两个右端为纯二次的一阶自治常微分方程的新代数可解组
New algebraically solvable systems of two autonomous first-order ordinary differential equations with purely quadratic right-hand sides
摘要：
我们确定了许多新的可解的一般动力系统的特点是两个自治的一阶常微分方程与纯二次右手边的子情况;这些动力系统的可解性相当于通过代数运算获得其初始值问题的解决方案的可能性。等价-通过考虑这些系统的解析延拓到复杂的时间-他们的代数可解字符对应的事实，即他们的一般解决方案是singlevalued或功能只有有限数量的代数分支点作为复杂的时间（自变量）的功能。因此，我们的研究结果提供了一个重大的扩大类的可解系统超出了那些单值一般解决方案确定的卡尼尔约60年前。这些新的动力系统的几个有趣的属性是他们的一般解决方案的基本特征，可识别的多项式的根明确获得的时间依赖系数。我们还提到，通过一个众所周知的时间依赖性变化，具有虚参数$% \mathbf{i} ω$的（因变量和自变量）变量（其中$ω$是任意严格正实数），我们识别的每个代数可解模型的自治变体可以显式地表现出来：这些变体都具有异黄酮的显著特性，即它们的一般解是周期性的，其周期是基本周期$T=2π/ω$的固定整数倍。
英文摘要：
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 $% \mathbf{i} ω$ (with $ω$ 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 $T=2π/ω$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
