Linear and non-linear ODEs, Math 134
Exam 3
英国数学代考 such that the origin is a Lyapunov stable but not attracting fixed point? If yes, please provide an example and draw the phase
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Exercise 1 (3+3 points) 英国数学代考
(a) Is there a 2-dimensional ODE system
x˙ =
y˙ =
such that the origin is a Lyapunov stable but not attracting fixed 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 fixed point, which is a saddle point?
If yes, please provide an example and draw the phase portrait. If not, please explain why.
Exercise 2 (1+3+6+3 points)
Consider the differential equation
x¨ = x 2 − 11x + 10
(a) Write the differential equation as a first order ODE system.
(b) Calculate all fixed points and classify them using linear stability analysis.
(c) Find a conserved quantity for the differential equation. Show that your quantity is indeed preserved. Classify the fixed points of the non-linear ODE.
(d) Draw a plausible phase portrait. Indicate the stable and unstable manifolds.
Exercise 3 (7 points)
Show that the differential equation
has a closed orbit. You may use that the origin is the only fixed point.
Exercise 4 (6 points)
Use the function
L(x, y) = x n + ay2,
for appropriate choices of a > 0 and n ∈ N, to show that the origin is the unique fixed point of the differential equation
Deduce the stability type of the origin.
Exercise 5 (4+4 points) 英国数学代考
For µ ∈ R consider the differential equation
(a) Draw the phase portrait for µ = 0. Remark. You do not need to compute the linearization at the fixed points.
(b) Is there a closed orbit that encloses the origin? Please explain.
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