State, without proof, the centre manifold theorem. Show that the fixed point at the origin of the system

$$\begin{aligned} &\dot{x}=y-x+a x^{3} \\ &\dot{y}=r x-y-y z \\ &\dot{z}=x y-z \end{aligned}$$

where $a \neq 1$ is a constant, is nonhyperbolic at $r=1$. What are the dimensions of the linear stable and (non-extended) centre subspaces at this point?

Make the substitutions $2 u=x+y, 2 v=x-y$ and $\mu=r-1$ and derive the resultant 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 $Z$ to $O\left(u^{2}\right)$ and $V$ to $O\left(u^{3}\right)$. Hence obtain the evolution equation on the centre manifold correct to $O\left(u^{3}\right)$, and identify the type of bifurcation distinguishing between the cases $a>1$ and $a<1$.

If now $a=1$, assume that $\mu=O\left(u^{4}\right)$ and extend your calculations of $Z$ to $O\left(u^{4}\right)$ and of the dynamics on the centre manifold to $O\left(u^{5}\right)$. Hence sketch the bifurcation diagram in the neighbourhood of $u=\mu=0$.