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

8.2 Using f1 as the bifurcation parameter 83
that is left, i.e. the HRS. We could also look at the equation that describes the rate
of change in the concentration of activated macrophages; cf. equation 5.8. There we
see the damping term given by −e2Ma(M + Ma) which becomes dominating when Ma
becomes high – this prevents the concentration from diverging.
As mentioned there is a theoretical possibility that other branches of fixed points
could exist in-between the NODf1-USB and the NODf1-LUB though the flow seems to
speak against it. Even so, we have done many simulations with different initial values
for Ma, and different values of f1 in the area that is bordered by the unstable NODf1- USB and the NODf1-LUB, and every single one of them exhibit a behavior that is in
qualitative agreement with figure 8.6, so we find it to be unlikely that there are any
additional fixed points in this area.
The fact that we find it unlikely that any stable fixed points exist in the area mentioned
above, and the behavior seen in figure 8.6 led us to wonder what else could be going
under in the area under scrutiny. We suspect that so-called heteroclinic orbits – that
is an orbit, that connects one saddle-point with another saddle-point – inhabits this
domain. Heteroclinic orbits arise when the stable manifold of a saddle-point intersects
the unstable manifold of another saddle-point (Guckenheimer and Holmes, 2002, p.22).
The behavior of the concentration of active macrophages shown in figure 8.6 corroborates
the existence of such orbits. Let us assume that there are indeed heteroclinic
orbits connecting the fixed points on the NODf1-LUB to those on the unstable part
of the NODf1-USB, and let us further assume that we could start our flow directly
on the unstable manifold of the fixed points on the NODf1-LUB then for t → ∞ the
flow would approach the unstable fixed points on the NODf1-USB. This is because the
unstable manifold is invariant, so the flow cannot leave it when it has started there
We have tried to illustrate this in figure 8.7. The figure represents a slice of the 5dimensional
phase space. 10 The stable fixed point, HRS, is represented by the full dot
while the saddle-points, NODf1-LUB and NODf1-USB, are represented as black circles
with white interior. There is a heteroclinic orbit from the NODf1-LUB to the NODf1-
USB, along with some representative manifolds, and arrows designating the flow. On
the orbit is a point that represents an optimal starting condition; i.e the point represents
a set of initial conditions that would let us establish the existence of heteroclinic orbits
graphically. But since we are not able to determine the unstable manifold analytically
it is next to impossible to start precisely on the unstable manifold of the NODf1-LUB,
therefore the flow initially tends to the unstable fixed points on the NODf1-USB, and
then becomes repelled from them11 the nearer it comes, and is led down to the HRS
along the unstable manifold of the NODf1-USB, which is depicted by the dashed line.
We have made the line dashed to illustrate that its path may not be in the 2-dimensional
plane.
The existence of heteroclinic orbits would be very interesting for one reason in particular:
the existence of heteroclinic orbits opens up for the possibility of a global bifurcation
(Guckenheimer and Holmes, 2002, p.290). Furthermore heteroclinic orbits would act as
a constant reminder that the flow is not 1-dimensional
10 Please be aware that this figure is for illustrative purposes only, and does not represent an exact
slice of the phase space! I.e. the figure only gives a qualitative gist of how the flow might behave.
11 To avoid confusion we would like to restate, that for each f1 there is only one fixed point on the
USB!