Limit cycles of slow-fast quintic Lienard systems

Abstract

For every fixed real quintic polynomial $F$, we prove that the slow--fast Liénard system $\dot{x}=y-F(x),\quad \dot{y}=-\varepsilon x$ admits at most two limit cycles whenever $0<\varepsilon<\varepsilon_0(F)$ for a suitable $\varepsilon_0(F)>0$. Using scalar comparison near the equilibrium and a logarithmic fold-passage correction, we exclude \rev{canard singular limits}. Every sequence of limit cycles with $\varepsilon\to0^+$ has a subsequence converging in the Hausdorff metric to a common slow--fast path, whose regular slow arcs are either all normally attracting or all normally repelling. We also establish uniform Floquet separation for fixed $F$. The global counting argument uses shared regular arcs, nesting, and stability alternation. These results give a global small-parameter bound and describe the singular-limit structure, complementing the local cyclicity analysis of Li and Lu(J. Differential Equations, 2014).