# 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?

## 1 Answer

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. ). 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 $\mathbb{R}^2$.