For what $r$ does the Lorenz attractor exist?

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 $\mathbb{R}^2$.