## Transcribed Text

Watch video on Bifurcation Basics. Here we review a) what a (local) bifurcation is, b) the three basic
bifurcations (saddle-node, transcritical, pitchfork) and c) discuss how to draw bifurcation diagrams in
model problems.
Watch video on Preservation of Equilibria. Here we use the Implicit Function Theorem to show that
if a fixed point has a Jacobian which is invertible then that fixed point will persists (locally) as a
parameter is varied. Therefore, bifurcations where the number of equilibria change must occur at
parameter values for which the equilbiria has a zero eigenvalue.
i Problem #1 Sketch a bifurcation diagram for the following differential equation.
- ,
a
+
x
with h > 0 the bifuration parameter and 0 < a < 1. Focus only on non-negative values of h and x.
Identify all bifurcation points on the diagram and classify them (you don't need explicit formulas for
the points).
To study bifurcations we have several approaches. The book first focuses on unfolding bifurcations.
This basically is to say that we start with a system that has a zero eigenvalue and add parameters
to the system to see how many different ways we could alter the dynamics by adding parameters.
You typically start with a large number of parameters and then show that only a few (called the co-
dimension of the bifurcation) are important. The second focus is on using center manifold techniques.
Since we are running short on time and we already have this machinery in place that is what I will
focus on this week. It is also typically what one uses in applications, as we have the parameters fixed
already and we want to know how the system changes as the parameters are varied.
For co-dimension one bifurcations (saddle-node, transcritical, pitchfork), it is generically the case that
one should expect saddle-node bifurcations. The reason is that we take the right hand side f (x,) and
Taylor expand it near the equilibrium point to get
f(x, = f(0,0) + + + + + h.o.t.
Since the origin is an equilibrium point f(0,0) = 0. Since there is a bifurcation we have that fx(0,0) =
0. Thus, we are left with,
+ + +
The first two terms in this expansion will be non-zero in general, which means that H x cx² will be the
curve of equilibrium nearby. However, even though this reasoning says that we should almost always
observe saddle-node bifurcations we often see transcritical and pitchfork bifurcations in applications.
For transcritical bifurcations, this follows from the fact that often x = 0 is an equilibrium point for
all parameters values in problems of interest. Thus, all terms involving only H will not appear in the
bifurcation equation and we can factor out an x. This leads to a leading order description consistent
with the transcritical bifurcation diagram. For pitchfork bifurcations, this follows from the existence
of a symmetry in the equation. If the symmetry is x -x, then this implies that a number of terms
must be zero; for example = 0 and fxx = 0.
Watch video on Center Manifold Approaches to Bifurcation problems.
Problem #2 Consider the following system of equations,
r = y-x =
y' = ax - y - x2
x' = - Z + xy.
When a = 1, the linearization at the origin has a zero eigenvalue with multiplicity one. Characterize
the bifurcation that occurs at this value as saddle-node, transcritical or pitchfork. Justify your answer.
One point for identifying the bifurcation correctly. One extra credit point for working out the reduced
equation on the center manifold.

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.