Coupled non linear ordinary differential equations.

[rkrsnan]

I have a set of ten dependent variables x1, x2 ... x10 and an independent variable t. The differential equations are of the form

d x1 /dt = f1(x1, x2... x10)
d x2 /dt = f2(x1, x2... x10)
......
d x10 /dt = f10(x1, x2... x10)

where f1, f2 ...f10 are non linear functions of x1, x2...x10

I am not interested in solving these equations, all I want is to find constants of motion, ie some functions of x1, x2 ... x10 which do not depend on t. I can find two independent constants of motion just by inspection. I am hoping that there are more, but is there any systematic way to find them?

 

[rkrsnan]

I guess I will explicitely give the equations here.
Lets rename the dependent variables $$x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, w_1, w_2$$.
The independent variable is $$t$$.
$$a$$ and $$b$$ are constants.

$$\frac{dx_1}{dt}= -2(a - 3 (x_1+y_1)) x_1 + 3 (x_1^2 - 2 x_2 x_3 - z_1)$$

$$\frac{dy_1}{dt}= -2(b - 3 (x_1+y_1)) y_1 + 3 (y_1^2 - 2 y_2 y_3 - z_1)$$

$$\frac{dx_2}{dt}= 2(a - 3 (x_1+y_1)) x_2 - 9 + 3 w_2$$

$$\frac{dy_2}{dt}= 2(b - 3 (x_1+y_1)) y_2 - 9 + 3 w_1$$

$$\frac{dx_3}{dt}= 3 x_3(-2(a - 3 (x_1+y_1)) + x_1-y_1)$$

$$\frac{dy_3}{dt}= 3 y_3(-2(b - 3 (x_1+y_1)) + y_1-x_1)$$

$$\frac{dz_1}{dt}= z_1(-2 (a +b - 6 (x_1+y_1)))$$

$$\frac{dz_2}{dt}= z_2( 2 (a +b - 6 (x_1+y_1)))$$

$$\frac{dw_1}{dt}= w_1(-2 (a -b +6 x_1))-6 x_1 + 6 x_3(z_2-x_2 y_2)$$

$$\frac{dw_2}{dt}= w_2(2 (a -b +6 y_1))-6 y_1 + 6 y_3(z_2-x_2 y_2)$$

From inspection you can see that $$\frac{d(z_1 z_2)}{dt} = 0 , \frac{d(x_3 y_3 z_2^3)}{dt} = 0, \frac{d(x_3 y_3 / z_1^3)}{dt} = 0$$. Out of the three constants of motion I have written only two are independent. The question is can you find more? Is there a systematic method to find the constants of motion?

Thanks a lot!