Trace's zeros

Numerator of the trace for the Minimal Template 

[Graphics:../Images/templator_gr_25.gif]
[Graphics:../Images/templator_gr_26.gif]
[Graphics:../Images/templator_gr_27.gif]
[Graphics:../Images/templator_gr_28.gif]

Notice the spaces after the + sign.  They stand for +1, but MATHEMATICA Ver. 3.0can not simplify the expressions!  Thus by hand we simplify the expression 

[Graphics:../Images/templator_gr_29.gif]
[Graphics:../Images/templator_gr_30.gif]
[Graphics:../Images/templator_gr_31.gif]
[Graphics:../Images/templator_gr_32.gif]
[Graphics:../Images/templator_gr_33.gif]
[Graphics:../Images/templator_gr_34.gif]

MATHEMATICA Ver 4, cannot solve the previous  equation or the following one 

[Graphics:../Images/templator_gr_35.gif]
[Graphics:../Images/templator_gr_36.gif]

Adjusting the definition by multipling by -1, c00 has the same sign as the trace 

[Graphics:../Images/templator_gr_37.gif]

Now we use the function ContourPlot to find the zero manifold of the function c00[fo,K] in parameter space 

[Graphics:../Images/templator_gr_38.gif]

[Graphics:../Images/templator_gr_39.gif]

We have to adjust the zero line for K. 

[Graphics:../Images/templator_gr_40.gif]

[Graphics:../Images/templator_gr_41.gif]

The values of the bifurcatiopn point are given by: 

[Graphics:../Images/templator_gr_42.gif]
[Graphics:../Images/templator_gr_43.gif]
[Graphics:../Images/templator_gr_44.gif]
[Graphics:../Images/templator_gr_45.gif]

Converted by Mathematica      November 14, 2000