answer me

2. Consider the system in polar coordinates (r, ) with r > 0: 7-1 2r (2) where w > 0 is a given parameter. (a)[5pts] Choose the positive r-axis as the cross section S. Let f denote the asso- ciated Poincare map for the r-coordinate of the trajectory on S: Inti = /(I. ). Show that In , is related to r, by -1=(-1)e-, I,>0. (3) which gives In+1 = /(x.) = V1+ (x -1)e-#, In 20. (4) (b)[1pt] Use the Poincare map in (4) to show that system (2) has a closed orbit. (c)[3pts] Use the Poincare map to show that the close orbit in part (b) is stable for any w > 0