MATHEMATICAL MODELLING: NUMERICAL INTEGRATION AND PHASE
PORTRAITS OF PLANAR VECTOR FIELDS AND 2D-BIFURCATIONS
SANSONETTO NICOLA
Exercise 1 (Overdamped bead on a rotating hoop). Consider a bead of mass m that slides along a
circular rigid wire hoop of radius r, which is constrained to rotate at a constant angular velocity ω
about its vertical axis. We also assume that there is a friction force, of constant coefficient µ, that
opposes to the motion. Study the motion of the bead, knowing that gravity, with constant gravity
acceleration g, acts on the system.
For the sake of simplicity, assume that the coefficient mr is << 1, and thus the term mrθ̈ in the
equation of motion is neglectable.
Exercise 2. Consider the following planar system of ODEs
x2
ẋ = −ax + y , ẏ = − by ,
1 + x2
where a, b are positive parameters.
i. Show that the system has three critical points for a < ac , with ac to be determined;
ii. Show that two of these critical points coalesce in a saddle-node bifurcation where a = ac ;
iii. Sketch the phase portrait for a < ac .
v. The previous system is a model for a genetic control system. The activity of a certain gene is
assumed to be directly induced by two copies of the protein for which it codes. In other words,
the gene is stimulated by its own product, potentially leading to an autocatalytic feedback
process. x and y are proportional to the concentrations of the protein and the messenger RNA
from which it is translated, respectively, and the positive parameters a and b govern the rate
of degradation of x and y. Give a biological interpretation of the results.
Exercise 3. Consider the following planar system of ODEs
ẋ = µx + y + sin x , ẏ = x − y ,
µ ∈ R.
i. Prove that a supercritical bifurcation occurs at the origin and determine the bifurcation value
µc of the parameter.
ii. Plot the phase portrait near the origin for µ slightly greater than µc .
Exercise 4. Consider the dynamics on the 2-torus T2 ∼
= R2 /2πZ2 described by:
ẋ1 = ω1 ẋ2 = ω2 , (ω2 , ω2 ) ∈ R2 .
Write the explicit solution and represent it on the square of edge 2π. Prove that if ω1 /ω2 ∈ Q then
the dynamics if periodic, while if ω1 /ω2 ∈
/ Q it is dense on the torus.
Exercise 5. Consider the vector field
x − y − x(x2 + y 2 )
X= .
2 2
x + y − y(x + y )
Date: April 2, 2024.
e-mail: nicola.sansonetto@univr.it.
1
Prove that X admits a limit cycle (using the theory of the trapping region), and draw the phase
portrait of the system
Exercise 6 (Van der Pol equation). Consider the Van der Pol vector field
y
X= .
−x − µ(x2 − 1)y
Draw the phase portrait of the system as the parameter µ varies.
Exercise 7. Consider the planer system
ṙ = r(µ − r2 ) , θ̇ = −1 µ ∈ R.
Draw the phase portrait of the system as the parameter µ is varied, using an animation, and observe
that a supercritical Hopf bifurcation occurs.
Exercise 8. Consider the planer system
ṙ = r(µ + r2 − r4 ) , θ̇ = −1 µ ∈ R.
Draw the phase portrait of the system as the parameter µ is varied, using an animation, and observe
that a subcritical Hopf bifurcation occurs.
