An Algorithm for Detecting "Linear" Solutions of Nonlinear Polynomial Differential Equations

Construction of a stabilizing control and solution to a problem about the center and focus for differential systems with a polynomial part on the right side

Stationary differential systems with polynomial right sides are considered. Necessary and sufficient conditions are formulated when a given domain is a domain of asymptotic stability and the origin of coordinates is either focus or center. The problem of construction of a stabilizing control in the form of polynomial is studied.

微分方程中法向双曲不变环面检测机制

A mechanism for detecting normally hyperbolic invariant tori in differential equations

本文提出一种检测微分方程中法向双曲不变环面的机制，通过引导系统的双曲极限环条件，证明非自治周期方程存在此类环面，并应用于 jerk 方程族。

Liouvillian integrability of polynomial differential systems

M.F. Singer [Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673--688] proved the equivalence between Liouvillian integrability and Darboux integrability for two dimensional polynomial differential systems. In this paper we will extend Singer's result to any finite dimensional polynomial differential systems. We prove that if an n--dimensional polynomial differential system has n-1 functionally independent Darboux Jacobian multiplier then it has n-1 functionally independent Liouvillian first integrals. Conversely if the system is Liouvillian integrabl…

在非光滑系统中检测极限环面：一种分析方法及其在三维分段线性系统中的应用

Detecting Limit Tori in Non-Smooth Systems: An Analytic Approach with Applications to 3D Piecewise Linear Systems

本文提出了一种分析方法，用于在非光滑系统中检测孤立不变环面，并在三维分段线性系统中应用，首次实现了对极限环面的分析检测。

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

We consider systems of ordinary differential equations with quadratic homogeneous right hand side. We give a new simple proof of a result already obtained in [8,10] which gives the necessary conditions for the existence of polynomial first integrals. The necessary conditions for the existence of a polynomial symmetry field are given. It is proved that an arbitrary homogeneous first integral of a given degree is a linear combination of a fixed set of polynomials.

Phase Portrait of the Riccati Quadratic Polynomial Differential Systems

In this paper we characterize the phase portrait of the Riccati quadratic polynomial differential systems $$\dot{x}= α_2(x),\quad\dot{y} = ky^2+β_1(x) y + γ_2(x), $$ with $(x,y)\in\mathbb{R}^2$, $γ_2(x)$ non-zero (otherwise the system is a Bernoulli differential system), $k\neq0$ (otherwise the system is a Lienard differential system), $β_ 1(x)$ a polynomial of degree at most $1$, $α_ 2(x)$ and $γ_ 2(x)$ polynomials of degree at most 2, and the maximum of the degrees of $ α_2(x)$ and $k y^2+β_1(x) y + γ_2(x)$ is 2. We give the complete description of their phase portraits in the Poincare disk