Related papers: Dynamics and integrability of polynomial vector fi…

In this paper, we consider the following two algebraic hypersurfaces $$S^1\times S^2=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2-a^2)^2 + x_3^2 + x_4^2 -1=0;~ a>1\}$$ and $$S^2\times S^1=\{(x_1,x_2,x_3,x_4)\in…

This is a survey on the Darboux theory of integrability for polynomial vector fields, first in $\R^n$ and second in the $n$-dimensional sphere $\sss^n$. We also provide new results about the maximum number of parallels and meridians that a…

We extend to the $n$-dimensional ellipsoid contained in $\R^{n+1},$ the Darboux theory of integrability for polynomial vector fields in the $n$-dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant…

In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere $\mathbb{S}^2 = \{(x, y, z) \in \mathbb{R}^3 ~|~ x^2+y^2+z^2 = 1\}$. We start by classifying all degree three polynomial vector fields on…

In this paper, we classify all polynomial vector fields in $\mathbb{R}^3$ of degree up to three such that their flow makes the torus $$\mathbb{T}^2=\{(x,y,z)\in \mathbb{R}^3:(x^2+y^2-a^2)^2+z^2-1=0\}~\mbox{with}~a\in (1,\infty)$$ invariant.…

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 Poincar\'e from dimension two to dimension…

In this paper, we characterize all polynomial Kolmogorov vector fields for which the standard $n$-sphere is invariant. We exhibit completely integrable Kolmogorov vector fields of degree $m$ on $\mathbb{S}^n$ for any $m >2$. Then, we show…

Let X be a homogeneous polynomial vector field of degree 2 on S^2. We show that if X has at least a non--hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X…

First we characterize all the polynomial vector fields in $\R^4$ which have the Clifford torus as an invariant surface. After we study the number of invariant meridians and parallels that such polynomial vector fields can have in function…

A vector field s on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised Cheeger-Gromoll metrics on TM with respect to which s is a harmonic section. If M is a simply-connected…

We study, theoretically and experimentally, a 1-parameter family of transformations and their limiting vector field on the space of plane polygons. These transformations are discrete analogs of completely integrable transformation on closed…

In this work we revisit and extend the method introduced by Lins Neto, Sad and Sc\'{a}rdua for detecting the non-existence of invariant algebraic curves other than some prescribed invariant nodal curve. We prove that, under the existence of…

A relationship between real, complex, and quaternionic vector fields on spheres is given by using a relationship between the corresponding standard inner products. The number of linearly independent complex vector fields on the standard…

We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine, twelve, fifteen and eighteen degree.

This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous…

We present a computational method for reconstructing a vector field on a convex polytope $\mathcal{P} \subset \mathbb{R}^d$ of arbitrary dimension from discrete samples. We specifically address the scenario where the vector field is subject…

We construct, for every even dimensional sphere $S^n$, $n >1$, and every odd integer $k$, a homogeneous polynomial map $f: S^{n}\to S^{n}$ of Brouwer degree $k$ and algebraic degree $2|k|-1$.

The study of proper rational mappings between balls in complex Euclidean spaces naturally leads to the relationship between the degree and imbedding dimension of such a mapping. The special case for monomial mappings is equivalent to the…

We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of distinct monomials for dimensions 2 and 3. We study the connection with monomial CR maps of hyperquadrics and prove similar bounds in this setup…

Let $S_{p,q}$ be the hypersurface in $\mathbb{R}^{p+q+1}$ defined by the following: $$ S_{p,q} := \left\lbrace (x_1,\ldots,x_{p+1},x_{p+2},\ldots,x_{p+q+1}) \in \mathbb{R}^{p+q+1} \big| \left( \sum_{i=1}^{p+1} x_i^2 - a^2 \right)^2 +…