\mathcal{H}_c(n)<\infty$, along with a growth bound for the Hamiltonian case: $\widehat{\mathcal{H}}_c(n) \ge (n\log n)/(2\log 2)$. They construct explicit sequences of piecewise vector fields—both general and Hamiltonian—whose degrees grow as $n_k=2^{k}-1$ or $n_k=3\cdot 2^{k}-1$ and possess, respectively, $(n_k^{2}-1)/4$ and $c_k=3k\,2^{k-1}+1$ crossing limit cycles, thereby delivering sharp asymptotic lower bounds and advancing the understanding of Hilbert-type questions in discontinuous systems. The results have implications for the complexity of limit-cycle configurations in piecewise dynamics and broaden the toolkit for analyzing crossing phenomena in Filippov systems.

Abstract

Motivated by the classical Hilbert's Sixteenth Problem, we extend some main developments obtained for Hilbert's number in the polynomial setting to the piecewise polynomial context. Specifically, we study the growth of the maximum number of crossing limit cycles in planar piecewise polynomial vector fields of degree $n$, denoted by $H_c(n)$. The best previously known general lower bound is $H_c(n)\geq 2n - 1$. In this work, we show that $H_c(n)$ grows at least as fast as $n^2/4.$ Furthermore, we prove that $H_c(n)$ is strictly increasing whenever it is finite, and that in such cases this maximum can be realized by piecewise polynomial systems whose crossing limit cycles are all hyperbolic. Finally, for the more restrictive class of piecewise polynomial Hamiltonian vector fields, we adapt the recursive construction of Christopher and Lloyd to demonstrate that the corresponding maximal number of crossing limit cycles, denoted by $\widehat{H}_c(n)$, grows at least as fast as $n\log n/(2\log 2)$, thereby improving previously established linear growth estimate.

Figures (3)

• Figure 1: Pseudo-Hopf bifurcation in the case $\ell=a=1$.
• Figure 2: Possible configuration of a limit cycle and the associate half-return maps.
• Figure 3: Level curves of $H_0$, $H_1$, and $H_2$, for $\varepsilon = 0$.

Theorems & Definitions (16)

• Theorem A
• Theorem B
• Proposition 1
• proof
• Remark 1
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4