Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems

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).…”

“…Then, by using Equation (2), we obtain the result where the cofactor of the invariant algebraic curve y − P(x) = 0 is K = −a. Consequently, system (1) has the Darboux invariant (7), which in our case becomes I = (y − P(x))e at . Hence, statement (a) is proved.…”

“…Note that the degree of the polynomial Liénard differential system is the degree of, i.e., the maximum of m + p and 2p − 1. Now, we assume that y − P(x) is an invariant algebraic curve of system (1) with cofactor K. Then, from (7), we obtain −P (x)y − y f m (x) − g n (x) = K(y − P(x)).…”