Chara Pantazi

Darboux integrability and invariant algebraic curves for planar polynomial systems

In this paper we study the normal forms of polynomial systems having a set of given generic invariant algebraic curves.

Inverse problems for invariant algebraic curves: explicit computations

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are non-degenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

Inverse problems for multiple invariant curves

Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals are of special interest. In the present paper we solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore we construct examples where the genericity assumption does not hold and indicate that the situation is different for these.

Limit cycles bifurcating from a degenerate center

We study the maximum number of limit cycles that can bifurcate from a degenerate center of a cubic homogeneous polynomial differential system. Using the averaging method of second order and perturbing inside the class of all cubic polynomial differential systems we prove that at most three limit cycles can bifurcate from the degenerate center. As far as we know this is the first time that a complete study up to second order in the small parameter of the perturbation is done for studying the limit cycles which bifurcate from the periodic orbits surrounding a degenerate center (a center whose linear part is identically zero) having neither a Hamiltonian first integral nor a rational one. This study needs many computations, which have been verified with the help of the algebraic manipulator Maple.

A note on the Darboux theory of integrability of non-autonomous polynomial differential systems

In this work, we unfold some differential algebraic aspects of Darboux first integrals of polynomial vector fields. An interesting improvement is that our approach can be applied both to autonomous and non-autonomous vector fields. We give a sufficient and necessary condition for the existence of a Darboux first integral of a specific form for a polynomial vector field with some known algebraic invariant hypersurfaces. For the autonomous case, the classical result of Darboux is obtained as a corollary. For the non-autonomous case our characterization improves a known criterium.

POLYNOMIAL VECTOR FIELDS IN R 3 WITH INFINITELY MANY LIMIT CYCLES

We provide a constructive method to obtain polynomial vector fields in ℝ3 having infinitely many limit cycles starting from polynomial vector fields in ℝ2 with a period annulus. We present two exam...

Morphisms and inverse problems for Darboux integrating factors

Polynomial vector fields which admit a prescribed Darboux integrat- ing factor are quite well-understood when the geometry of the underlying curve is nondegenerate. In the general setting morphisms of the affine plane may remove degeneracies of the curve, and thus allow more structural insight. In the present paper we establish some properties of integrating factors subjected to morphisms, and we discuss in detail one particular class of morphisms related to finite reflection groups. The results indicate that degeneracies for the underlying curve generally impose restrictions on the nontrivial vector fields which admit a given integrating factor.

Analytic integrability of the Bianchi Class A cosmological models with 0 k < 1

Many works study the integrability of the Bianchi class A cosmologies with k = 1, where k is the ratio between the pressure and the energy density of the matter. Here we characterize the analytic integrability of the Bianchi class A cosmological models when 0 ⩽ k < 1. We conclude that Bianchi types VI0, VII0, VIII and IX can exhibit chaos whereas Bianchi type I is not chaotic and Bianchi type II is at most partially chaotic.

On the absence of analytic integrability of the Bianchi Class B cosmological models

We follow Bogoyavlenskys approach to deal with Bianchi class B cosmological models. We characterize the analytic integrability of such systems.