Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Lienard Systems

In this paper we deal with the Lienard system (x) over dot = y, (y) over dot = - 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 = <= n <= [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.