On the circular limit cycles of polynomial Liénard systems and of second-order polynomial differential equations

Supriyo Jana and Jaume Llibre

Abstract. It is known that the polynomial differential systems of degree \(n\) in the plane \(\mathbb{R}^2\) can exhibit at most \(n-1\) algebraic limit cycles formed by circles. A natural question is to ask: How many algebraic limit cycles formed by circles can the subclass of polynomial Liénard differential systems in the plane \(\mathbb{R}^2\) exhibit? We prove that the polynomial Liénard differential systems do not admit circular limit cycles. We also prove that the maximal number of circular limit cycles of the polynomial differential systems of the form \(\dot{x}= y,~\dot{y}=Q(x,y)\) of degree \(n\) is \(\lfloor \frac{n-1}{2}\rfloor\), with \(\lfloor z\rfloor\) being the integer part function of \(z\), and this upper bound is reached.