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Given trigonometric monomials A 1 , A 2 , A 3 , A 4 , such that A 1 , A 3 have the same signs as sin ⁡ t , and A 2 , A 4 the same signs as cos ⁡ t , and natural numbers n , m > 1 , we study the family of Abel equations x ′ = ( a 1 A 1 ( t ) + a 2 A 2 ( t ) ) x m + ( a 3 A 3 ( t ) + a 4 A 4 ( t ) ) x n , a 1 , a 2 , a 3 , a 4 ∈ R . The center variety is the set of values a 1 , a 2 , a 3 , a 4 such that the Abel equation has a center (every bounded solution is periodic). We prove that the codimension of the center variety is one or two. Moreover, it is one if and only if A 1 = A 3 and A 2 = A 4 and it is two if and only if the family has non-trivial limit cycles (different from x ( t ) ≡ 0 ) for some values of the parameters. [ABSTRACT FROM AUTHOR]

We study the global behaviour of the period function on the period annulus of degenerate centres for two families of planar polynomial vector fields. These families are the quasi-homogeneous vector fields and the vector fields given by the sum of two quasi-homogeneous Hamiltonian ones. In the first case we prove that the period function is globally decreasing, extending previous results that deal either with the Hamiltonian quasi-homogeneous case or with the general homogeneous situation. In the second family, and after adding some more additional hypotheses, we show that the period function of the origin is either decreasing or has at most one critical period and that both possibilities may happen. This result also extends some previous results that deal with the situation where both vector fields are homogeneous and the origin is a non-degenerate centre. [ABSTRACT FROM AUTHOR]

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: We consider the Abel equation , where and are trigonometric polynomials of degree n and m, respectively, and we give lower bounds for its number of isolated periodic orbits for some values of n and m. These lower bounds are obtained by two different methods: the study of the perturbations of some Abel equations having a continuum of periodic orbits and the Hopf-type bifurcation of periodic orbits from the solution . [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]

The aim of this paper is to study the existence of limit cycles for a family of generalized Abel equations x ′ = A (t) x m + B (t) x n , m , n ≥ 2. Under certain assumptions, it is proved that there exists a non-trivial limit cycle. This limit cycle has the characteristic that it arises from neither a Hopf bifurcation nor a perturbation of periodic orbits in a period annulus around the centre at the origin. [ABSTRACT FROM AUTHOR]