Mathematics Homework Help
University of California Los Angeles Differential Equation Exercises
Exercise 1
(a) Is there a 2-dimensional ODE system
x_ =
y_ =
such that the origin is a Lyapunov stable but not attracting xed point?
If yes, please provide an example and draw the phase portrait. If not, please
explain why.
(b) Is there a 2-dimensional ODE system
x_ =
y_ =
with a closed orbit and single xed point, which is a saddle point?
If yes, please provide an example and draw the phase portrait. If not, please
explain why.
Exercise 2
Consider the dierential equation
x = x2 .. 11x + 10
(a) Write the dierential equation as a rst order ODE system.
(b) Calculate all xed points and classify them using linear stability analysis.
(c) Find a conserved quantity for the dierential equation. Show that your quantity
is indeed preserved.
Classify the xed points of the non-linear ODE.
(d) Draw a plausible phase portrait. Indicate the stable and unstable manifolds.
Exercise 3
Show that the dierential equation
x_ = 4x .. 2×5 + 2y .. 4x3y2;
y_ = ..2x + 4y .. 4x4y .. 2y5
has a closed orbit. You may use that the origin is the only xed point.
Exercise 4
Use the function
L(x; y) = xn + ay2;
for appropriate choices of a > 0 and n 2 N, to show that the origin is the unique xed
point of the dierential equation
x_ = 3y 3×3 3xy2;
y_ = x y3 x2y:
Deduce the stability type of the origin.
Exercise 5
For 2 R consider the dierential equation
r_ = r(r 1 + cos )
_ = sin 2
(a) Draw the phase portrait for = 0:
Remark. You do not need to compute the linearization at the xed points.
(b) Is there a closed orbit that encloses the origin? Please explain.
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