The Lorenz equations are

$$\begin{aligned} &\dot{x}=\sigma(y-x) \\ &\dot{y}=r x-y-x z \\ &\dot{z}=x y-b z \end{aligned}$$

where $r, \sigma$ and $b$ are positive constants and $(x, y, z) \in \mathbb{R}^{3}$.

(i) Show that the origin is globally asymptotically stable for $0<r<1$ by considering a function $V(x, y, z)=\frac{1}{2}\left(x^{2}+A y^{2}+B z^{2}\right)$ with a suitable choice of constants $A$ and $B$

(ii) State, without proof, the Centre Manifold Theorem.

Show that the fixed point at the origin is nonhyperbolic at $r=1$. What are the dimensions of the linear stable and (non-extended) centre subspaces at this point?

(iii) Let $\sigma=1$ from now on. Make the substitutions $u=x+y, v=x-y$ and $\mu=r-1$ and derive the resulting equations for $\dot{u}, \dot{v}$ and $\dot{z}$.

The extended centre manifold is given by

$$v=V(u, \mu), \quad z=Z(u, \mu)$$

where $V$ and $Z$ can be expanded as power series about $u=\mu=0$. What is known about $V$ and $Z$ from the Centre Manifold Theorem? Assuming that $\mu=O\left(u^{2}\right)$, determine $Z$ correct to $O\left(u^{2}\right)$ and $V$ to $O\left(u^{3}\right)$. Hence obtain the evolution equation on the extended centre manifold correct to $O\left(u^{3}\right)$, and identify the type of bifurcation.