Exercise 1 (3+3 points)
Consider the dierential equation
x_ = (x .. 6)(x .. 1)2:
(a) Draw the associated phase portrait and graphically deduce the stability of the
xed points.
(b) For t 0 draw the graph of the solution…
… x1(t) with x1(0) = 2;
… x2(t) with x2(0) = 5;
… x3(t) with x3(0) = 7:
2
Exercise 2 (3+3 points)
For r > 0 and x > 1 consider the dierential equation
x_ = rx
x
1 + x
:
(a) Find all xed points and draw the bifurcation diagram.
(b) What type of bifurcation occurs and what is the bifurcation point?
Exercise 3 (3+3 points)
Let f : R ! R be a dierentiable function with continuous derivative. Suppose the
dierential equation x_ = f(x) has precisely two xed points x1 < x2:
(a) Suppose the xed point x1 is semi-stable and the xed point x2 is a stable. Can
a solution x(t) with x(0) < x1 approach the xed point x2?
If yes, please provide an example. If not, please explain why.
(b) Is it possible that both xed points x1; x2 are unstable?
If yes, please provide an example. If not, please explain why.
4
Exercise 4 (3+3+3 points)
For a parameter r 2 R consider the dierential equation
x_ = r + ex
(a) Let r 0. Show that every solution x(t) with x(0) > 0 blows up in nite time.
(b) Draw the phase portrait for r 0: Do solutions x(t) with x(0) < 0 also blow up
in nite time?
(c) Draw the bifurcation diagram for r 2 R: Describe the type of bifurcation.
Exercise 5 (3+3 points)
(a) For x > 3 consider the dierential equation
x_ = x4 1 + (1 x4) ln(x + 3):
Compute all xed points and use linear stability analysis to determine their
stability.
(b) Let f : R ! R be an innitely often dierentiable function. Let x be a xed
point of the dierential equation x_ = f(x) with f0(x) = 0 and f00(x) < 0:
Determine the stability of x. Brie
y explain your answer.
Exercise 6 (2+4+1 points)
For r < 0 and x > 0 consider the dierential equation
x_ = ..
1
x
+ rx + 2:
(a) Calculate the unique bifurcation point (r; x):
(b) Approximate the ODE near the bifurcation point and use a normal form to
classify the bifurcation.
Specify the order of the approximation as an error in ” > 0 and the corresponding
scalings of r; x; e.g. jx .. xj < “:
(You do not need to change coordinates.)
(c) Draw the local bifurcation diagram near (r; x):
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