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

86 Bifurcation Diagrams and Analysis
Flow tends to the
USB for all initial
values in this area
Line of unstable fixed points. The
flow tends to the HRS for all initial
values above this.
Flow tends to the
USB for all initial
values in this area
Hopf bifurcation pt.
Flow tends to
the HRS for all
initial values in
this area
Figure 8.9 Shows a modified version of figure 8.1, where we have added a curve that
approximates the UUB. We have also added legends to facilitate the overview.
a curve that approximates the UUB. 12 We have also added text to explain how the
flow behaves for different initial Ma-concentrations. We see that the UUB and the
NODf1-USB ultimately coincide. This appears in the eigenvalues of the NODf1-USB approximately with the advent of positive eigenvalues. We dare not state that it is
exactly when the eigenvalues become positive. If the UUB is in fact a line of unstable
fixed points then the collision between the NODf1-USB and the UUB is basically a
(presumably degenerate) sub-critical pitchfork bifurcation – an unstable and a stable
fixed point coalesce to produce one unstable fixed point.
We have speculated if the UUB constitutes a section of the unstable limit cycle that
12 Notice that though we used matlab to approximate the demarcation values, the UUB-curve shown
in figure 8.9 is not made with matlab it was drawn using Adobe Illustrator CS3.