On the Number of Hyperelliptic Limit Cycles of Liénard Systems

Abstract

In this paper, we study the maximum number, denoted by H(m, n), of hyperelliptic limit cycles of the Liénard systemṡwhere, 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: We obtain the upper bound and lower bound of H(m, n) in all the cases with n = 2m + 1. When n = 2m + 1, we derive the lower bound of H(m, n). Furthermore, these upper bound can be reached in some cases.

Invariant Algebraic Curves of Generalized Liénard Polynomial Differential Systems

“…Given P and Q polynomials, an algebraic curve of the form (y + P(x)) 2 − Q(x) = 0 is called hyperelliptic curve (see for instance [5][6][7][8]). In such works, hyperelliptic curves are used to determine the algebraic limit cycles of generalized Liénard systems (1).…”

“…Assume now that P(x) = b. Therefore, g(x) = −b f (x), and the differential system becomes ẋ = y and ẋ = −(y + b) f (x), which has the Darboux first integral H = e y+F(x) (y − b) b if b = 0, and the Darboux first integral H = y + F(x) if b = 0, as it is easy to verify using (6).…”