The Lins-de Melo-Pugh conjecture for quintic Lienard systems: at most two limit cycles

Theorem 1.1: if $\deg F\le5$, with no parity or coefficient-sign assumption, the system $\dot x=y-F(x)$, $\dot y=-x$ has at most two limit cycles in $\mathbb R^2$, counted as geometric images regardless of multiplicity, stability or hyperbolicity, and some member of the class has exactly two. This confirms the conjectured value $\lfloor (5-1)/2\rfloor=2$. It concerns only the classical form with restoring force exactly $x$; generalized Lienard systems and other degrees are not treated. The paper also shows that a proposed degree-five four-cycle example of Hernandez Rosales has only two cycles.

Theorem 1.1: if $\deg F\le5$, with no parity or coefficient-sign assumption, the system $\dot x=y-F(x)$, $\dot y=-x$ has at most two limit cycles in $\mathbb R^2$, counted as geometric images regardless of multiplicity, stability or hyperbolicity, and some member of the class has exactly two. This confirms the conjectured value $\lfloor (5-1)/2\rfloor=2$. It concerns only the classical form with restoring force exactly $x$; generalized Lienard systems and other degrees are not treated. The paper also shows that a proposed degree-five four-cycle example of Hernandez Rosales has only two cycles.