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

80 Bifurcation Diagrams and Analysis
NODf 1 -USB
f1 ∈ (0;3.4 × 10 −2 ) (3.4 × 10 −2 ; 2.15) (2.15; 2.57) (2.57; 2.8) (2.8; 3.45)
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 S SS USS SP
s )
Dim(W
5 5 5 3 3
u ) 0 0 0 2 2
Table 8.1 Gives an overview of the eigenvalues, and the behavior of the fixed points on the
NODf1-USB. By C(Re(λ)) we mean the sign of the real part of the complex eigenvalues. “S”
stands for stable, “SS” stands for stable spiral, “USS” stands for unstable spiral, and “SP”
stands for saddle-point. 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 8.5 In the upper figure we see (real) values of the eigenvalues of the NODf1-LUB.
At f1 ≈ 1.165 × 10 −5 two of the eigenvalues become complex, and stay complex up until
f1 = 3.45 × 10 −5 where the NODf1-LUB coalesces with the NODf1-USB (that consists of
saddle-points at this point); cf. figure 8.1. The values of the complex part of these two
eigenvalues are plotted in the lower figure. The complex eigenvalues retain negative real parts
through the entire interval.