Related papers: Applications of algebraic methods in solving the c…

The Abel differential equation $y'=p(x)y^3 + q(x) y^2$ with polynomial coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$. The problem…

Abel equations of the form $x'(t)=f(t)x^3(t)+g(t)x^2(t)$, $t \in [-a,a]$, where $a>0$ is a constant, $f$ and $g$ are continuous functions, are of interest because of their close relation to planar vector fields. If $f$ and $g$ are odd…

The Abel differential equation $y'=p(x)y^3 + q(x) y^2$ with meromorphic coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$. The problem…

In this paper we study the center algebras of multilinear forms. It is shown that the center of a nondegenerate multilinear form is a finite dimensional commutative algebra and can be effectively applied to its direct sum decompositions. As…

In this paper, I have proved that for a class of polynomial differential systems of degree n+1 ( where n is an arbitrary positive integer) the composition conjecture is true. I give the sufficient and necessary conditions for these…

We provide a sufficient condition for solvability of a system of real quadratic equations $p_i(x)=y_i$, $i=1, \ldots, m$, where $p_i: {\mathbb R}^n \longrightarrow {\mathbb R}$ are quadratic forms. By solving a positive semidefinite…

A center of a differential system in the plane $\mathbb{R}^2$ is an equilibrium point $p$ having a neighborhood $U$ such that $U\setminus \{p\}$ is filled of periodic orbits. A center $p$ is global when $\mathbb{R}^2\setminus \{p\}$ is…

The Abel differential equation $y'=p(x)y^2+q(x)y^3$ with $p,q\in \mathbb R[x]$ is said to have a center on a segment $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(b)=y(a)$. The problem of…

We continue the study of the center problem for the ordinary differential equation $v'=\sum_{i=1}^{\infty}a_{i}(x)v^{i+1}$ started in our earlier papers. In this paper we present the highlights of the algebraic theory of centers.

A difficult classical problem in the qualitative theory of differential systems in the plane $\mathbb{R}^2$ is the center-focus problem, i.e. to distinguish between a focus and a center. Another difficult problem is to distinguish inside a…

We study a planar polynomial differential system, given by \dot{x}=P(x,y), \dot{y}=Q(x,y). We consider a function I(x,y)=\exp \{h_2(x) A_1(x,y) \diagup A_0(x,y) \} h_1(x) \prod_{i=1}^{\ell} (y-g_i(x))^{\alpha_i}, where g_i(x) are algebraic…

We consider a system of the form x'=P_n(x,y)+xR_m(x,y), y'=Q_n(x,y)+yR_m(x,y), where P_n(x,y), Q_n(x,y) and R_m(x,y) are homogeneous polynomials of degrees n, n and m, respectively, with n<=m. We prove that this system has at most one limit…

The Polynomial Abel differential equations are considered a model problem for the classical Poincar\'e center--focus problem for planar polynomial systems of ordinary differential equations. Last decades several works pointed out that all…

We explain how algebraic geometry comes into play in the study of non-linear mean field (singular Liouville) equations $$ \triangle u + e^u = 4\pi \sum_{i = 1}^N \ell_i \delta_{p_i} $$ on a flat torus $E = \Bbb C/\Lambda$, where $N, \ell_1,…

For the planar $N$-centre problem $$ \ddot x = - \sum_{i=1}^N \frac{m_i (x-c_i)}{| x - c_i|^{\alpha+2}}, \qquad x \in \mathbb{R}^2 \setminus \{ c_1,\ldots,c_N \}, $$ where $m_i > 0$ for $i=1,\ldots,N$ and $\alpha \in [1,2)$, we prove the…

In this paper, we give a direct method to study the isochronous centers on center manifolds of three dimensional polynomial differential systems. Firstly, the isochronous constants of the three dimensional system are defined and its…

We study the existence of fully nontrivial solutions to the system $$-\Delta u_i+ \lambda_iu_i = \sum\limits_{j=1}^\ell \beta_{ij}|u_j|^p|u_i|^{p-2}u_i\ \hbox{in}\ \Omega, \qquad i=1,\ldots,\ell,$$ in a bounded or unbounded domain $\Omega$…

In this paper by exploiting critical point theory, the existence of two distinct nontrivial solutions for a nonlinear algebraic system with a parameter is established. Our goal is achieved by requiring an appropriate behavior of the…

The paper deals with the following system of nonlinear difference equations \begin{equation*} x_{n+1}=ax_{n}^{2}y_{n}+bx_{n}y_{n}^{2},\ y_{n+1}=cx_{n}^{2}y_{n}+dx_{n}y_{n}^{2},\ n\in \mathbb{N}_{0}, \end{equation*} where the initial values…

This paper study the planar semi-quasi homogeneous polynomial differential systems (short for PSQHPDS), which can be regard as a generalization of semi-homogeneous and of quasi-homogeneous systems. By using the algebraic skills, several…