nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

7 A Brief Introduction to Bifurcation
Analysis and Numerical Methods
A natural question to ask one self when working with models of physiological phenomena
is how sensitive the model is to changes in parameter values. This is interesting from
a purely mathematical point of view, because an analysis of how the stability depends
on one or several parameter values is a bifurcation analysis. A bifurcation analysis also
provides insight into the behavior of the mathematical model, which of course should
reflect the biology it models. Thus the bifurcation analysis is also interesting from the
modelling/biological perspective.
In this chapter we provide the reader with an introduction to (local) bifurcations in
general, and proceed on to choose bifurcation parameters and appertaining intervals.
After this we briefly describe how we have found, and determined the stability of, the
eigenvalues. All of these pieces are combined into bifurcation diagrams in chapter 8
where we also perform an analysis of the DuCa model.
But first let us introduce the concept of bifurcation, and shed some light on its significance
in modelling of biological system in general and the DuCa model in particular.
7.1 Generic Bifurcations
Assume that we have a one-parameter system of differential equations, f(x, r), where
r ∈ R is the parameter, and f(x) := ˙x ∈ R n , where the dot denotes d/dt. We want to
find out what happens to the behavior for different values of r.
Case 1 is: different values of r shifts the location of the equilibria, but does not change
their stability or create new ones. Like changing p ∈ R + \0 in g(x) = x 2 − p will change
the roots for g, but not the number of solutions. In this case the flows corresponding
to different r’s are topologically equivalent.
Case 2 is: different values of r does induce a change in the flow, i.e. change in stability
of, or creation of, equilibria. In this case there is no longer a topological equivalence
between the flows that arise for the different r values, and we say that a bifurcation
has occurred. It may be easier to think of a bifurcation as a sudden change in the
qualitative behavior of the solutions to a system of differential equations. 1
Thus a bifurcation analysis is the investigation of how the qualitative behavior of a
system depends on a parameter.
When a bifurcation occurs upon shifting only one parameter, the bifurcation is sometimes
called a codimension one bifurcation. Similarly if two parameters must be varied
1 Here system is used in the general sense, so one differential equation alone can be a system.
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