Searcharxiv⌕ Search

arXiv subjects

Chara Pantazi

Publications and source records attributed to Chara Pantazi.

12 recordsLinked to original sources
arXiv2026

Dynamics of Kahan-Hirota-Kimura maps with rational invariant fibrations

We present a simple method to study the dynamics of planar Kahan-Hirota-Kimura (KHK) maps preserving rational fibrations. Using this approach, we show that integrable KHK maps may exhibit complex dynamics, even when obtained from vector fields with trivial behavior. As an application, we study the KHK map associated with a quadratic planar vector field with an isochronous center. This map preserves the original first integral and admits the vector field as a Lie symmetry. Moreover, for a dense set of values of the integration step, it is globally periodic and exhibits all possible periods except 2. We also provide evidence of non-integrability for KHK maps associated with other quadratic vector fields possessing isochronous centers. To overcome this issue, we introduce the notion of pseudo-KHK maps, as alternative integrable discretizations for vector fields with isochronous centers. These maps are constructed to preserve the first integrals of the original vector field and to ensure that the vector field itself is a Lie symmetry of the map. The construction can be extended to isochronous centers of degree greater than two.

math.DS↗
arXiv2025

Liouvillian integrability of three dimensional vector fields

We consider a three dimensional complex polynomial, or rational, vector field (equivalently, a two-form in three variables) which admits a Liouvillian first integral. We prove that there exists a first integral whose differential is the product of a rational 1-form with a Darboux function, or there exists a Darboux Jacobi multiplier. Moreover, we prove that Liouvillian integrability {in any dimension $n\geq 3$} always implies the existence of a first integral that is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field.

nlin.SI↗
arXiv2025

Liouvillian integrability of vector fields in higher dimensions

We consider complex rational vector fields in dimension $n>2$ (equivalently, differential forms of degree $n-1$ in $n$ variables) which admit a Liouvillian first integral. Extending a classical result by Singer for $n=2$, our main result states that there exists a first integral which is obtained by two successive integrations from one-forms with coefficients in a finite algebraic extension of the rational function field. The proof uses Puiseux series in a novel way to simplify computations. We also apply this method to give elementary proofs of Singer's theorem for rational one-forms, and of the Prelle-Singer theorem on elementary integrability of rational vector fields.

nlin.SI↗
arXiv2025

Liouvillian integrability of rational vector fields: The case of algebraic extensions

As shown in a previous paper, whenever a rational vector field on $\mathbb C^n$, $n>2$, is Liouvillian integrable, then it admits a first integral obtained by two successive integrations from a one-form with coefficients in a finite algebraic extension $L$ of the rational function field $K$. In the present work we discuss and characterize exceptional vector fields in this class, for which -- by definition -- the choice $L=K$ is not possible. In particular we show that exceptional vector field exist, giving explicit constructions in dimension three.

math.RA↗
arXiv2025

Kahan-Hirota-Kimura maps preserving original cubic hamiltonians

We study the class of cubic Hamiltonian vector fields whose associated Kahan-Hirota-Kimura (KHK) maps preserve the original Hamiltonian function. Our analysis focuses on these fields in $\mathbb{R}^2$ and $\mathbb{R}^4$, extending to a family of fields in $\mathbb{R}^6$. Additionally, we investigate various properties of these fields, including the existence of additional first integrals of a specific type, their role as Lie symmetries of the corresponding KHK map, and the symplecticity of these maps.

nlin.SI↗
arXiv2019

Invariant algebraic surfaces of polynomial vector fields in dimension three

We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields.

math.DS↗
arXiv2017

Differential Galois Theory and non-Integrability of Planar Polynomial Vector Fields

We study a necessary condition for the integrability of the polynomials 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 wether 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 integrablity with some higher transcendent functions, like the error function.

math.DS↗
arXiv2012

Darboux Integrals for Schrödinger Planar Vector Fields via Darboux Transformations

In this paper we study the Darboux transformations of planar vector fields of Schrödinger type. Using the isogaloisian property of Darboux transformation we prove the "invariance" of the objects of the "Darboux theory of integrability". In particular, we also show how the shape invariance property of the potential is important in order to preserve the structure of the transformed vector field. Finally, as illustration of these results, some examples of planar vector fields coming from supersymmetric quantum mechanics are studied.

quant-ph↗
arXiv2012

The Picard Fuchs equations for complete hyperelliptic integrals of even order curves, and the actions of the generalized Neumann system

We consider a family of genus 2 hyperelliptic curves of even order and obtain explicitly the system of 5 linear ODEs for periods of the corresponding Abelian integrals of first, second, and third kind, as functions of the parameters of the curves. The system is an extention of the well studied Picard--Fuchs equations for periods of complete integrals of first and second kind on odd hyperelliptic curves. The periods we consider are linear combinations of the action variables of several integrable systems, in particular the generalized Neumann system with polynomial separable potentials. Thus the solutions of the extended Picard--Fuchs equations can be used to study various properties of the actions.

math-ph↗
arXiv2012

On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached.

math.DS↗