Rational Limit Cycles on Abel Polynomial Equations - Claudia Valls (Mathematics, 2020)
In this paper we deal with Abel equations of the form <semantics> d y / d x = A 1 ( x ) y + A 2 ( x ) y 2 + A 3 ( x ) y 3 </semantics> , where <semantics> A 1 ( x ) , A 2 ( x ) </semantics> and <semantics> A 3 ( x ) </semantics> are real polynomials and <semantics> A 3 ≢ 0 </semantics> . We prove that these Abel equations can have at most two rational (non-polynomial) limit cycles when <semantics> A 1 ≢ 0 </semantics> and three rational (non-polynomial) limit cycles when <semantics> A 1 ≡ 0 </semantics> . Moreover, we show that these upper bounds are sharp. We show that the general Abel equations can always be reduced to this one.