(a) Prove that the origin of the system dx dt = −

x 2 + 2x2y, dy dt = x − y − x3 is asymptotically stable using the Lyapunov function V = x2 + 2y2.

[6]

(b) Solve the differential equations dr dt = −r2, dθ

dt = 1, given that r(0) = 1 and θ(0) = 0. Hence show that the return map, say, P, mapping points, say, rn, on the positive x-axis to itself is given by rn+1 = P(rn) =

rn 1 + 2πrn

.

[14]