Transcritical bifurcations homework (SOLVED)

Your answers must be uploaded on Canvas, as a single PDF, by
5pm Eastern Time.
This homework covers 1. Transcritical bifurcations; 2. Pitchfork bifurcations (super- and subcritical); 3.
Hysteresis; and 4. Imperfect bifurcations.
These topic are covered in §3.2-3.6 in Strogatz.
For your bifurcation diagrams:
i. Indicate stable fixed points with a solid line and unstable fixed points with a dashed line
ii. Show your calculations for how you determined the fixed points
iii. Explain how you determined stability and/or show your calculations
iv. Clearly indicate any bifurcation(s) (if they exist)
v. Clearly identify and label bifurcation(s) (saddle-node, transcritical, pitchfork, if they exist)
1. Consider the equation
x˙ = ax − x(1 − x)
2
(1)
a) Draw a bifurcation diagram for this equation as a varies.
b) In the neighborhood of all bifurcation(s), if they exist, transform Eq. 1 into the normal form.
2. Consider the following bifurcation diagram, showing fixed points (x
∗
) as a function of parameter, p, for
an equation of the form ˙x = f(x). Note that there are three different parameter values indicated, p1, p2,
and p3.
x*
p
p p p
1 2 3
Figure 1: Bifurcation diagram; stable fixed points are drawn as a solid line, unstable as a dashed line.
a) Label and identify all bifurcations in the figure.
b) Draw phase portraits consistent with the bifurcation curve at each of the parameter values, p1, p2 and
p3. (You should draw one plot for each parameter value, a total of three phase portraits).
c) Can this system exhibit hysteresis (according to the definition used in class)?
d) How would your answer to c) change if the stability in Fig. 1 were flipped (i.e., each stable fixed point
were unstable, and each unstable fixed point were stable)? Explain.
3
3. Here is the equation for an imperfect transcritical bifurcation
x˙ = ax + x
2 + ε (2)
a) Sketch bifurcation diagrams for 1) ε = 0, 2) ε > 0, 3) ε < 0. You may assume ε is small in the latter
two cases. On each of the three bifurcation diagrams, indicate stable fixed points with a solid line, unstable
fixed points with a dashed line, and label all bifurcations.
b) Sketch a stability diagram (Recall that a stability diagram will have a and ε as axes, and will indicated
regions where there are differing numbers of fixed points).
4. Turn in a completed version of worksheet 3, which you worked on during class on September 15.
4