On the existence of polynomial first integrals of quadratic homogeneous systems of ordinary differential equations

Phase portraits for quadratic homogeneous polynomial vector fields on S^2

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 on S^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16^{th} Hilbert's problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homo…

Algebraically integrable quadratic dynamical systems

We consider in C^n the class of symmetric homogeneous quadratic dynamical systems. We introduce the notion of algebraic integrability for this class. We present a class of symmetric quadratic dynamical systems that are algebraically integrable by the set of functions h_1(t), ..., h_n(t) where h_1(t) is any solution of an ordinary differential equation of order n and h_k(t) are differential polynomials in h_1(t), k = 2, ..., n. We describe a method of constructing this ordinary differential equation. We give a classification of symmetric quadratic dynamical systems and describe the maximal subg…

新的可解系统：2个一阶非线性耦合常微分方程组

New Solvable System OF 2 First-Order Nonlinearly-Coupled Ordinary Differential Equations

本文提出一个由2个非线性耦合一阶常微分方程组成的自治系统，其初值问题可代数显式求解，并给出含4个因变量、9个参数的等时变体。

Planar Semi-quasi Homogeneous Polynomial differential systems with a given degree

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 important properties of PSQHPDS are derived and are employed to establish an algorithm for obtaining all the explicit expressions of PSQHPDS with a given degree. Afterward, we apply this algorithm to research the center problem of quadratic and cubic PSQHPDS. It is proved that the quadratic one hasn't center, and, that the cubic one has center if and only if it…

Uniform stability and chaotic dynamics in nonhomogeneous linear dissipative scalar ordinary differential equations

The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor.

$S$-整二次型与齐次动力学

$S$-integral quadratic forms and homogeneous dynamics

本文利用齐次动力学研究多变量整二次型，给出 $S$-整等价判据与 $S$-整正交群有限生成集，并以系数大小的多项式界推广已有结果。

秩2齐性空间中的有效等分布与二次型的取值

Effective equidistribution in rank 2 homogeneous spaces and values of quadratic forms

本文在绝对秩2的拟分裂几乎单线性代数群商空间中建立了带多项式误差率的幂幺轨道有效等分布定理，并据此给出不定三元二次型整点取值分布的定量结果，从而有效且定量地证明了Oppenheim猜想。

发现非线性常微分方程中的多项式和二次结构

Discovering Polynomial and Quadratic Structure in Nonlinear Ordinary Differential Equations

针对非线性常微分方程，本章总结有限维动力系统多项式和二次表示发现的最新进展，回顾存在性结果、讨论两种自动化算法并结合实例，助力系统分析与学习。

实数域上三维齐次二次方程整数零点的动力学

Dynamics of integer zeroes of homogeneous quadratic equations over $\mathbb{R}^3$

研究实数域上三维齐次二次方程整数零点相关动力学系统。依Romik方法，由Berggren树计算一维动力系统，用Keane方法确定其绝对连续不变测度。

Linnik's problem in fiber bundles over quadratic homogeneous varieties

We compute the statistics of $SL_{d}(\mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(\mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of radially projected integral vectors from a large sphere into the unit sphere. Using the above result we generalize the work of Aka, Einsiedler and Shapira in various directions. For example, we compute the joint distribution of the residue classes modulo $q$ and the properly normalized orthogonal lattices of primitive integral vectors lying on the level set $-…

由区域数量效应引发的集聚：一个带有二次子效用的NEG模型

Agglomeration triggered by the effect of the number of regions: A model in NEG with a quadratic subutility

本文将Ottaviano-Tabuchi-Thisse (2002)模型扩展至多区域情形，发现在一维周期空间中，当区域数量为2和3且运输成本足够高时齐次稳态解稳定，而为4的倍数时在任何运输成本下均不稳定。

Values of pairs involving one quadratic and one linear form at S-integral points

We prove the existence of S-integral solutions of simultaneous diophantine inequalities for pairs (Q,L) involving one quadratic form and one linear form satisfying some arithmetico-geometric conditions. The proof uses strong approximation in algebraic groups and Ratner's topological rigidity of unipotent actions on homogeneous spaces.

Bifurcating solutions in a non-homogeneous boundary value problem for a nonlinear pendulum equation

Motivated by recent studies of bifurcations in liquid crystals cells [1,2] we consider a nonlinear pendulum ordinary differential equation in the bounded interval $(-L, L)$ with non-homogeneous mixed boundary conditions (Dirichlet an one end of the interval, Neumann at the other) and study the bifurcation diagram of its solutions having as bifurcation parameter the size of the interval, $2L$, and using techniques from phase space analysis, time maps, and asymptotic estimation of integrals, complemented by appropriate numerical evidence.

Quadratic Differential Systems on $\mathbb{R}^{3}$ Having a Semisimple Derivation

The classification, up to a center-affinity, of the homogeneous quadratic differential systems defined on $\mathbb{R}^{3}$ that have at least a semisimple nonsingular derivation, is achieved. It is proved that there exist four affine-equivalence classes of such systems, only.

Classification of Quadratic Differential Systems on $\mathbb{R}^{3}$ Having a Semisimple Derivation with a one-Dimensional Kernel

The classification, up to a center-affinity, of the homogeneous quadratic differential systems defined on $\mathbb{R}^{3}$ that have at least a semisimple derivation with one-dimensional kernel, is achieved. It is proved that there exist 35 families of affine equivalence classes of such systems.