论文标题
两个耦合的一阶ODES的可解决系统与均匀的立方右侧
Solvable systems of two coupled first-order ODEs with homogeneous cubic polynomial right-hand sides
论文作者
论文摘要
解决方案$ x_n \ left（t \ right）$，$ n = 1,2，\ textIt {jirtial-values}的$ textit {promosous}系统的$ 2 $ 2 $耦合的一阶ODES的系统与\ textit { c_ {n1} \ left（x_1 \ oyt）^3 + c_ {n2} \ left（x_1 \ right）^2 x_2 + c_ {n3} x_1 \ left（x_2 \ oright（x_2 \ oright）^2 + c_ {n4} \ end {eqnarray}当$ 8 $（时间无关）系数$ c_ {n \ ell} $按适当定义的定义，以$ 7 $ \ textit {nutionary}参数，然后还标识该模型的解决方案。还研究了这些关系的反转，即如何从$ 8 $系数$ c_ {n \ ell}，$ 7 $的参数来获得该模型的解决方案的$ 7 $参数；和$ 2 $ \ textIt {约束}是\ textIt {明确}确定的，如果通过$ 8 $ parameters $ c_ {n \ ell}满足，则保证该动态系统的代数操作} \ textit {solvisit {solvicitiation}。还确定了一个相关的，\ textIt {适当修改}，（通常\ textit {complex}）系统，读取\ begin {eqnarray} \ dot {\ tilde {x} _ {x} _ {n}}} = \ mathbf {i} \ tilde {i}〜\ tilde { c_ {n1} \ left（\ tilde {x} _ {1} \ right） ^{3} + c_ {n2} \ left（\ tilde {x} _ {1} _ {1} \ right） \ tilde {x} _2 \ right）^2 + c_ {n4} \ left（\ tilde {\ tilde {x} _2 \ right）^3 \，\ quad n = 1,2 \，\ nonumber \ nonumber \ ennumber \ end end end end e eqnArray} with $ \ m mathbf inther with unumber \ nonumber \ nonumber \ nonumber \ nonumber。引人注目的属性为\ textIt {等级}，即它们的\ textIt {generic}解决方案是 - 作为\ textIt {remainition} - \ textit {完全周期性}的函数，其周期为每个模型，a \ textit {pidexit {pidectit} {pidectit} \ textit {inte} inte { $ \ tilde {t} =2π/\ left \vertΩ\ right \ vert $。
The solution $x_n\left(t\right)$, $n=1,2,$ of the \textit{initial-values} problem is reported of the \textit{autonomous} system of $2$ coupled first-order ODEs with \textit{homogeneous cubic polynomial} right-hand sides, \begin{eqnarray} \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\ ,\quad n=1,2\ , \nonumber \end{eqnarray} when the $8$ (time-independent) coefficients $c_{n\ell}$ are appropriately defined in terms of $7$ \textit{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 $8$ coefficients $c_{n\ell},$ the $7$ parameters characterizing the solution of this model; and $2$ \textit{constraints} are \textit{explicitly} identified which, if satisfied by the $8$ parameters $c_{n\ell },$ guarantee the \textit{solvability by algebraic operations} of this dynamical system. Also identified is a related, \textit{appropriately modified}, class of (generally \textit{complex}) systems, reading \begin{eqnarray} \dot{\tilde{x}_{n}} = \mathbf{i}ω\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\ ,\quad n=1,2\ , \nonumber \end{eqnarray} with $\mathbf{i}ω$ an \textit{arbitrary imaginary} parameter, which feature the remarkable property to be \textit{isochronous}, namely their \textit{generic} solutions are -- as functions of \textit{real time} -- \textit{completely periodic} with a period which is, for each of these models, a \textit{fixed} \textit{integer multiple} of the basic period $\tilde{T}=2π/\left\vert ω\right\vert$.
