(a) State the properties defining a Lyapunov function for a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$. State Lyapunov's first theorem and La Salle's invariance principle.

(b) Consider the system

$$\begin{aligned} &\dot{x}=y \\ &\dot{y}=-\frac{2 x\left(1-x^{2}\right)}{\left(1+x^{2}\right)^{3}}-k y \end{aligned}$$

Show that for $k>0$ the origin is asymptotically stable, stating clearly any arguments that you use.

$$\left[\text { Hint: } \frac{d}{d x} \frac{x^{2}}{\left(1+x^{2}\right)^{2}}=\frac{2 x\left(1-x^{2}\right)}{\left(1+x^{2}\right)^{3}} \cdot\right]$$

(c) Sketch the phase plane, (i) for $k=0$ and (ii) for $0<k \ll 1$, giving brief details of any reasoning and identifying the fixed points. Include the domain of stability of the origin in your sketch for case (ii).

(d) For $k>0$ show that the trajectory $\mathbf{x}(t)$ with $\mathbf{x}(0)=\left(1, y_{0}\right)$, where $y_{0}>0$, satisfies $0<y(t)<\sqrt{y_{0}^{2}+\frac{1}{2}}$ for $t>0$. Show also that, for any $\epsilon>0$, the trajectory cannot remain outside the region $0<y<\epsilon$.