Generic polynomial vector fields are not integrable

We study some generic aspects of polynomial vector fields or polynomial derivations with respect to their integration. In particular, using a well-suited presentation of Darboux polynomials at some Darboux point as power series in local Darboux coordinates, it is possible to show, by algebraic means only, that the Jouanolou derivation in four variables has no polynomial first integral for any integer value s ≥ 2 of the parameter.Using direct sums of derivations together with our previous results we show that, for all n ≥ 3 and s ≥ 2, the absence of polynomial first integrals, or even of Darboux polynomials, is generic for homogeneous polynomial vector fields of degree s in n variables.     

Zeroes of complete polynomial vector fields

We prove that a complete polynomial vector field on has at most one zero, and analyze the possible cases of those with exactly one which is not of Poincaré-Dulac type. We also obtain the possible nonzero first jet singularities of the foliation at infinity and the nongenericity of completeness. Connections with the Jacobian Conjecture are established.

     

On the isoclines of polynomial vector fields

Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 6, pp. 1390–1396, November–December, 1994.     

On C classifications of hyperbolic vector fields

In this paper we study smooth classification of hyperbolic vector fields based on their linear approximations only and obtain the following. On Rⁿ, n?5, with only two kinds of exceptions, any two hyperbolic vector fields with generic nonlinear parts and where A_i are n×n matrices, are C¹ conjugate to each other if and only if A₁ and A₂ are strictly similar, and they are C¹ orbitally equivalent if and only if A₁ and A₂ are similar.     

Complete polynomial vector fields on the complex plane

We give a full classification, up to polynomial automorphisms, of complete polynomial vector fields in two complex variables.     

Quantities at infinity in translational polynomial vector fields

In this paper, we study quantities at infinity and the appearance of limit cycles from the equator in polynomial vector fields with no singular points at infinity. We start by proving the algebraic equivalence of the corresponding quantities at infinity (also focal values at infinity) for the system and its translational system, then we obtain that the maximum number of limit cycles that can appear at infinity is invariant for the systems by translational transformation. Finally, we compute the singular point quantities of a class of cubic polynomial system and its translational system, reach with relative ease expressions of the first five quantities at infinity of the two systems, then we prove that the two cubic vector fields perturbed identically can have five limit cycles simultaneously in the neighborhood of infinity and construct two systems that allow the appearance of five limit cycles respectively. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones, The calculation can be readily done with using computer symbol operation system such as Mathematics.     

Explicit volume-preserving splitting methods for polynomial divergence-free vector fields

We present new, explicit, volume-preserving splitting methods for polynomial divergence-free vector fields of arbitrary degree (both positive and negative). The main idea is to decompose the divergence polynomial by means of an appropriate basis for polynomials: the monomial basis. For each monomial basis function, the split fields are then identified by collecting the appropriate terms in the vector field so that each split vector field is volume preserving. We show that each split field can be integrated exactly by analytical methods. Thus, the composition yields a volume preserving numerical method. Our numerical tests indicate that the methods compare favorably to standard integrators both in the quality of the numerical solution and the computational effort.     

Configurations of limit cycles and planar polynomial vector fields

We show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field. Moreover, the realization can be made by algebraic limit cycles, and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles.     

Local cyclicity in low degree planar piecewise polynomial vector fields

In this work, we are interested in isolated crossing periodic orbits in planar piecewise polynomial vector fields defined in two zones separated by a straight line. In particular, in the number of limit cycles of small amplitude. They are all nested and surrounding one equilibrium point or a sliding segment. We provide lower bounds for the local cyclicity for planar piecewise polynomial systems, $M_p^c\left(n\right),$ with degrees 2, 3, $4,$ and 5. More concretely, $M_p^c\left(2\right)\ge 13,$ $M_p^c\left(3\right)\ge 26,$ $M_p^c\left(4\right)\ge 40,$ and $M_p^c\left(5\right)\ge 58.$ The computations use parallelization algorithms.     

Differential Galois theory and non-integrability of planar polynomial vector fields

We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrability with some higher transcendent functions, like the error function.     

Planar polynomial vector fields having a polynomial first integral can be obtained from linear systems

We consider in this work planar polynomial differential systems having a polynomial first integral. We prove that these systems can be obtained from a linear system through a polynomial transformation of variables.     

Functions which are smooth along vector fields

Translated from Matematicheskie Zametki, Vol. 48, No. 1, pp. 95–102, July, 1990.     

Darboux integrability of real polynomial vector fields on regular algebraic hypersurfaces

In this paper we extend the Darboux theory of integrability in ℝ ⁿ to the regular algebraic hypersurfaces. Then we apply the extended theory first to the 3-dimensional generalized cylinders ×ℝ^3−r of ℝ⁴ forr=0, 1, 2, 3; and after to then-dimensional sphere of ℝ ⁿ⁺¹.     

Complex polynomial vector fields having an orbit with finite total curvature

In this paper we address the following questions: (i) Let \({C \subset \mathbb{C}^2}\) be an orbit of a polynomial vector field which has finite total Gaussian curvature. Is C contained in an algebraic curve? (ii) What can be said of a polynomial vector field which has a finitely curved transcendent orbit? We give a positive answer to (i) under some non-degeneracy conditions on the singularities of the projective foliation induced by the vector field. For vector fields with a slightly more general class of singularities we prove a classification result that captures rational pull-backs of Poincaré-Dulac normal forms.     

Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields

We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian nilpotent centers of linear plus cubic homogeneous planar polynomial vector fields.     

The Lie algebra of polynomial vector fields and the Jacobian conjecture

The Jacobian conjecture for polynomial maps :K ⁿ K ⁿ is shown to be equivalent to a certain Lie algebra theoretic property of the Lie algebra of formal vector fields inn variables. To be precise, let be the unique subalgebra of codimensionn (consisting of the singular vector fields),H a Cartan subalgebra of ,H the root spaces corresponding to linear forms onH and . Then every polynomial map :K ⁿ K ⁿ with invertible Jacobian matrix is an automorphism if and only if every automorphism of with (A) satisfies (A)=A.     

Planar polynomial vector fields having first integrals and algebraic limit cycles

With the help of Abel differential equations we obtain a new class of Darboux integrable planar polynomial differential systems, which have degenerate infinity. Moreover such integrable systems may have algebraic limit cycles. Also we present the explicit expressions of these algebraic limit cycles for quintic systems.     

Hamiltonian nilpotent saddles of linear plus cubic homogeneous polynomial vector fields

We completely characterize the global phase portraits in the Poincaré disk for all planar Hamiltonian vector fields with linear plus cubic homogeneous terms having a nilpotent saddle at the origin.     

Correction and linearization of resonant vector fields and diffeomorphisms

We extend the classical Siegel-Brjuno-Rüssmann linearization theorem to the resonant case by showing that under A. D. Brjuno's diophantine condition, any resonant local analytic vector field (resp. diffeomorphism) possesses a well-defined correction which (1) depends on the chart but, in any given chart, is unique (2) consists solely of resonant terms and (3) has the property that, when substracted from the vector field (resp. when factored out of the diffeomorphism), the vector field or diffeomorphism thus “corrected” becomes analytically linearizable (with a privileged or “canonical” linearizing map). Moreover, in spite of the small denominators and contrary to a hitherto prevalent opinion, the correction's analyticity can be established by pure combinatorics, without any analysis. Received January 7, 1997; in final form April 22, 1997