(i) Define the terms stable manifold and unstable manifold of a hyperbolic fixed point $\mathbf{x}_{0}$ of a dynamical system. State carefully the stable manifold theorem.

Give an approximation, correct to fourth order in $|\mathbf{x}|$, for the stable and unstable manifolds of the origin for the system

$$\left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right)=\left(\begin{array}{c} x+x^{2}-y^{2} \\ -y+x^{2} \end{array}\right) .$$

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

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

where $a$ is a constant, is non-hyperbolic at $r=1$.

Using new coordinates $v=x+y, w=x-y$, find the centre manifold in the form

$$w=\alpha v^{3}+\ldots, \quad z=\beta v^{2}+\gamma v^{4}+\ldots$$

for constants $\alpha, \beta, \gamma$ to be determined. Hence find the evolution equation on the centre manifold in the form

$$\dot{v}=\frac{1}{8}(a-1) v^{3}+\left(\frac{(3 a+1)(a+1)}{128}+\frac{(a-1)}{32}\right) v^{5}+\ldots$$

Ignoring higher order terms, give conditions on $a$ that guarantee that the origin is asymptotically stable.