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

8.2 Using f1 as the bifurcation parameter 87
should surround the stable spirals that arise at f1 ≈ 2.15 × 10 −5 , but we have not been
able to verify this through simulations. When we discovered the UUB we realized that
USB
HRS
f_1 is increased
LUB
Figure 8.10 The flow of Ma represented on a circle with two stable and two unstable fixed
points. The fixed point without a label is the UUB. As f1 increases the NODf1-USB and the
UUB draw closer together (depicted on the right circle).
the flow in terms of Ma, can be envisioned as the flow on a circle with four fixed points.
Two stable and two unstable. We have depicted this in figure 8.10. From the left circle
to the right f1 is increased. Increasing f1 further would make the NODf1-USB and the
UUB meet, to become a saddle-point, that is repelling in the direction of the HRS, and
attracting from the side of the NODf1 -LUB.
A final thing that should be mentioned about figure 8.1 is that it tells us that the
system is irreversible with respect to changes in f1. In other words hysteresis occurs.
This happens as f1 exceeds a threshold value that is identical to the Hopf bifurcation
value. This suggests that we have a subcritical Hopf bifurcation (Strogatz, 2000, p.252).
However, if we want to make sure that this is indeed the case we must bring our system
to normal-form and find the adhering coefficients. This is a substantial undertaking for
a 5-dimensional system which we have not ventured, though it is not impossible; cf.
e.g. Yu (1997).
The Balb/c bifurcation diagram
Now let us turn to the bifurcation diagram in which f2 = 5 × 10 −5 . This bifurcation
diagram differs from figure 8.1 (as well as those to come) by having two additional
unstable curves of fixed points, illustrated by the dotted line (black) and the dasheddotted
line (magenta). Figures B.8 and B.9 in appendix B.2 provide the eigenvalue
USB
HRS
LUB