Nonlinear Systems
Problem Sheet 4
Oxford, Hilary Term 2017
Prof. Alain Goriely
March 9, 2017
Note: Problems or sub-problems marked with a star* are meant to challenge you beyond
the regular course (problems with ** are particularly challenging). It is up to you to decide
if you want to try them. I suggest that you give them a try and think about these problems
as a way to gain a deeper understanding of the material.
1
Bifurcations
1.1 More on Hopf: Analyze the Hopf bifurcation set µ = 0 for ν < 0 and µ = ν for ν > 0
for the system
ẍ + µẋ + νx + x2 ẋ + x3 = 0.
What happens at µ = ν = 0?
1.2 *Bifurcations: Consider the following vector fields
ẋ = x − σy − y(x2 + y 2 )
ẏ = σx + y − y(x2 + y 2 ) − γ
(1)
(2)
Find the bifurcation set in (σ, γ). Use Sotomayor’s and Hopf’s theorems to find the points
at which different bifurcations take place. (Note: the difficulty of this exercise is due to
the analysis of a cubic equation for the fixed points. You should probably use symbolic
computation to do the analysis)
2
Maps
2.1 A Binary expansions: Consider the mapping F : [0, 1] → [0, 1]
F (x) = 2x mod 1
(i) Show that if x ∈ [0, 1] has the binary expansion
x = .s1 s2 . . . =
∞
X
si
i=0
with si ∈ {0, 1}, then
1
2i
F n (x) = .sn+1 sn+2 . . .
(ii) Prove the existence of a countable infinity of periodic orbits.
(iii) Prove the existence of a uncountable infinity of nonperiodic orbits.
(iv) Show that the system has sensitive dependence to initial conditions.
3
Melnilkov
3.1 Existence of homoclinic orbit Consider the perturbed Duffing equation
ẋ = y
ẏ = x − x3 + [αy + βx2 y]
(3)
(4)
Draw the phase portrait for = 0. Use Melnikov’s analysis to find a condition on the
parameters α and β such that two homoclinic orbit exist for small enough. Draw the
perturbed phase portrait for α < 0.
3.2 Bubble again Consider a gas of bubble of volume 43 πa3 in the presence of a timeperiodic, axisymmetric uniaxial extensional flow of a fluid. The bubble shape is given by
r = r(θ, t). Introduce a scalar measure of deformation:
Z π
r(θ, t)P2 (cos θ) sin θ dθ
x=
0
where P2 is a Legendre polynomial. The variable x measures the deviation of the bubble
from sphericity.
The evolution of x, in rescaled variables, is:
ẍ = w − 2x + x2 − (µẋ + δ cos ωt),
where w is related to the Weber number:
W0 =
2ρE02 a3
,
γ
w=
4 × 144 × 755
.
672 W0 a2
where γ is the surface tension, E0 the principal strain rate, ρ the fluid density and µ its
viscosity.
After translation, the system reads:
ü + ω02 u + u2 = (−µu̇ + δ cos ωt),
(5)
√
where ω02 = 2 1 − w.
For this system, compute the Melnikov function for the existence of transverse homoclinic
points. For a fixed forcing amplitude, find the optimal forcing frequency for bubble breakup.
2