7.4.2 Subcritical Saddle-Centre Bifurcation
The system of differential equations corresponding to this bifurcation has a form
similar to Eq. (7.65), except for the sign of x
2 in the second equation:
_
x ¼ Ày
_
y ¼ l þ x
2
&
ð7:66Þ
The transposed Jacobian of (7.66) is
~
= ab ¼
0
À1
þ 2x 0
Accordingly, there are two equilibrium points for l 0, respectively in
ð
ffiffiffiffiffi ffi
jlj
p ; 0Þ and ðÀ
ffiffiffiffiffi ffi
jlj
p ; 0Þ (coalescing at a degenerate critical point for l ¼ 0). The
corresponding matrices are
~
= ab ð
ffiffiffiffiffi ffi
jlj
p ; 0Þ ¼
0
À1
2
ffiffiffiffiffi ffi
jlj
p
0
;
~
= ab ðÀ
ffiffiffiffiffi ffi
jlj
p ; 0Þ ¼
0
À1
À2
ffiffiffiffiffi ffi
jlj
p
0
Hence
• the point ð
ffiffiffiffiffi ffi
jlj
p ; 0Þ, with corresponding pure imaginary eigenvalues k 1;2 ¼
Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
À2
ffiffiffiffiffi ffi
jlj
p
q
is a centre,
• the point ðÀ
ffiffiffiffiffi ffi
jlj
p ; 0Þ, with corresponding real eigenvalues of opposite sign,
k 1;2 ¼ Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ffiffiffiffiffi ffi
jlj
p
q
, is a saddle.
The scenario is inverted with respect to that of the supercritical bifurcation of
Sect. 7.4.1, see the right column of Fig. 7.4.
7.4.3 Transcritical Saddle-Centre Bifurcation
(Supercritical)
This bifurcation corresponds to the system
_
x ¼ Ày
_
y ¼ lx À x
2
&
ð7:67Þ
7 Topology of Quantum Mechanical Current Density …
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