The solution $x_n\left(t\right)$, n = 1, 2, of the initial-value problem is reported for the autonomous system of two coupled first-order ordinary differential equations with homogeneous cubic polynomial right-hand sides, ${\dot{x}}_n=c_{n1}{\left(x_1\right)}^3+c_{n2}{\left(x_1\right)}^2{x}_2+c_{n3}{x}_1{\left(x_2\right)}^2+c_{n4}{\left(x_2\right)}^3,n=1,2$, when the eight (time-independent) coefficients c_nℓ are appropriately defined in terms of seven arbitrary parameters, which then also identify the solution of this model. The inversion of these relations is also investigated, namely, how to obtain, in terms of the eight coefficients c_nℓ, the seven parameters characterizing the solution of this model, and two constraints are explicitly identified, which, if satisfied by the eight parameters c_nℓ, guarantee the solvability by algebraic operations of this dynamical system. Also identified is a related, appropriately modified, class of (generally complex) systems, reading $\dot{{\stackrel{̃}{x}}_n}=\mathbf{i}\omega {\tilde{x}}_n+c_{n1}{\left({\tilde{x}}_1\right)}^3+c_{n2}{\left({\tilde{x}}_1\right)}^2{\tilde{x}}_2+c_{n3}{\tilde{x}}_1{\left({\tilde{x}}_2\right)}^2+c_{n4}{\left({\tilde{x}}_2\right)}^3,n=1,2$, with iω being an arbitrary imaginary parameter, which features the remarkable property to be isochronous, namely, their generic solutions are—as functions of real time—completely periodic with a period that is, for each of these models, a fixed integer multiple of the basic period $\tilde{T}=2\pi /\left|\omega \right|$.

中文翻译：

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具有齐次三次多项式右侧的两个耦合一阶ODE的可解系统

解决方案 $X_{ñ}\left(Ť\right)$，n = 1，2，是两个具有齐次三次多项式右边的耦合一阶常微分方程的自治系统的初值问题，${\dot{X}}_{ñ}=C_{ñ\mathrm{1个}}{\left(X_{\mathrm{1个}}\right)}^3+C_{ñ2}{\left(X_{\mathrm{1个}}\right)}^2{X}_2+C_{ñ3}{X}_{\mathrm{1个}}{\left(X_2\right)}^2+C_{ñ4}{\left(X_2\right)}^3，ñ=\mathrm{1个}，2$，当八（与时间无关的）系数Ç_标准升在七个方面适当地限定任意参数，其然后也识别该模型的解。这些关系的反转还研究，即如何获得，在八个系数方面Ç_标准升，七个参数表征这种模式的解决方案，以及两个约束被明确认定，这，如果满意由八个参数Ç _nℓ，通过该动力系统的代数运算来保证可解性。还确定了一个相关的，经过适当修改的类（通常复杂的）系统，阅读$\dot{{\stackrel{̃}{X}}_{ñ}}=\mathbf{一世}\omega {\tilde{X}}_{ñ}+C_{ñ\mathrm{1个}}{\left({\tilde{X}}_{\mathrm{1个}}\right)}^3+C_{ñ2}{\left({\tilde{X}}_{\mathrm{1个}}\right)}^2{\tilde{X}}_2+C_{ñ3}{\tilde{X}}_{\mathrm{1个}}{\left({\tilde{X}}_2\right)}^2+C_{ñ4}{\left({\tilde{X}}_2\right)}^3，ñ=\mathrm{1个}，2$与我ω是一个任意的假想参数，它采用了显着的特性是同步，即，它们的通用解决方案是，作为功能实时-完全周期，其周期即，对于每个模型，一个固定的整数倍基本时期$\widetilde{Ť}=2\pi /\left|\omega \right|$。