Section 3 exercises: Bifurcations
Section 3 exercises: Bifurcations
Complete these as you are going through part 3. Try and do questions from this regularly.
Many of these questions come from or are inspired by the questions in Strogatz  Nonlinear dynamics and Chaos.
1
.Questions on saddlenode bifurcations:i) Sketch the different vector field types that appear when you vary in the following differential equations. ii) Show that there is a saddlenode bifurcation (though there may more fixed points than just the two), and find the critical point of the bifurcation iii) and plot the bifurcation diagram. For instance, if you were given the equation =+r, you would write:i) ii) There is a critical point at where the stable and unstable fixed points come together after which there is no fixed point. This is the definition of a saddlenode bifurcation.iii) The bifurcation diagram looks like:
r
x
2
x
r=
r
c
1
.2
.3
.4
.2
.For the example of the equation =+r, plot the potential for different values of and see what the behaviour looks like there in that picture.
x
2
x
r
3
.We mentioned Normal Forms in the notes. The point is that close to the bifurcation point, ALL saddlenode bifurcations look like =r+ or =r. For instance, for the equation =rx,there is a critical point at =1 at . We can expand the right hand side close to it gives us: =rx1x+...=(r1)+.... Up to a scaling in and a shift in , this is the same as =r. Go through the above equations from exercise 1 and show that when you expand around the bifurcation point, they all take on the same normal form. If you need to expand a function, you can use Wolfram Alpha, for instance, for which you can then plug into the right hand side of =rcosh(x) and see what happens for
x
2
x
x
2
x
x
x
e
r
x=0
x=0and
x
2
x
2!
2
x
2
x
r
x
2
x
cosh(x):
x
xcloseto0.
4
.Now do the same thing as questions 1 and 2 for the following transcritical bifurcations:
1
.
x
2
x
+
2
x

2
x
2
.3
.4
.5
.5
.Now do the same thing as questions 1 and 2 for the following pitchfork bifurcations. This time also classify them as subcritical or supercritical.
1
.
x
3
x
2
.3
.
x
3
x
4
.
x
rx
1+
2
x
6
.For these flows on the line, find the value of r for which a bifurcation occurs, classify the bifurcation type, and draw the bifurcation diagram
1
.
x
2
x
2
.
x
x
1+x
3
.
x

2
x
e
4
.
x
x
1+
2
x
5
.
x
6
.
x
3
x
1+
2
x