(i) Define carefully what is meant by a Hopf bifurcation in a two-dimensional dynamical system. Write down the normal form for this bifurcation, correct to cubic order, and distinguish between bifurcations of supercritical and subcritical type. Describe, without detailed calculations, how a general two-dimensional system with a Hopf bifurcation at the origin can be reduced to normal form by a near-identity transformation.

(ii) A Takens-Bogdanov bifurcation of a fixed point of a two-dimensional system is characterised by a Jacobian with the canonical form

$$A=\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right)$$

at the bifurcation point. Consider the system

$$\begin{aligned} &\dot{x}=y+\alpha_{1} x^{2}+\beta_{1} x y+\gamma_{1} y^{2} \\ &\dot{y}=\alpha_{2} x^{2}+\beta_{2} x y+\gamma_{2} y^{2} \end{aligned}$$

Show that a near-identity transformation of the form

$$\begin{aligned} &\xi=x+a_{1} x^{2}+b_{1} x y+c_{1} y^{2} \\ &\eta=y+a_{2} x^{2}+b_{2} x y+c_{2} y^{2} \end{aligned}$$

exists that reduces the system to the normal (canonical) form, correct up to quadratic terms,

$$\dot{\xi}=\eta, \quad \dot{\eta}=\alpha_{2} \xi^{2}+\left(\beta_{2}+2 \alpha_{1}\right) \xi \eta .$$

It is known that the general form of the equations near the bifurcation point can be written (setting $p=\alpha_{2}, q=\beta_{2}+2 \alpha_{1}$ )

$$\dot{\xi}=\eta, \quad \dot{\eta}=\lambda \xi+\mu \eta+p \xi^{2}+q \xi \eta .$$

Find all the fixed points of this system, and the values of $\lambda, \mu$ for which these fixed points have (a) steady state bifurcations and (b) Hopf bifurcations.