The question need to be done step by step all information is given please solve it. Thank you

4. (a) A two-dimensional flow in Cartesian coordinates is given by * = ( v ity - 1) x -Hy. (5) where a is a real-valued parameter. (i) Use linearisation to classify the stationary point (0, 0). (3 marks) (ii) Show that equations (5) can be expressed in polar form as i = r(r - /). (6) 0 = 1. (6 marks) (iii) Find the circular limit cycle of equations (6), determine restrictions on / that ensures its existence and classify its stability in terms of p. (3 marks) (iv) Classify the bifurcation that arises at (0, 0). (1 mark) (v) Sketch the three qualitatively different phase portraits that arise as a is varied. (5 marks) (b) A two-dimensional flow is defined by (7) 1 = -ry-y'+s. (i) Show that (0, 0) is a stationary point. (1 mark) (ii) By considering a Lyapunov function of the form V(x, y) = ar" + by" where a, b, n and m are positive integers you must determine, show that equation (7) has no limit cycles. (6 marks) [Q4: 25 marks] [Total: 100 marks 5