Coexistence of limit cycles and invariant algebraic curves for a Kukles system Original Research Article

We consider a Kukles system of the form View the MathML sourcex˙=-y,y˙=f(x,y) where f(x,y)f(x,y) is a polynomial with real coefficients of degree d without y as a divisor. We study the maximum number of small-amplitude limit cycles for these kind of systems which can coexist with invariant algebraic curves. We give all the possible distributions of invariant straight lines for a Kukles system and we give some bounds for the number of limit cycles. We also give some necessary conditions for the existence of an invariant algebraic curve of degree ⩾2⩾2 and we study the possible coexistence of this curve and a limit cycle. Finally, we give two examples of cubic Kukles systems both with an invariant hyperbola. In the first example the hyperbola coexists with a center and in the second one it coexists with two small-amplitude limit cycles. These two examples contradict a previous result given in Ann. Differential Equations 7(3) (1991) 323.