Electronic Journal of Qualitative Theory of Differential Equations 2019, No. 3, 1–6; https://doi.org/10.14232/ejqtde.2019.1.3 www.math.u-szeged.hu/ejqtde/ Period annulus of the harmonic oscillator with zero cyclicity under perturbations with a homogeneous polynomial field Isaac A. García  and Susanna Maza Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Spain Received 2 August 2018, appeared 14 January 2019 Communicated by Alberto Cabada Abstract. In this work we prove, using averaging theory at any order in the small per- turbation parameter, that the period annulus of the harmonic oscillator has cyclicity zero (no limit cycles bifurcate) when it is perturbed by any fixed homogeneous polyno- mial field. Keywords: averaging theory, periodic orbits, Poincaré map. 2010 Mathematics Subject Classification: 37G15, 37G10, 34C07. 1 Introduction and main result Consider an arbitrary polynomial planar vector field ˙ x = −y + ε P( x, y, ε), ˙ y = x + ε Q( x, y, ε), (1.1) where P, Q ∈ R{ε}[ x, y] are polynomials in the state variables x and y with coefficients de- pending analytically on the small perturbation parameter ε ∈ R. Here the dot denotes, as usual, derivative with respect to the time independent variable. The unperturbed system (1.1) with ε = 0 is the harmonic oscillator which has a period annulus P given by the punctured phase plane P = R 2 \{(0, 0)}. Limit cycle bifurcations for the vector fields (1.1) can be produced either from the open set P or from its boundary ∂P = {(0, 0)}∪ L ∞ , where L ∞ is the line at infinity (equator of the Poincaré compactification). In this paper we do not pay attention to the Hopf bifurcations at the origin neither to the bifurcations at infinity (see Remark 1 of [5] for a simple example of limit cycle bifurcation at L ∞ ). Let X ε be the vector field associated to system (1.1). We denote by Cycl(X ε , P ) the cyclicity of P under the perturbations (1.1) with | ε|≪ 1, that is, the maximum number of limit cycles of (1.1) bifurcating from the circles that foliates P .  Corresponding author. Email: garcia@matematica.udl.cat