具有纯二次边的两个自治一阶常微分方程的新代数可解系统

F. Calogero, R. Conte, F. Leyvraz
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引用次数: 6

摘要

我们确定了以两个独立的一阶常微分方程为特征的一般动力系统的许多新的可解子情况;这些动力系统的可解性意味着可以通过代数运算获得其初值问题的解。同样,通过考虑这些系统到复时间的解析延拓,它们的代数可解特征对应于这样一个事实,即它们的通解要么是单值的,要么只有有限数量的代数分支点作为复时间(自变量)的函数。因此,我们的结果提供了一个主要的扩展类的可解系统,超出了那些具有单值通解的伽尼尔大约60年前确定。这些新动力系统的一个有趣的性质是它们的通解的基本特征,可以用具有显式可得的时间相关系数的多项式的根来识别。我们还提到,通过众所周知的随时间变化的(因变量和自变量)变量,其特征是虚参数$% \mathbf{i} \omega $(其中$\omega $是任意严格正实数),我们识别的每个代数可解模型都可以显式地展示自治变体:这些变体都具有显著的等时性,即它们的一般解是周期的,其周期是基本周期$T=2\pi/\ ω $的固定整数倍。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
New algebraically solvable systems of two autonomous first-order ordinary differential equations with purely quadratic right-hand sides
We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides; the solvable character of these dynamical systems amounting to the possibility to obtain the solution of their initial value problem via algebraic operations. Equivalently---by considering the analytic continuation of these systems to complex time---their algebraically solvable character corresponds to the fact that their general solution is either singlevalued or features only a finite number of algebraic branch points as functions of complex time (the independent variable). Thus our results provide a major enlargement of the class of solvable systems beyond those with singlevalued general solution identified by Garnier about 60 years ago. An interesting property of several of these new dynamical systems is the elementary character of their general solution, identifiable as the roots of a polynomial with explicitly obtainable time-dependent coefficients. We also mention that, via a well-known time-dependent change of (dependent and independent) variables featuring the imaginary parameter $% \mathbf{i} \omega $ (with $\omega $ an arbitrary strictly positive real number), autonomous variants can be explicitly exhibited of each of the algebraically solvable models we identify: variants which all feature the remarkable property to be isochronous, i.e. their generic solution is periodic with a period that is a fixed integer multiple of the basic period $T=2\pi/\omega$.
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