On the number of hyperelliptic limit cycles of Li\'{e}nard systems

TL;DR

This paper investigates the maximum number of hyperelliptic limit cycles in Liéard systems, providing bounds and conditions under which these bounds are attainable, thereby advancing understanding of their cyclic behavior.

Contribution

The paper establishes bounds for the number of hyperelliptic limit cycles in Liéard systems based on polynomial degrees, and identifies cases where these bounds are achieved.

Findings

Derived upper and lower bounds for H(m,n)

Identified conditions for the bounds to be sharp

Enhanced understanding of hyperelliptic limit cycles in Liéard systems

Abstract

In this paper, we study the maximum number, denoted by $H(m,n)$, of hyperelliptic limit cycles of the Li\'enard systems $$\dot x=y, \qquad \dot y=-f_m(x)y-g_n(x),$$ where, respectively, $f_m(x)$ and $g_n(x)$ are real polynomials of degree $m$ and $n$, $g_n(0)=0$. The main results of the paper are as follows: In term of $m$ and $n$ of the system, we obtain the upper bound and lower bound of $H(m,n)$ in all the possible cases. Furthermore, these upper bound can be reached in some cases.