How to solve the polynomial ordinary differential equations

This paper discussed how to solve the polynomial ordinary differential equations. At first, we construct the theory of the linear equations about the unknown one variable functions with constant coefficients. Secondly, we use this theory to convert the polynomial ordinary differential equations into the simultaneous first order linear ordinary differential equations with constant coefficients and quadratic equations. Thirdly, we work out the general solution of the polynomial ordinary differential equations which is no longer concerned with the differential. Finally, we discuss the necessary and sufficient condition of the existence of the solution.     

An approach to solving systems of polynomials via modular arithmetics with applications

The objective of this paper is twofold. First, we describe a method to solve large systems of polynomial equations using modular arithmetics. Then, we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations.     

An approach to solving systems of polynomials via modular arithmetics with applications

The objective of this paper is twofold. First, we describe a method to solve large systems of polynomial equations using modular arithmetics. Then, we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations.     

On a Quadratic Eigenvalue Problem and its Applications

We investigate the eigenvalues and eigenvectors of a special quadratic matrix polynomial and use the results obtained to solve the initial value problem for the corresponding linear system of differential equations.     

Polynomial inverse integrating factors for quadratic differential systems

We characterize all the quadratic polynomial differential systems having a polynomial inverse integrating factor and provide explicit normal forms for such systems and for their associated first integrals. We also prove that these families of quadratic systems have no limit cycles.     

Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives.First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, …Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincaré compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincaré disc for the separable quadratic polynomial differential systems.     

Phase portraits of the quadratic vector fields with a polynomial first integral

In this work we classify the phase portraits of all quadratic polynomial differential systems having a polynomial first integral. IfH(x, y) is a polynomial of degreen+1 then the differential systemx′=−∂H/∂y,y′=∂H/∂x is called a Hamiltonian system of degreen. We also prove that all the phase portraits that we obtain in this paper are realizable by Hamiltonian systems of degree 2.     

Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12

A quadratic polynomial differential systemcan be identified with a single point of ?¹² through its coefficients. The phase portrait of the quadratic systems having a rational first integral of degree 3 have been studied using normal forms. Here using the algebraic invariant theory, we characterize all the non-degenerate quadratic polynomial differential systems in ?¹² having a rational first integral of degree 3. We show that there are only 31 different topological phase portraits in the Poincaré disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide representatives of every class modulo an affine change of variables and a rescaling of the time variable. Moreover, each one of these 31 representatives is determined by a set of algebraic invariant conditions and we provide for it a first integral.     

本刊英文版Vol.34(2018),No.5论文摘要（英文）北大核心CSCD

Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve Jaume LLIBRE Claudia VALLS Abstract In this paper,we study a new class of quadratic systems and classify all its phase portraits.More precisely,we characterize the class of all quadratic polynomial differential     

Computing the partial fraction decomposition of rational functions with irreducible quadratic factors in the denominators

In this note, a new method for computing the partial fraction decomposition of rational functions with irreducible quadratic factors in the denominators is presented. This method involves polynomial divisions and substitutions only, without having to solve for the complex roots of the irreducible quadratic polynomial or to solve a system of linear equations. Some examples of its applications are included.     

Polynomial first integrals of quadratic vector fields

We classify all quadratic polynomial differential systems having a polynomial first integral, and provide explicit normal forms for such systems and for their first integrals.     

Circles and Quadratic Maps Between Spheres

Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence relation on roundings and prove that any rounding, whose differential at 0 has rank at least 2, is equivalent to a fractional quadratic rounding. A fractional quadratic map is just the ratio of a quadratic map and a quadratic polynomial. We also show that any rounding gives rise to a quadratic map between spheres. The known results on quadratic maps between spheres have some interesting implications concerning roundings. Partially supported by CRDF RM1-2086.     

POINCAR BIFURCATION FOR QUADRATIC SYSTEMS WITH A CENTER REGION AND AN UNBOUNDED TRIANGULAR REGION

In this paper, we discuss the Poincare bifurcation for a class of quadratic systems with an unbounded triangular region and a center region. It is proved, by Poincare bifurcation, that inside the center region quadratic system perturbed by quadratic polynomial perturbation may generate three limit cycles.     

Polynomial dynamical systems and ordinary differential equations associated with the heat equation

We consider homogeneous polynomial dynamical systems in n-space. To any such system our construction matches a nonlinear ordinary differential equation and an algorithm for constructing a solution of the heat equation. The classical solution given by the Gaussian function corresponds to the case n = 0, while solutions defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n = 2. We explicitly describe the family of ordinary differential equations arising in our approach and its relationship with the wide-known Darboux-Halphen quadratic dynamical systems and their generalizations.     

A combinatorial formula of Leibniz type with application to Gegenbauer's polynomials

We establish a combinatorial formula of Leibniz type, which is an identity for a certain differential polynomial. The formula leads to new quadratic relations between Gegenbauer's orthogonal polynomials.

     

Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line

We classify the phase portraits of the quadratic polynomial differential systems having an invariant parabola, an invariant straight line, and a Darboux first integral produced by these two invariant curves.     

The Differential Counting Polynomial

The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common generalization of the dimension polynomial and the (algebraic) counting polynomial. Under mild additional assumptions, the differential counting polynomial decides whether a given set of solutions of a system of differential equations is the complete set of solutions.     

Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse

We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse.More precisely,a quadratic system having an invariant ellipse can be written into the form x=x2＋y2-1＋y（ax＋by＋c）,y=x（ax＋by＋c）,and the ellipse becomes x2＋y2=1.We prove that（i） this quadratic system is analytic integrable if and only if a=0;（ii） if x2＋y2=1 is a periodic orbit,then this quadratic system is Liouvillian integrable if and only if x2＋y2=1 is not a limit cycle;and（iii） if x2＋y2=1 is not a periodic orbit,then this quadratic system is Liouvilian integrable if and only if a=0.     

Quadratic systems with a rational first integral of degree 2: A complete classification in the coefficient space $$\mathbb{R}^{12} $$

A quadratic polynomial differential system can be identified with a single point of through the coefficients. Using the algebraic invariant theory we classify all the quadratic polynomial differential systems of having a rational first integral of degree 2. We show that there are only 24 topologically different phase portraits in the Poincaré disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide a unique representative of every class modulo an affine change of variables and a rescalling of the time variable. Moreover, each one of these 24 representatives is determined by a set of invariant conditions and each respective first integral is given in invariant form directly in The authors are partially supported by a MEC/FEDER grant MTM2005-06098-C02-01, and a CONACIT grant number 2005SGR-00550. Partially supported by CRDF-MRDA CERIM-1006-06     

On the norm theorem for semisingular quadratic forms

The aim of this paper is to prove some results concerning the norm theorem for semisingular quadratic forms, i.e., those which are neither nonsingular nor totally singular. More precisely, we will give necessary conditions in order that an irreducible polynomial, possibly in more than one variable, is a norm ofa semisingular quadratic form, and we prove that our conditions are sufficient if the polynomial is given by a quadratic form which represents 1. As a consequence, we extend the Cassels-Pflster subform theorem to the case of semisingular quadratic forms.