Finding the stability of the origin fixed point in a Lorentz system.
The Lorentz equation:
$\dot{x}=\sigma\left ( y-x \right )$
$\dot{y}=rx-y-xz$
$\dot{z}=-bz+xy$
The Jacobian evaluated at the fixed point $\left ( 0,0,0 \right )$:
$J\mid _{\left ( 0,0,0 \right )}=\begin{bmatrix} -\sigma &\sigma &0 \\ r &-1 &0 \\ 0&0 &-b \end{bmatrix}$
We seek $det\left ( J \mid _{\left ( 0,0,0 \right )}-\lambda I \right )=0$
We compute the $3\times 3$ matrix:
This gives: to save you helpful souls from working it out
Setting the above result to 0, and rearranging the term to $\lambda^{3}$, $\lambda^{2}$, $\lambda$ and the constants, there is no integer valued solutions to be found.
How should I get around this conundrum?
Any explanation to expand my understanding is greatly appreciated.
Hint: Don't expand the brackets. The first cofactor gives you one of the eigenvalues precisely, and the second cofactor is a simple quadratic equation which roots could be analyzed by Vieta's formula.