The transposed Jacobian of (7.67) is
~
= ab ¼
0
À1
l À 2x 0
There are two equilibrium points, ð0; 0Þ and ðl; 0Þ, with corresponding
Jacobians
~
= ab ð0; 0Þ ¼
0 À1
l 0
;
~
= ab ðl; 0Þ ¼
0 À1
Àl 0
The eigenvalues are k 1;2 ¼ Æ
ffiffiffiffiffiffi ffi
Àl
p
and k 1;2 ¼ Æ
ffiffiffi
l
p . Hence,
• the (0, 0) point is respectively a saddle for l\0 and a centre for l [ 0,
• the ðl; 0Þ point is a centre for l\0 and a saddle for l [ 0.
Therefore, on varying the value of the parameter from negative to positive within
the interval À1 l 1, an inversion of the bifurcation scenario with exchange of
stability [94] is observed. The bifurcation occurs at the degenerate critical point for
l ¼ 0. The trajectories are displayed on the left of Fig. 7.5. The flow is characterized by the C 2v ðC s Þ 2mm magnetic symmetry, see Sect. 7.5.
7.4.4 Transcritical Saddle-Centre Bifurcation (Subcritical)
The system of differential equations is
_
x ¼ Ày
_
y ¼ lx þ x
2
&
ð7:68Þ
The transposed Jacobian has the form
~
= ¼
0
À1
l þ 2x 0
The equilibrium points are (0, 0) and ðÀl; 0Þ. Hence,
• the eigenvalues at (0, 0) are k 1;2 ¼ Æ
ffiffiffiffiffiffi ffi
Àl
p : a saddle and a centre are observed
for l\0 and l [ 0, respectivley,
• the eigenvalues at ðÀl; 0Þ are k 1;2 ¼ Æ
ffiffiffi
l
p : a saddle and a centre are observed
for l [ 0 and l\0, respectively.
The trajectories are displayed on the right column of Fig. 7.5. Also in this case,
on varying the parameter within the interval À1 l 1 there is an inversion of the
bifurcation scenario with exchange of stability.
174
P. Lazzeretti