(ii) Using a Liapunov function of the form V (x, y) = Ax2 + By2 ,where A and C are constants, examine the stability of the critical point(s) for the following systems:
dx dt = −x + y − xy2 , dy dt = −2x − y − x 2 y,  (b) dx dt = −x 3 + y 4 , dy dt = −y 3 + y 4 .  (iii) (a) For the system dx dt = f(x) + by dy dt = cx + dy where (f(0) = 0), establish that V given by V (x, y) = (dx − by) 2 + 2d ∫ x 0 f(u) du − bcx2 is a strong Liapunov function for the zero solution (0, 0), when in some neighbourhood of the origin (0, 0), the following relation holds d f(x) x − bc > 0, f(x) x + d < 0 for x ̸= 0.  (b) Using the above to deduce that for initial conditions in the circle x 2 + y 2 < 1 the solution (x(t), y(t)) of the system dx dt = −x 3 + x 4 + y dy dt = −x tends to (0, 0) as t → ∞.
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