Describe the direct method of Liapunov (sometimes spelt Lyapunov) as an alternative to obtaining stability results for solutions to ordinary differential equations. Give the necessary definitions definitions as well as state without proof the appropriate theorems. [2]

[2] (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, [4] (b) dx dt = −x 3 + y 4 , dy dt = −y 3 + y 4 . [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. [4] (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 → ∞.