Investigate the stability of the critical points at the origin for the following systems:

(a) ˙ x = −y − x3, ˙ y = x − y3, using the Lyapunov function V (x, y) =

x2 + y2;

(b) ˙ x = x(x − α), ˙ y = y(y − β), using the Lyapunov function V (x, y) =

x

α

2

+

y

β

2

;

(c) ˙ x = y, ˙ y = y − x3, using the Lyapunov function V (x, y) = ax4 +

bx2 + cxy + dy2.