A Note on ‘‘Uniqueness of Limit Cycles in a Li´enard-Type System’’

Huang and Sun w J. Math. Anal. Appl. 184 1994., 348]359xproposed a theorem to guarantee the uniqueness of limit cycles for the generalized Li´enard system dxrdtsh y.yF x., dyrdtsyg x.. We will give a counterexample to their theorem. It will be shown that their theorem is valid only if F x. is monotone on certain intervals. For this case we give a shorter proof and we also show that the limit cycle is hyperbolic. If the condition on the monotonicity of F x. is violated then we need an additional condition to guarantee the uniqueness of the limit cycle. Examples are provided to illustrate our results.