Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems

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In this paper we deal with the Lienard system $$\dot{x}=y, \dot{y}=-f_m(x)y-g_n(x),$$ where $$f_m(x)$$ and $$g_n(x)$$ are real polynomials of degree m and n, respectively. We call this system the Lienard system of type (m, n). For this system, we proved that if $$m+1\le n\le [\frac{4m+2}{3}]$$ , then the maximum number of hyperelliptic limit cycles is $$n-m-1$$ , and this bound is sharp. This result indicates that the Lienard system of type $$(m,m+1)$$ has no hyperelliptic limit cycles. Secondly, we present examples of irreducible algebraic curves of arbitrary high degree for Lienard systems of type $$(m,2m+1)$$ . Moreover, these systems have a rational first integral. Finally, we proved that the Lienard system of type (2, 5) has at most one hyperelliptic limit cycle, and this bound is sharp.