7.4.5 Supercritical Pitchfork Bifurcation
This bifurcation pattern is more complex than the previous ones. In fact, on varying
the parameter, we pass from one equilibrium point for l\0 to three equilibrium
points for l [ 0. The autonomous system is
_
x ¼ Ày
_
y ¼ lx À x
3
&
ð7:69Þ
The corresponding transposed Jacobian is
~
= ¼
0
À1
l À 3x
2
0
For l\0 there is only the (0, 0) equilibrium point, for 0\l\ þ 1 also the
solutions ðÆ
ffiffiffi
l
p ; 0Þ emerge. Hence the phase portraits are
• a saddle for l\0 and a centre for l [ 0 at point (0, 0);
• two saddles at points ðÆ
ffiffiffi
l
p ; 0Þ (only for l [ 0).
Within this scenario the saddle flow bifurcates into a centre and two adjacent
saddles. This happens at the critical degenerate point corresponding to l ¼ 0, at
which the unique solution of the differential system starts splitting into three
solutions. The phase portrait is neither a saddle nor a centre. The orbits are shown in
the left column of Fig. 7.6.
The Gomes theorem [56] is fulfilled. In fact, using the index +1 (−1) for a centre
(saddle), Eq. (7.59) is satisfied passing from top to bottom of the left column of
Fig. 7.6: the index value −1 is conserved.
7.4.6 Subcritical Pitchfork Bifurcation
This is analogous to the previous one, except for the sign of the nonlinear term,
_
x ¼ Ày
_
y ¼ lx þ x
3
&
ð7:70Þ
The transposed Jacobian is
~
= ¼
0
À1
l þ 3x
2
0
Hence,
• the root (0, 0) corresponds to a centre for l [ 0 and a saddle for l\0,
• the roots ðÆ
ffiffiffiffiffi ffi
jlj
p ; 0Þ correspond to centres for l\0.
176
P. Lazzeretti