Exercise 2
1. For the systems below sketch all qualitatively different
vector fields that occur as r is varied (that is, the vector
field on the x-axis for at least 3 judiciously chosen values).
Show that a bifurcation occurs at a critical value of r, to be
determined. Sketch the bifurcation diagram of fixed points
x* versus r and determine the type of bifurcation.
2
2
a) ẋ = 1 + rx + x (2p) b)
*
()
c) "̇ = 5 − &'
(2p)
ẋ = rx + x
(2p)
(Turn page.)
2. Consider the system "̇ = $" − sin ".
a) For the case $ = 0, find and classify all the fixed points
and sketch the vector fields. (1p)
b) Show that when $ > 1, there is only one fixed point. What
kind of fixed point is it? (2p)
c) As $ decreases from ∞ towards 0, find and classify the first
bifurcation that occurs. (3p)
(Bifurcations coming after the first one are all saddle-node
bifurcations, which you may check, if you are interested.
This is not part of the exercise)
(Turn page.)
3. Imperfect bifurcation; a perturbation to the supercritical
pitchfork. Consider the system ẋ = rx + ax2 x3 ,
where 1 < a < 1. When a = 0, we have the normal
form for the supercritical pitchfork. Study the effects of the
new parameter a. For each a, there is a bifurcation diagram
of x* vs. r. As a varies, these bifurcation diagrams can
undergo qualitative changes. Sketch all the qualitatively
different bifurcation diagrams that can be obtained by
varying a. (Hint: Diagrams for a = -1, 0, and 1 will suffice.)
(6p)