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Is there a specific name to the following type of non linear ODEs

$\begin{array}{c} \dot{x}_1 &= c_1 \, x_2\, x_3 \\ \dot{x}_2 &= \, c_2 x_1 x_3 \\ \dot{x}_3 &= c_3 \, x_2 x_1 \end{array} $

where $c_i$ are real constants. More specifically, for a higher dimensional version,

$\dot{x}_i = \sum_{\begin{array}{c}r,s \\ r \neq s \neq i \end{array}} c_{ir} x_r x_s$

Are there any known transformations to simplify things? I conjecture the solutions will be elliptic functions of some sort, but you might not want to write them down.

For the three dimensional case one can obtain

$ \frac{d}{dt}(\frac{1}{2} ( x_2^2 + x_3^3) ) = 0$

$ \implies x_2^2 + x_3^3 = 2 C$

And use this to transform some of the equations, eventually finding elliptic solutions for all the $x_i$. In higher dimensional case I would expect a similar situation.

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$$x_1x_2x_3=\frac{1}{c_1}x_1x_1'=\frac{1}{c_2}x_2x_2'=\frac{1}{c_3}x_3x_3'$$

$$\frac{1}{c_1}x_1^2+constant=\frac{1}{c_2}x_2^2+constant=\frac{1}{c_3}x_3^2+constant$$

$$x_2^2=\frac{c_2}{c_1}x_1^2+A$$ $$x_3^2=\frac{c_3}{c_1}x_1^2+B$$ where $A,B$ are arbitrary constants.

Bringing them back into the first equation leads to : $$x_1'^2=c_1^2x_2^2x_3^2=c_1^2 \left(\frac{c_2}{c_1}x_1^2+A\right) \left(\frac{c_3}{c_1}x_1^2+B\right)$$

$$x_1'^2=c_1 \left(c_2x_1^2+Ac_1\right) \left(c_3x_1^2+Bc_1\right)$$

This is an ODE of the separable kind. The integration involves the $sn(t|m)$ Jacobi elliptic function.

The "higher dimensional version" is of different kind because there is not a so simple relationship between the functions $x_1(t), x_2(t), x_3(t) , x_4(t), ...$. I suspect that the functions involved might be special functions of higher level than the elliptic functions. More likely, no closed form could be found.

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  • $\begingroup$ I haven't seen you in a while JJacquelin. It is great to see your work again. Even if the question is not up to your usual difficulty level :) +1 $\endgroup$ – Chinny84 Aug 8 '15 at 10:47

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