Example Sheet 2

TIF155/FIM770 Problem Set 2 due 13:15 24 April, 2015
1
Problem 1 Consider the problem
x˙ = (λ + 1)x +
3y
y˙ =
−2x
+ (λ − 1)y
• (a) Solve this problem analytically in terms of x(0) and y(0) for arbitrary value of λ and
plot representative trajectories for λ = −1/10, λ = 0 and λ = 1/10.
• (b) For λ = 0, show that invariant orbits are ellipses and analytically compute the period,
the ratio of major and minor axis of the ellipse as well as their direction in the (x, y) plane.
Now consider the generalized problem x˙ = ax − by and y˙ = cx − ay and a2 < bc and let
"
m=
a −b
c −a
#
Prove that the dynamics is an ellipse, compute the frequency in terms of a, b and c and compute
a 2×2 matrix η obeying η · η = −1 so that the ellipse obeys the equation x (η · m) x = constant
Problem 2 Consider the problem
x˙ = (λ + 3)x + 4y
y˙ = − (9/4) x + (λ − 3)y
and define the problem by x˙ = Mλ x where x = (x, y).
For λ = ±1 and 0 classify the fixed point, compute all the eigenvalues, eigenvectors and, if it
exists, the inverse matrix to Mλ .
Next, solve this problem analytically for all values of λ. Then
• for λ = −1 plot a set of representative trajectories and analytically compute the direction
of the stable manifold.
• for λ = 0 plot and interpret a set of representative trajectories in terms of the eigenvector(s) and eigenvalue(s). Compute the line of fixed points that separate the two regions.
• for λ = +1 plot a set of representative trajectories and analytically compute the direction
of the unstable manifold.
Now consider the generalized problem x˙ = (λ − cd)x + d2 y and y˙ = −c2 x + (λ + cd)y Classify
the dynamics as a function of λ and compute the direction of the three manifolds mentioned
in the previous part of the problem.
TIF155/FIM770 Problem Set 2 due 13:15 24 April, 2015
2
Problem 3 Consider the damped rotator
mr2 θ¨ = −mgr sin θ − krθ˙
(a) Show that by proper rescaling this can be reduced to the two-dimensional flow
x˙ =
y
y˙ = − sin x − γy
(b) For arbitrary γ ≥ 0, calculate and classify the fixed points as a function of γ.
(c) For γ = (0, .3, 1, 1.5, 2, 2.5) plot and interpret a set of representative trajectories.
You may use StreamPlot to get an idea of what you are looking at, but it is not accurate
enough to be an acceptable solution.
Problem 4a (6.8.2 - 6.8.5 ) For each of the following systems, locate the fixed points and
obtain the index of each fixed point and draw a (rough!) phase plot. You may use StreamPlot
but you will need to fill in the streamlines near the fixed points which you may do by hand if
you wish. The hand-fixed part does not have to be included in the urkund submitted version.
• (a) x˙ = x2 , y˙ = y
• (b)x˙ = y − x , y˙ = x2
• (c)x˙ = y 3 , y˙ = x
• (d)x˙ = xy , y˙ = x + y
A resonably accurate drawing of the streamlines near the fixed point is sufficient, since the
result must be integers so integrals do not need to be done numerically.
Problem 5: higher index fixed points Construct a dynamics that has a fixed point with
index ±2, ±3 and draw the flow. Perturb the flow slightly, describe the bifurcation that occurs
and classify the fixed points that occur in the perturbed flow.
TIF155/FIM770 Problem Set 2 due 13:15 24 April, 2015
Problem 6
3
This is based on problem 6.5.15 to 6.5.18 in the book.
Consider the bead on a rotating hoop
mrφ¨ = −bφ˙ − mg sin φ + mrω 2 sin φ cos φ
The case b = 0
In this case, show that the equations can be cast into the form
φ¨ = sin φ(cos φ − γ −1 )
where γ = rω 2 /g and the diffrentiation is with respect to τ = ωt.
• (a) Qualitatively sketch phase portraits in the variables φ and φ˙ for γ < 1, γ = 1 and
γ > 1. (Again, you may use StreamPlot and fill in the important details by hand)
• (b) Derive a constant of the motion.
The case b > 0 In the case when four fixed points exist, make a representative phase portrait
for small (undercritical) damping, choosing parameters so that the four fixed points are clearly
visible. It may be useful to extend the range of φ to −2π ≤ φ ≤ 2π to see the motion that can
occur. Again, if you use StreamPlot , the details near the fixed points should be filled in by
hand.