• supercritical, or normal bifurcations, characterized by opposite sign of the terms
appearing in the polynomial expansion of f l . In such a case, the nonlinear terms
in x
2 or x
3 within this expansion have an effect opposing to the instability
generated by the term of the lower order [94, 95].
• subcritical, or inverse bifurcations, in which the nonlinear term, of the same sign
as that of lower order, has also a destabilizing effect.
7.4.1 Supercritical Saddle-Centre Bifurcation
This bifurcation can be modelled by the system of differential equations
_
x ¼ Ày
_
y ¼ l À x
2
&
ð7:65Þ
obtained by adding the first equation to the canonical one-dimensional normal form
in Table I, p. 297 of Ref. [94]. The field is characterized by the C 2t ðC s Þ 2mm
magnetic symmetry, see Sect. 7.5. The transposed Jacobian of (7.65) is
~
= ab ¼
0
À1
À2x 0
For l\0 the dynamic system has no equilibrium points, for l ! 0 there is a
couple of equilibrium points (coinciding in a critical degenerate point for l ¼ 0),
respectively in ð
ffiffiffi
l
p ; 0Þ ðÀ
ffiffiffi
l
p ; 0Þ. The corresponding matrices are
~
= ab ð
ffiffiffi
l
p ; 0Þ ¼
0
À1
À2
ffiffiffi
l
p
0
; ~
= ab ðÀ
ffiffiffi
l
p ; 0Þ ¼
0
À1
2
ffiffiffi
l
p
0
The eigenvalues are determined by solving the equation k
2
À 2x ¼ 0, with roots
k 1;2 ¼ Æ
ffiffiffiffiffi
2x
p
. Hence
• for the point ð
ffiffiffi
l
p ; 0Þ the solutions are k 1;2 ¼ Æ
ffiffiffiffiffiffiffiffiffi ffi
2
ffiffiffi
l
p
p
, i.e., k 1 [ 0 and k 2 \0:
the equilibrium point is a saddle,
• for the point ðÀ
ffiffiffi
l
p ; 0Þ the solutions are pure imaginary, k 1;2 ¼ Æi
ffiffiffiffiffiffiffiffiffi ffi
2
ffiffiffi
l
p
p
: the
equilibrium point is a centre.
The trajectories on the ðx; yÞ plane do not exhibit equilibrium points for l\0,
see top left of Fig. 7.4. For l ¼ 0, a cusp is observed in the proximity of the (0, 0)
degenerate point, at which the bifurcation takes place. For l [ 0 the typical phase
portraits of a saddle and a vortex are observed. Homoclinic orbits, or trajectories, of
the flow of this dynamical system join a saddle equilibrium point to itself.
7 Topology of Quantum Mechanical Current Density …
171
appearing in the polynomial expansion of f l . In such a case, the nonlinear terms
in x
2 or x
3 within this expansion have an effect opposing to the instability
generated by the term of the lower order [94, 95].
• subcritical, or inverse bifurcations, in which the nonlinear term, of the same sign
as that of lower order, has also a destabilizing effect.
7.4.1 Supercritical Saddle-Centre Bifurcation
This bifurcation can be modelled by the system of differential equations
_
x ¼ Ày
_
y ¼ l À x
2
&
ð7:65Þ
obtained by adding the first equation to the canonical one-dimensional normal form
in Table I, p. 297 of Ref. [94]. The field is characterized by the C 2t ðC s Þ 2mm
magnetic symmetry, see Sect. 7.5. The transposed Jacobian of (7.65) is
~
= ab ¼
0
À1
À2x 0
For l\0 the dynamic system has no equilibrium points, for l ! 0 there is a
couple of equilibrium points (coinciding in a critical degenerate point for l ¼ 0),
respectively in ð
ffiffiffi
l
p ; 0Þ ðÀ
ffiffiffi
l
p ; 0Þ. The corresponding matrices are
~
= ab ð
ffiffiffi
l
p ; 0Þ ¼
0
À1
À2
ffiffiffi
l
p
0
; ~
= ab ðÀ
ffiffiffi
l
p ; 0Þ ¼
0
À1
2
ffiffiffi
l
p
0
The eigenvalues are determined by solving the equation k
2
À 2x ¼ 0, with roots
k 1;2 ¼ Æ
ffiffiffiffiffi
2x
p
. Hence
• for the point ð
ffiffiffi
l
p ; 0Þ the solutions are k 1;2 ¼ Æ
ffiffiffiffiffiffiffiffiffi ffi
2
ffiffiffi
l
p
p
, i.e., k 1 [ 0 and k 2 \0:
the equilibrium point is a saddle,
• for the point ðÀ
ffiffiffi
l
p ; 0Þ the solutions are pure imaginary, k 1;2 ¼ Æi
ffiffiffiffiffiffiffiffiffi ffi
2
ffiffiffi
l
p
p
: the
equilibrium point is a centre.
The trajectories on the ðx; yÞ plane do not exhibit equilibrium points for l\0,
see top left of Fig. 7.4. For l ¼ 0, a cusp is observed in the proximity of the (0, 0)
degenerate point, at which the bifurcation takes place. For l [ 0 the typical phase
portraits of a saddle and a vortex are observed. Homoclinic orbits, or trajectories, of
the flow of this dynamical system join a saddle equilibrium point to itself.
7 Topology of Quantum Mechanical Current Density …
171