QNo.1:
For each of the following systems, use a quadratic Lyapunov function candidate
to show that the origin is asymptotically stable:
(1) $\dot{x}_1 = -x_1 + x_1x_2$, $\dot{x}_2 = -x_2$
(2) $\dot{x}_1 = -x_2 - x_1(1 - x_1^2 - x_2^2)$, $\dot{x}_2 = x_1 - x_2(1 - x_1^2 - x_2^2)$
(3) $\dot{x}_1 = x_2(1 - x_1^2)$, $\dot{x}_2 = -(x_1 + x_2)(1 - x_1^2)$
(4) $\dot{x}_1 = -x_1 - x_2$, $\dot{x}_2 = 2x_1 - x_2^3$
Investigate whether the origin is globally asymptotically stable.
