User:Nedtheprotist


 * $$\frac{dx}{dt} = a (-x + y)$$


 * $$\frac{dy}{dt} = b x - y -x z$$


 * $$\frac{dz}{dt} = xy - c z$$


 * $$\ ax - ay = 0$$


 * $$\ b x - y -x z - 0$$


 * $$\ xy - c z = 0$$


 * $$\ x(b-1-z) = 0$$


 * $$\ -cz+x^2 = 0$$


 * $$\ 3\lambda^3+41\lambda^2+8(b+10)\lambda+160(b-1)=0$$


 * $$\ C_1 = (0,0,0)$$


 * $$\ C_2 = (\sqrt{c(b-1)},\sqrt{c(b-1)},b-1)$$


 * $$\ C_3 = (-\sqrt{c(b-1)},-\sqrt{c(b-1)},b-1)$$

$$ \begin{pmatrix} x \\ y \\ z \end{pmatrix}'= \begin{pmatrix} -10 & 10 & 0 \\ b & -1 & 0 \\ 0 & 0 & -8/3 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix}.$$

$$\begin{vmatrix} -10-\lambda & 10 & 0\\b & -1-\lambda & 0\\0 & 0 & -8/3-\lambda \end{vmatrix}=-(8/3+\lambda)[\lambda^2+11\lambda-10(b-1)]=0.$$

$$\lambda_1=-\frac{8}{3},  \lambda_2=\frac{-11-\sqrt{81+40b}}{2},   \lambda_3=\frac{-11+\sqrt{81+40b}}{2}$$

$$ \begin{pmatrix} u \\ v \\ w \end{pmatrix}'= \begin{pmatrix} -10 & 10 & 0 \\ 1 & -1 & -\sqrt{8(b-1)/3} \\ \sqrt{8(b-1)/3} & \sqrt{8(b-1)/3} & -8/3 \end{pmatrix} \begin{pmatrix} u \\ v \\ w \end{pmatrix}.$$