(i) State Liapunov's First Theorem and La Salle's Invariance Principle. Use these results to show that the system

$$\ddot{x}+k \dot{x}+\sin x=0, \quad k>0$$

has an asymptotically stable fixed point at the origin.

(ii) Define the basin of attraction of an invariant set of a dynamical system.

Consider the equations

$$\dot{x}=-x+\beta x y^{2}+x^{3}, \quad \dot{y}=-y+\beta y x^{2}+y^{3}, \quad \beta>2$$

(a) Find the fixed points of the system and determine their type.

(b) Show that the basin of attraction of the origin includes the union over $\alpha$ of the regions

$$x^{2}+\alpha^{2} y^{2}<\frac{4 \alpha^{2}\left(1+\alpha^{2}\right)(\beta-1)}{\beta^{2}\left(1+\alpha^{2}\right)^{2}-4 \alpha^{2}} .$$

Sketch these regions for $\alpha^{2}=1,1 / 2,2$ in the case $\beta=3$.