(i) A linear system in $\mathbb{R}^{2}$ takes the form $\dot{\mathbf{x}}=\mathrm{Ax}$. Explain (without detailed calculation but by giving examples) how to classify the dynamics of the system in terms of the determinant and the trace of A. Show your classification graphically, and describe the dynamics that occurs on the boundaries of the different regions on your diagram.

(ii) A nonlinear system in $\mathbb{R}^{2}$ has the form $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}), \mathbf{f}(0)=0$. The Jacobian (linearization) $A$ of $\mathbf{f}$ at the origin is non-hyperbolic, with one eigenvalue of $A$ in the left-hand half-plane. Define the centre manifold for this system, and explain (stating carefully any results you use) how the dynamics near the origin may be reduced to a one-dimensional system on the centre manifold.

A dynamical system of this type has the form

$$\begin{aligned} &\dot{x}=a x^{3}+b x y+c x^{5}+d x^{3} y+e x y^{2}+f x^{7}+g x^{5} y \\ &\dot{y}=-y+x^{2}-x^{4} \end{aligned}$$

Find the coefficients for the expansion of the centre manifold correct up to and including terms of order $x^{6}$, and write down in terms of these coefficients the equation for the dynamics on the centre manifold up to order $x^{7}$. Using this reduced equation, give a complete set of conditions on the coefficients $a, b, c, \ldots$ that guarantee that the origin is stable.