Abstract
In this paper we deal with Abel equations of the form d y / d x = A 1 ( x ) y + A 2 ( x ) y 2 + A 3 ( x ) y 3 , where A 1 ( x ) , A 2 ( x ) and A 3 ( x ) are real polynomials and A 3 ≢ 0 . We prove that these Abel equations can have at most two rational (non-polynomial) limit cycles when A 1 ≢ 0 and three rational (non-polynomial) limit cycles when A 1 ≡ 0 . Moreover, we show that these upper bounds are sharp. We show that the general Abel equations can always be reduced to this one.
Highlights
Introduction and Statement of the ResultsIn this paper we study the existence of rational limit cycles of the Abel polynomial equations.The Abel polynomial equations are equations of the form dy= A1 ( x ) y + A2 ( x ) y2 + A3 ( x ) y3, dx (1)where x, y are real variables and A1 ( x ), A2 ( x ) and A3 ( x ) are polynomials with A3 6≡ 0.A periodic solution of Equation (1) is a solution y( x ) defined in the closed interval [0, 1] such that y(0) = y(1)
We say that a limit cycle is a periodic solution isolated in the set of periodic solutions of a differential Equation (1)
The rational function y = q( x )/p( x ) with p( x ) non-constant is a periodic solution of system (1) if and only if q( x ) = c ∈ R \ {0}, p(0) = p(1) and p( x ) has no zero in [0, 1] and p ( x ) A2 ( x ) +
Summary
In this paper we study the existence of rational (non-polynomial) limit cycles of the Abel polynomial equations. The polynomial limit cycles of these equations have been intensively investigated (see for instance [1,2]). We are interested in the rational limit cycles of Equation (1) (when the functions Ai ( x ) are polynomials).
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