# Stability of origin for Lorenz system and the nature of local bifurcation for certain control parameter

The Lorenz equation:

$\dot{x}=\sigma\left ( y-x \right )$

$\dot{y}=rx-y-xz$

$\dot{z}=-bz+xy$

The fixed points are $\left ( 0,0,0 \right )$ and $\left ( \pm \sqrt{\left ( r-1 \right )b}, \pm \sqrt{\left ( r-1 \right )b},r-1 \right ), \forall r>1$.

To compute the stability at the origin, Strogaz recommends removing the non-linear terms and ignoring the decoupled equation so as to perform a linearisation.

Thus,

$\dot{x}=\sigma\left ( y-x \right )$

$\dot{y}=rx-y-xz$

$\dot{z}=-bz+xy$

is reduced to

$\dot{x}=\sigma\left ( y-x \right )$

$\dot{y}=rx-y$

Now, we have:

$\begin{bmatrix} \dot{x}\\ \dot{y} \end{bmatrix}$ =$\begin{bmatrix} -\sigma &\sigma \\ r&-1 \end{bmatrix}$ $\begin{bmatrix} x\\ y \end{bmatrix}$

which from the matrix

$\begin{bmatrix} -\sigma &\sigma \\ r&-1 \end{bmatrix}$

one can directly compute the trace and determinant to predict the eigenvalues and therefore the stability.

Am I in the right direction thus far?

Now, if the control parameter $r \mapsto 1$:

The non-zero fixed point for this Lorenz equation tends to $\left ( 0,0,0 \right )$. It coalesce at the origin.

I would like to analyse and understand what the local bifurcation is for r=1. To my knowledge, it should be a saddle-node bifurcation, since $\left ( \pm \sqrt{\left ( r-1 \right )b}, \pm \sqrt{\left ( r-1 \right )b},r-1 \right ), \forall r>1$ and it only coalesce at the origin for r=1.

Would someone illuminate my understanding?

Well, it could be done this way, but I don't agree that it's easier. If you want to find a linearized system for an equilibrium at the origin, it is it: $$\dot{x}=\sigma\left ( y-x \right ),$$ $$\dot{y}=rx-y,$$ $$\dot{z}=-bz ,$$ just dropping nonlinear terms. The linearization matrix has block-diagonal structure, and it's really easy to compute its characteristic equation and its roots. The last equation is decoupled from others and it gives this $-b$ root of characteristic equation instantaneously, and since the first two equations doesn't include $z$, you can consider them separately. I don't agree that the stability could be determined only by them unless we know something about $b$. It is true for $b > 0$, but completely false for $b < 0$ — in that case equilibrium at the origin has an eigenvalue with positive real part and it's definitely unstable.
I would like to analyse and understand what the local bifurcation is for $r=1$.