Explain the difference between a stationary bifurcation and an oscillatory bifurcation for a fixed point $\mathbf{x}_{0}$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x} ; \mu)$ in $\mathbb{R}^{n}$ with a real parameter $\mu$.

The normal form of a Hopf bifurcation in polar coordinates is

$$\begin{aligned} &\dot{r}=\mu r-a r^{3}+O\left(r^{5}\right) \\ &\dot{\theta}=\omega+c \mu-b r^{2}+O\left(r^{4}\right) \end{aligned}$$

where $a, b, c$ and $\omega$ are constants, $a \neq 0$, and $\omega>0$. Sketch the phase plane near the bifurcation for each of the cases (i) $\mu<0<a$, (ii) $0<\mu, a$, (iii) $\mu, a<0$ and (iv) $a<0<\mu$.

Let $R$ be the radius and $T$ the period of the limit cycle when one exists. Sketch how $R$ varies with $\mu$ for the case when the limit cycle is subcritical. Find the leading-order approximation to $d T / d \mu$ for $|\mu| \ll 1$.