A new criterion for controlling the number of limit cycles of some generalized liénard equations

We consider a class of planar differential equations which include the Liénard differential equations. By applying the Bendixson-Dulac Criterion for ℓ-connected sets we reduce the study of the number of limit cycles for such equations to the condition that a certain function of just one variable does not change sign. As an application, this method is used to give a sharp upper bound for the number of limit cycles of some Liénard differential equations. In particular, we present a polynomial Liénard system with exactly three limit cycles. © 2002 Elsevier Science (USA).