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.
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
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?
Assume a < 0, d > 0
x' = a x
y' = d y
Discuss the differences for a < d, a=d, a > d.
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.
x'=a-x^2
x' = a x -x^2
x' = a + b x -x^3
x' = y, y' = - y -x -x^3
x' = y, y' = -y + x -x^3
x' = x -y, y' = y^2-x
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, .