Mathematics Homework Help

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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