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

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.

💡 Research Summary

The paper investigates the maximal number of crossing limit cycles that can appear in planar piecewise‑polynomial vector fields of a given degree n, denoted by H_c(n). A crossing limit cycle is a closed orbit that transversally intersects the switching line (the line of discontinuity) and is defined according to Filippov’s convention. The authors first recall the classical Hilbert’s 16th problem for smooth polynomial vector fields, where the Hilbert number H(n) is the supremum of the number of limit cycles for degree n systems. In the discontinuous setting, the analogue is H_c(n) = sup π_c(P,Q) with deg P, deg Q ≤ n, where π_c counts crossing limit cycles.

Previously, only a linear lower bound H_c(n) ≥ 2n − 1 was known, together with isolated results for low degrees (e.g., H_c(1) ≥ 3, H_c(2) ≥ 12). The paper’s first major contribution, Theorem A, improves this dramatically: it proves that lim inf_{n→∞} H_c(n)/n² ≥ 1/4, i.e. H_c(n) grows at least quadratically, specifically H_c(n) ≥ n²/4 for infinitely many n. Moreover, if H_c(n) is finite for some n, then the sequence is strictly increasing: H_c(n+1) ≥ H_c(n)+1. The proof proceeds in three steps. Proposition 1 adapts the Christopher–Lloyd recursive construction (originally for smooth systems) to the piecewise context. By applying the singular transformation (x,y)↦(x²−A, y²−A) to a polynomial system that already possesses many limit cycles, the degree doubles while four copies of the original system appear in the four quadrants, creating additional centers along the coordinate axes. A simultaneous pseudo‑Hopf bifurcation (a discontinuous analogue of the Hopf bifurcation) then generates extra crossing limit cycles around the new sliding segments. This yields a family of piecewise systems of degree n_k = 2ᵏ−1 with at least (n_k²)/4 crossing limit cycles, establishing the quadratic lower bound.

Proposition 2 shows that when H_c(n) is finite, the supremum can be realized by a system whose crossing limit cycles are all hyperbolic. The authors analyze the first return map associated with each crossing cycle, decompose it into analytic, strictly decreasing half‑return maps, and study the displacement function δ(y,b) as the switching line is shifted by a small parameter b. Depending on the parity of the zero’s multiplicity, three unfolding scenarios (type O, E⁺, E⁻) occur. By choosing a suitable sign of b, one can guarantee that each original cycle either persists hyperbolically or splits into two hyperbolic cycles, ensuring that the perturbed system has at least H_c(n) hyperbolic crossing cycles. Since no system of the same degree can have more than H_c(n) cycles, the perturbed system must have exactly H_c(n) hyperbolic cycles.

Proposition 3 establishes the strict monotonicity: starting from a degree‑n system that attains H_c(n) hyperbolic cycles, a small polynomial perturbation of degree n+1 creates a new sliding segment and, via a pseudo‑Hopf bifurcation, produces at least one additional hyperbolic crossing cycle. Hence H_c(n+1) ≥ H_c(n)+1.

The second main result, Theorem B, concerns the more restrictive class of piecewise‑polynomial Hamiltonian systems, where each side of the switching line is Hamiltonian. Denoting the corresponding maximal number by \widehat H_c(n), the authors adapt the Christopher–Lloyd recursion while preserving the Hamiltonian structure. The transformation still doubles the degree, but the number of new centers now grows proportionally to n log n instead of n². Consequently they prove lim inf_{n→∞} \widehat H_c(n)/(n log n) ≥ 1/(2 log 2), i.e. \widehat H_c(n) ≥ n log n/(2 log 2). This improves the previously known linear lower bound \widehat H_c(n) ≥ n−1.

The paper is organized as follows. Section 2 contains the detailed proof of Theorem A, including the definition of the pseudo‑Hopf bifurcation, the construction of the recursive family, and the analysis of the return maps. Section 3 presents the adaptation of the Christopher–Lloyd method to the Hamiltonian piecewise setting and establishes Theorem B. The final discussion highlights the significance of these results: they extend classical Hilbert‑type lower bounds to discontinuous systems, demonstrate that maximal numbers can be achieved with hyperbolic cycles, and open the way for further investigations of upper bounds, multi‑switching lines, and higher‑dimensional piecewise polynomial dynamics.