Two cyclists are racing on a circular track. The positions of each cyclist on the track, relative to some xed point: are determined by the angles 6'1, 5'2. Their angular velocities are Ell] = w1+ f}[4 + 3CDS(31 32} SiIEHl 232)] -. d2 = mg + b[4 + 3003092 5'1} sinflg 231)] , where in], mg, f.- Zb [l are constants. {a} Find a dimensionless time variable 1" and parameter r so that the angle between the cyclists, Ii! 2 5'1 5'2, satises the non-dimensionalized equation it? E = r sin[2r,). (b) Sketch all qualitatively different vector elds (on the circle) for different choices of the parameter 1'. Find all bifurcation points (r*,*}. (c) In a pursuit race, the cyclists start at opposite sides of the track. Assuming that ml 2:- mg, nd the value of 'r for which the faster cyclist catches the slower cyclist at dimensionless time 1" 2 :rr