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Coexistence of limit cycles and invariant algebraic curves for a Kukles system. (English) Zbl 1076.34029

This paper deals with the so-called Kukles systems, \[ \dot x=-y,\qquad \dot y=f(x,y), \] \(f\) being a polynomial of degree \(d.\) Firstly, the authors study the number and distribution of invariant straight lines that this type of systems can have. For instance, among other results, they prove that necessary and sufficient conditions for a Kukles system to have two invariant straight lines cutting in a finite singular point have to be affine equivalent to a system with \(f(x,y)=abx+(a+b)y+(ax+y)(bx+y)g(x,y),\) where \(g\) is a polynomial of degree \(d-2\) and \(a\) and \(b\) are real numbers satisfying \(ab(a-b)\neq0.\) Afterwards, the authors prove several criteria of nonexistence of limit cycles for systems having invariant straight lines. For instance, they prove that if the polynomial \(g(x,y)\) above depends only on \(x\), then the Kukles system has no limit cycles. The second part of the paper is devoted to get necessary conditions for a Kukles system to have invariant algebraic curves of degree greater or equal 2. In particular, the paper shows examples of cubic Kukles systems exhibiting an invariant hyperbola and either a center or two small-amplitude limit cycles.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
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