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Abstract: We study the uniqueness of limit cycles (periodic solutions that are isolated in the set of periodic solutions) in the scalar ODE in terms of , , . Our main result characterizes, under some additional hypotheses, the exponents , , , such that for any choice of the equation has at most one limit cycle. The obtained results have direct application to rigid planar vector fields, thus, planar systems of the form , , where . Concretely, when the set has at least three elements (or exactly one) and another technical condition is satisfied, we characterize the exponents , such that the origin of the rigid system is a center for any choice of and also when there are no limit cycles surrounding the origin for any choice of . [Copyright &y& Elsevier]

Abstract: In this work we focus on the configuration (location and stability) of simple critical points of polynomial differential equations of the form , . The case where all the critical points are of center type is studied in more detail finding several new center configurations. One of the main tools in our approach is the 1-dimensional Euler–Jacobi formula. [Copyright &y& Elsevier]

Abstract: In this article we give two criteria for bounding the number of non-contractible limit cycles of a family of differential systems on the cylinder. This family includes Abel equations as well as the polar expression of several types of planar polynomial systems given by the sum of three homogeneous vector fields. [Copyright &y& Elsevier]

Abstract: A solution of the Abel equation such that is called a periodic orbit of the equation. Our main result proves that if there exist two real numbers a and b such that the function is not identically zero, and does not change sign in then the Abel differential equation has at most one non-zero periodic orbit. Furthermore, when this periodic orbit exists, it is hyperbolic. This result extends the known criteria about the Abel equation that only refer to the cases where either or does not change sign. We apply this new criterion to study the number of periodic solutions of two simple cases of Abel equations: the one where the functions and are 1-periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials. Finally, we give an upper bound for the number of isolated periodic orbits of the general Abel equation , when , and satisfy adequate conditions. [Copyright &y& Elsevier]

Abstract: It is well known that the normal form theory can be applied to solve the center–focus problem for monodromic planar nilpotent singularities. In this paper we see how this theory can also be applied to generate limit cycles from this type of singularities. [Copyright &y& Elsevier]