For what $r$ does the Lorenz attractor exist? The Lorenz equations are given by:
\[\dot X=\sigma(Y-X)\]
\[\dot Y=rX-Y-XZ\]
\[\dot Z=XY-bZ\]
I know that for $r\gt r_H$ all trajectories go over to a strange attractor, but this does not necessarily mean that the attractor exists for only $r\gt r_H$. My question is therefore, what values of $r$ does this attractor exist and why?
 A: Let us compute the equilibrium points such that
$$
\left\lbrace\begin{aligned}
&\sigma \left(Y-X\right) = 0 \, ,\\
&r (X - Y - XZ) = 0 \, ,\\
&XY-bZ = 0 \, ,
\end{aligned}\right.
\qquad\text{viz.}\qquad
\left\lbrace\begin{aligned}
&Y = X \, ,\\
&\left(r-1-Z\right) X = 0 \, ,\\
&X^2 - bZ =0 \, .
\end{aligned}\right.
$$
Therefore, the following equilibria are obtained:
$$
(X,Y,Z) \in \left\lbrace (0, 0, 0), \left(\pm\sqrt{b (r-1)}, \pm\sqrt{b (r-1)}, r-1\right) \right\rbrace .
$$
The Jacobian matrix of the dynamical system is given by
$$
J(X,Y,Z) =
\left[
\begin{array}{ccc}
-\sigma & \sigma & 0\\
r-Z & -1 & -X\\
Y & X & -b
\end{array}
\right] .
$$


*

*At the equilibrium point $(0,0,0)$, the eigenvalues of the Jacobian matrix are
$$
\mathrm{Sp}\, J(0,0,0) = \left\lbrace -b, \frac{1}{2}\left(\pm\sqrt{4 r \sigma + (\sigma-1)^2} - \sigma - 1\right) \right\rbrace .
$$
This equilibrium is stable as long as all eigenvalues have a non-positive real part, i.e. $r\leq 1$. For $r\leq 1$, the point $(0,0,0)$ is the only equilibrium of the dynamical system. No convection occurs, and no strange attractor exists.

*For $r>1$, the equilibrium $(0,0,0)$ becomes unstable, and the pair of equilibria starts existing: a (supercritical) pitchwork bifurcation occurs. Computing the eigenvalues of the Jacobian matrix, one can show that they have non-positive real parts if $\sigma > b+1$ and $r\leq r_H$, where
$$ r_H = \sigma \frac{\sigma + b + 3}{\sigma - b - 1}\, . $$
We are in the convection regime, and no strange attractor exists.

*For $r> r_H$, the pair of stable equilibria becomes unstable: a Hopf bifurcation occurs. All trajectories with initial condition appart from an equilibrium point will give the Lorenz attractor. Note that there can be periodic orbits (see e.g. [1]). However, these features are hard to analyze.
One reason why we can have such chaotic solutions relates to the Poincaré-Bendixson theorem. Indeed,  the Lorenz system is a differentiable real dynamical system on $\mathbb{R}^3$, and not on $\mathbb{R}^2$.
