multiple time scale dynamics with two fast variables and one slow ...

is negative for all values h∈(0,φ( √ 91
15 )] where x1= √ 91
15 is the second fold point
of ¯C0 besides x1= 0. In our case we have
R(h)=
xm(h)
xl(h)
( √ 91
5 x1− 3x2 1 )2
x− √ 91
10 x2
1
+ x3
1
with xl(h)∈[− √ 91
30 , 0) and xm(h)∈(0, √ 91
]. Figure 3.4 shows a numerical plot of
15
the function R(h) for the relevant values of h which confirms the required result.
0.02
0.04
0.06
0.08
0.10
R
dx
0.02 0.04 0.06 0.08 0.10 0.12
Figure 3.4: Plot of the function R(h) for h∈(0,φ( √ 91
15 )].
Remark: We have computed an explicit algebraic expression for R ′ (h) with a
computer algebra system. This expression yields R ′ (h)