1. Show that the following system * = f(u) +, =g(2) + p has no periodic...

  

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1. Show that the following system * = f(u) +, =g(2) + p has no periodic solutions for > 0. 2. Consider the system . Show that the system i=1+ry-I-Y. y=ry++x-y. (a) Transform the system into polar co-ordinates (r, 0) and show that it has exactly one periodic orbit of constant radius. (b) Classify the stability of the periodic orbit. i=x+y=x (x + 3y), y=-x+y-y (x + 3y), [2 marks] = Hint: Consider the scalar product of the tangent f n = [ry] to the circle x + y = R. [4 marks] has at least one closed path on the annulus 1/3 < x + y < 1. You may assume that the origin is the only equilibrium point. [ty] and outward normal [4 marks] 1. Show that the following system * = f(u) +, =g(2) + p has no periodic solutions for > 0. 2. Consider the system . Show that the system i=1+ry-I-Y. y=ry++x-y. (a) Transform the system into polar co-ordinates (r, 0) and show that it has exactly one periodic orbit of constant radius. (b) Classify the stability of the periodic orbit. i=x+y=x (x + 3y), y=-x+y-y (x + 3y), [2 marks] = Hint: Consider the scalar product of the tangent f n = [ry] to the circle x + y = R. [4 marks] has at least one closed path on the annulus 1/3 < x + y < 1. You may assume that the origin is the only equilibrium point. [ty] and outward normal [4 marks]