Algebraic particular integrals, integrability and the problem of the center

Abstract

Abstract.In this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields. We first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center: these curves control the changes in the systems as parameters vary. The bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered, namely the real four-dimensional projective space. Next, we consider the known four algebraic conditions for the center for quadratic vector fields. One of them says that the system is Hamiltonian, a condition which has a clear geometric meaning. We determine the geometric meaning of the remaining other three algebraic conditions (I), (II), (III). We show that a quadratic system with a weak focus F , possessing algebraic particular integrals not passing through F of the following types, satisfies in some coordinate axes the condition (I), (II) or (III) respectively and hence has a center at F : either a parabola and an irreducible cubic particular integral having only one point at infinity, coinciding with the one of the parabola; or a straight line and an irreducible conic curve; or distinct straight lines (possibly with complex coefficients). We show that each one of these geometric properties is generic for systems satisfying the corresponding algebraic condition for the center. Another version of this result in terms of real algebraic curves is given. These results make clear the many facets of the problem of the center in the quadratic case, in particular the question of integrability and form a basis for analogous investigations for the general problem of the center for cubic systems.

Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields

“…The class SymC of symmetric systems with a center; the family L-V-C of the so called Lotka-Volterra systems with center defined further below; the class Ham of the Hamiltonian systems with center and the family Cod4 of the codimension 4 systems with center, which are characterized by having two invariant algebraic curves: an irreducible singular cubic curve and a conic, more precisely a parabola whose point at infinity is the singular point of the cubic (see [51]). …”

“…To visualize the bifurcation diagram of QC we work with each case separately (see [51]). In each one of the cases SymC, L-V-C, Ham, using the above normal form, we get a parameter space which is the real projective plane PR 2 .…”

On the Darboux Integrability of Polynomial Differential Systems

“…The proof of the classification of all centers of planar polynomial differential systems of degree 2 can be strongly simplified using the Darboux first integrals, see Schlomiuk [38] and the chapter 8 of Dumortier, Llibre and Artés [16].…”

Center conditions at infinity for Abel differential equations

“…This problem together with a closely related second part of Hilbert's 16-th problem (asking for the maximal possible number of isolated closed trajectories of (1.2) with F .x; y/ and G.x; y/ polynomials of a given degree) have resisted until now all the attacks. Many deep partial results have been obtained (see [7], [8], [28]- [30], [34], [38], [42], [44], [45], [48], [55], [58], [63], [65], [72] and references therein) but general center conditions are not known even for F .x; y/ and G.x; y/ polynomials of degree three.…”

“…For some special classes of the system (1.2), like the Liénard equation and some others Center conditions can be given explicitly (see [63], [65], [28]- [33], [3]- [5]). However, in a general situation, and especially for F .x; y/ and G.x; y/ polynomials of high degree, only a part of Center configurations can be described while even a reasonable approximation of the entire Center set is not available.…”