http://www.gsd.uab.cat POLYNOMIAL DIFFERENTIAL SYSTEMS IN R 3 HAVING AN INVARIANT QUADRIC AND A DARBOUX INVARIANT JAUME LLIBRE 1 , MARCELO MESSIAS 2 AND ALISSON C. REINOL 3 Abstract. We give the normal forms of all polynomial differential systems in R 3 which have a non–degenerate or degenerate quadric as an invariant algebraic surface. We also characterize among these systems those which have a Darboux invariant constructed uniquely using the invariant quadric, giving explicitly their expressions. As an example we apply the obtained results in the determination of the Darboux invariants for the Chen system with an invariant quadric. 1. Introduction and statement of the main results Let K[x, y, z ] be the ring of the polynomials in the variables x, y and z with coefficients in K, where K is either R or C. We consider the polynomial differential system in R 3 defined by (1) ˙ x = P (x, y, z ), ˙ y = Q(x, y, z ), ˙ z = R(x, y, z ), where P, Q, R ∈ R[x, y, z ] are relatively prime polynomials. The dot denotes deriv- ative with respect to the independent variable t usually called the time. In what follows we shall write system (1) simply as ˙ x = P ,˙ y = Q,˙ z = R. We say that m = max{deg P, deg Q, deg R} is the degree of system (1) and we can naturally associate to this system the vector field X = P ∂ ∂x + Q ∂ ∂y + R ∂ ∂z . Systems like (1) appear frequently in the literature due to their theoretical im- portance as well as to their use in applied mathematics, since polynomial systems are usually used to model natural phenomena, arising in Physics, Biology, Chem- istry and other sciences, see for instance [23, 24] and references therein. In this way many books and hundreds of papers have been published aiming to describe the dynamics of the solutions of system (1). However this dynamics is far from being completely understood, even in the quadratic case, i.e. when system (1) has de- gree m = 2. Indeed the dynamics generated by the flow of system (1) with degree m ≥ 2 is, in general, very complex and difficult to be studied. Beyond singular points, periodic, homoclinic and heteroclinic orbits, which is commonly encoun- tered in the phase space of planar polynomial vector fields, 3–dimensional systems 2010 Mathematics Subject Classification. 34C05, 34C20. Key words and phrases. Polynomial differential systems, invariant quadrics, Darboux integra- bility, Darboux invariant. 1 This is a preprint of: “Normal Forms for Polynomial Differential Systems in R 3 Having an Invariant Quadric and a Darboux Invariant”, Jaume Llibre, Marcelo Messias, Alison C. Reinol, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 25(1), 1550015, 2015. DOI: [10.1142/S0218127415500157]