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

8.3 Using f2 as the bifurcation parameter 91
NODf 2 -USB
f2 ∈ (0;1.4 × 10 −1 ) (1.4 × 10 −1 ; 2.15) (2.15; 8.544) (8.544; 9.184) (9.184; 9.2205)
Nr. λ ∈ R +
0 0 0 0 2
Nr. λ ∈ R −
5 3 1 1 1
Nr. λ ∈ C(Re(λ)) 0 2(-) 4(-) 2(-),2(+) 2(-)
Fixed pt.
Dim(W
S SS SS SP SP
s )
Dim(W
5 5 5 3 3
u ) 0 0 0 2 2
Table 8.3 Gives an overview of the eigenvalues, and the behavior of the fixed points on the
NODf2-USB. “S” stands for stable, “SS” stands for stable spiral, and “SP” stands for saddlepoint.
Dim(W s ) is the dimension of the stable manifold, and Dim(W u ) is the dimension of
the unstable manifold. All intervals are in the order of 10 −5 , and all values are approximate.
Figure 8.14 The real (upper subplot) and imaginary (lower subplot) parts of the eigenvalues
of the fixed point along the NODf2-LUB.
features similar to the NODf1-LUB. The behavior of the eigenvalues is less erratic for
the eigenvalues along the NODf2-LUB, than along the NODf2-USB, as we see when we
compare table 8.4 to table 8.3. The bifurcation diagram for f2 with Balb/c parameters
is given in figure 8.15. We clearly see that for the value of f2 that Balb/c-mice have
inflammation is non-persistent, in agreement with what we would expect. The way the
concentration of active macrophages decreases along the Balb/cf2-USB makes it look
similar to figure 8.12, but we see that the shift between a stable and an unstable USB
comes at a much lower f2-value when we use the Balb/c-value for f1.
As we did in the case of the NOD-mice we have also searched for additional fixed