The Euler Top is the system $$\begin{cases}x_1' = \alpha_1 x_2 x_3 \\ x_2' = \alpha_2 x_3 x_1 \\ x_3' = \alpha_3 x_1 x_2\end{cases}$$
The HK discretization scheme is given by: HK
I am reading this part of a paper: (Petrera2011)
I want to figure out how $(6.4)$ is obtained exactly by replicating the calculation, but I get stuck. So far I have solved for $x_3^{\sim}$, but what I find doesn't look like it will lead me to what they have. Can anyone take a look?
Write it down systematically, $$ \begin{bmatrix} 1&-hα_1x_3&-hα_1x_2\\ -hα_2x_3&1&-hα_2x_1\\ -hα_3x_2&-hα_3x_1&1 \end{bmatrix} \begin{bmatrix} \tilde x_1\\\tilde x_2\\\tilde x_3 \end{bmatrix} = \begin{bmatrix} x_1\\x_2\\x_3 \end{bmatrix} $$ Now apply Cramer's rule to get rational expressions for the solution. In a numerical implementation you would call a linear solver