On the Darboux Integrability of Polynomial Differential Systems

Abstract

Abstract. This is a survey on recent results on the Darboux integrability of polynomial vector fields in R n or C n with n ≥ 2. We also provide an open question and some applications based on the existence of such first integrals. IntroductionIn many branches of applied mathematics, physics and, in general, in applied sciences appear nonlinear ordinary differential equations. If a differential equation or vector field defined on a real or complex manifold has a first integral, then its study can be reduced by one dimension. Therefore a natural question is: Given a vector field on a manifold, how to recognize if this vector field has a first integral defined on an open and dense subset of the manifold? In general this question has no a good answer up to now.In this survey we provide sufficient conditions for the existence of a first integral for polynomial vector fields in R n or C n with n ≥ 2. An open question which essentially goes back to Poincaré is presented. Finally some applications of the existence of this kind of first integrals to physical problems, centers, foci, limit cycles and invariant hyperplanes are mentioned. Darboux theory of integrabilitySince any polynomial differential system in R n can be thought as a polynomial differential system inside C n we shall work only in C n . If our initial differential system is in R n , once we get a complex first integral of this system inside C n the real and the imaginary parts of it are real first integrals. Moreover if that complex first integral is rational, the same occur for its real and imaginary parts. In short in the rest of the paper we shall work in C n .In this section we study the existence of first integrals for polynomial vector fields in C n through the Darboux theory of integrability. The algebraic theory of integrability is a classical one, which is related with the first part of the Hilbert's 16th problem [19]. This kind of integrability is usually called Darboux integrability, and it provides a link between the integrability of polynomial vector fields and the number of invariant algebraic hypersurfaces that they have (see Darboux [13] and Poincaré [36]). This theory shows the fascinating relationships between integrability (a topological phenomenon) and the existence of exact algebraic invariant hypersurfaces formed by solutions for the polynomial vector field. This theory is now known as the Darboux theory of integrability, see for more details the Chapter 8 of [16].2010 Mathematics Subject Classification. 34A34, 34C05, 34C14.

Liouvillian Integrability Versus Darboux Polynomials

“…Darboux theory of integrability was started in 1878 by Darboux [7,8], who provided a method to construct first integral using Darboux polynomials. This theory has been greatly developed in different aspects, see for instance [2,5,6,10,11,13,22,28,25,26,27,39] and the references therein. The Darboux theory of integrability has also been broadly applied to study dynamics of differential systems, for instance the center problem, limit cycles and global dynamical analysis, see for example [5,9,17,18,31,35,36,37] and the references quoted there.…”

Integrability of the Rucklidge system

“…For a present state of this theory see the Chapter 8 of [3], the paper [5], and the references quoted in them. Moreover, we also study the analytic integrability of system (1), i.e., the existence of a global analytic first integral H : R 3 → R.…”

“…In this last case, the first integrals associated to integrating factors given by Darboux functions are the Liouvillian first integrals, see for more details [3,5].…”

“…In fact to every Darboux polynomial there is associated some invariant algebraic hypersurface (see again section 2), and the exponential factors appear when an invariant algebraic surface has multiplicity larger than 1, for more details see [1,3,5]. As far as we know is the unique theory of integrability which is developed for studying the first integrals of polynomial differential systems.…”

Local Darboux first integrals of analytic differential systems

“…For details and further results we refer to the survey [9] of Darboux integrability for polynomial systems.…”