On the existence and uniqueness of limit cycles for generalized Liénard systems

Abstract In this paper, we consider a generalized Lienard system d x d t = ϕ ( y ) − F ( x ) , (0.1) d y d t = − g ( x ) , where F is continuous and differentiable on an open interval ( b 1 , a 1 ) with − ∞ ⩽ b 1 0 a 1 ⩽ + ∞ . Assume that there exist a and b with b 1 b 0 a a 1 such that x F ( x ) 0 as b x a , and x F ( x ) > 0 as a x a 1 or b 1 x b . A new uniqueness theorem of limit cycles for the Lienard system (0.1) is obtained. An example is given to show the application of the theorem.

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