论文标题
通过广义反演确切可溶解的二次微分方程系统
Exactly solvable quadratic differential equation systems through generalized inversion
论文作者
论文摘要
我们研究形式的二次微分方程的自主系统$ \ dot {x} _i（t）= \ Mathbf {x}（t）^t \ t \ Mathbf {a} _i \ Mathbf {x}（x}（x}（x}） $ \ mathbf {x}（t）=（x_1（t），x_2（t），\ dots，x_i（t），\ dots）$，通常无法准确求解。在本文中，我们提出了一个可分析可解决的二次系统的子类，其解决方案是通过对反转的多维概括来实现的，该反转将二次系统转换为线性系统。我们提供了一种建设性算法，一方面，它可以通过反转转换在分析上解决微分方程系统，另一方面提供了解决方案。提出的结果适用于任意的，有限的变量。
We study the autonomous systems of quadratic differential equations of the form $\dot{x}_i(t)=\mathbf{x}(t)^T \mathbf{A}_i \mathbf{x}(t) + \mathbf{v}_i^T \mathbf{x}(t)$ with $\mathbf{x}(t) = (x_1(t),x_2(t),\dots,x_i(t),\dots)$ which, in general, cannot be solved exactly. In the present paper, we present a subclass of analytically solvable quadratic systems, whose solution is realized through a multi-dimensional generalization of the inversion which transforms a quadratic system into a linear system. We provide a constructive algorithm which, on one hand, decides whether the system of differential equations is analytically solvable with the inversion transformation and, on the other hand, provides the solution. The presented results apply for arbitrary, finite number of variables.
