Abstract. We consider the 1-parameter family of planar quintic systems,ẋ = y 3 −x 3 , y = −x + my 5 , introduced by A. Bacciotti in 1985. It is known that it has at most one limit cycle and that it can exist only when the parameter m is in (0.36, 0.6). In this paper, using the Bendixon-Dulac theorem, we give a new unified proof of all the previous results, we shrink this to (0.547, 0.6), and we prove the hyperbolicity of the limit cycle. We also consider the question of the existence of polycycles. The main interest and difficulty for studying this family is that it is not a semi-complete family of rotated vector fields. When the system has a limit cycle, we also determine explicit lower bounds of the basin of attraction of the origin. Finally we answer an open question about the change of stability of the origin for an extension of the above systems.

International audienceWe study the number of limit cycles and the bifurcation diagram in the Poincaré sphere of a one-parameter family of planar differential equations of degree five x˙=Xb(x) which has been already considered in previous papers. We prove that there is a value b∗>0 such that the limit cycle exists only when b∈(0,b∗) and that it is unique and hyperbolic by using a rational Dulac function. Moreover we provide an interval of length 27/1000 where b∗ lies. As far as we know the tools used to determine this interval are new and are based on the construction of algebraic curves without contact for the flow of the differential equation. These curves are obtained using analytic information about the separatrices of the infinite critical points of the vector field. To prove that the Bendixson--Dulac Theorem works we develop a method for studying whether one-parameter families of polynomials in two variables do not vanish based on the computation of the so called double discriminant

The Harmonic Balance Method provides a heuristic approach for finding truncated Fourier series as an approximation to the periodic solutions of ordinary differential equations. Another natural way for obtaining these types of approximations consists in applying numerical methods. In this paper we recover the pioneering results of Stokes and Urabe that provide a theoretical basis for proving that near these truncated series, whatever is the way they have been obtained, there are actual periodic solutions of the equation. We will restrict our attention to one-dimensional non-autonomous ordinary differential equations, and we apply the obtained results to a concrete example coming from a rigid cubic system.

In this paper we consider several families of potential non-isochronous systems and study their associated period functions. Firstly, we prove some properties of these functions, like their local behavior near the critical point or infinity, or their global monotonicity. Secondly, we show that these properties are also present when we approach to the same questions using the Harmonic Balance Method.

Abstract. We introduce several applications of the use of the double resultant through some examples of computation of different nature: special level sets of rational first integrals for rational discrete dynamical systems; remarkable values of rational first integrals of polynomial vector fields; bifurcation values in phase portraits of polynomial vector fields; and the different topologies of the offset of curves.