(i) Define the Floquet multiplier and Liapunov exponent for a periodic orbit $\hat{\mathbf{x}}(t)$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$. Show that one multiplier is always unity, and that the other is given by

$$\exp \left(\int_{0}^{T} \nabla \cdot \mathbf{f}(\hat{\mathbf{x}}(t)) d t\right)$$

where $T$ is the period of the orbit.

The Van der Pol oscillator $\ddot{x}+\epsilon \dot{x}\left(x^{2}-1\right)+x=0,0<\epsilon \ll 1$ has a limit cycle $\hat{x}(t) \approx 2 \sin t$. Show using $(*)$ that this orbit is stable.

(ii) Show, by considering the normal form for a Hopf bifurcation from a fixed point $\mathbf{x}_{0}(\mu)$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}, \mu)$, that in some neighbourhood of the bifurcation the periodic orbit is stable when it exists in the range of $\mu$ for which $\mathbf{x}_{0}$ is unstable, and unstable in the opposite case.

Now consider the system

$$\left.\begin{array}{l} \dot{x}=x(1-y)+\mu x \\ \dot{y}=y(x-1)-\mu x \end{array}\right\} \quad x>0$$

Show that the fixed point $(1+\mu, 1+\mu)$ has a Hopf bifurcation when $\mu=0$, and is unstable (stable) when $\mu>0(\mu<0)$.

Suppose that a periodic orbit exists in $\mu>0$. Show without solving for the orbit that the result of part (i) shows that such an orbit is unstable. Define a similar result for $\mu<0$.

What do you conclude about the existence of periodic orbits when $\mu \neq 0$ ? Check your answer by applying Dulac's criterion to the system, using the weighting $\rho=e^{-(x+y)}$.