Due Friday Sept 11

• A. For each of these vector fields, sketch the phase-plane, determine the stability of the origin, and if the point is a saddle-point, draw the stable and unstable invariant sets.
  1. Consider
     x' = a x + b y

     y' = c x + d y

     • a=1,b=2,c=3,d=1
     • a=-1,b=2, c=-3, d=1
     • a=-1,b=2, c=-3, d=2
  2. Assume a < 0, d > 0
     x' = a x

     y' = d y

     Discuss the differences for |a|< d, |a|=d, |a|> d. Is the unstable manifold attracting and does it depend on these 3 cases?
  3. Assume a < 0, d > 0
     x' = a x

     y' = d y

     Discuss the differences for a < d, a=d, a > d.
  4. Assume a < 0, b > 0
     x' = a x - b y

     y' = b x + a y

     How do the trajectories depend on the relative magnitudes of a,b. What happens when a=0, or b=0 ?
• B. For each of the following, find all the fixed points, determine stability and sketch a phaseplane or phase-line as well as you can. For those with a parameter, distinguish the different possible cases.
  1. x'=a-x^2
  2. x' = a x -x^2
  3. x' = a + b x -x^3
  4. x' = y, y' = - y -x -x^3
  5. x' = y, y' = -y + x -x^3
  6. x' = x -y, y' = y^2-x
  7. x' = 1, y' = 1 + sin(x-y), (x,y) on circle. (Hint: look at the differential equation for z=x-y .
• Use the Poincare Bendixson theorem to prove that there is at least one stable periodic solution to the differential equation

  x' = -x + tanh(10 (x - y)), y'=-y + tanh(2x)

  First sketch the nullclines. Note that there is only one fixed point. Determine its stability. Show that all trajectories go into the box |x| < 1, |y| < 1, .