Chemistry Reference
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where Trand Det are, respectively, the Trace and the Determinant of the Jacobian
matrix (5.107). The first condition is always satisfied, hence the stability conditions,
after some algebraic manipulations, reduce to
A.c
1
C
c
2
C
A/
a
1
a
2
B
2
6.2c
1
c
2
C
A/
a
1
B
6.2c
2
c
1
C
A/
a
2
B
C
36 > 0
(5.109)
and
A.c
1
C
c
2
C
A/
a
1
a
2
B
2
3.2c
1
c
2
C
A/
a
1
B
3.2c
2
c
1
C
A/
a
2
B
<0: (5.110)
These two inequalities define a
region of stability
(we may also call it a
learning
region
) in the space of the parameters. Moreover, the conditions (5.109) and (5.110)
taken as equalities, define bifurcation hypersurfaces. This means that when one or
more parameters are varied so that the equilibrium " becomes unstable, if (1) the
stability loss is due to a change of sign of the left hand side of (5.109), then a flip (or
period doubling) bifurcation occurs, and if (2) the stability loss is due to a change
of sign of the left hand side of (5.110), then a Neimark-Hopf bifurcation occurs. It
is useful to represent the learning region by projecting it into the two-dimensional
parameter plane .a
1
=B;a
2
=B/, where the bifurcation curves that bound the region
of stability are equilateral hyperbolas (see Fig. 5.6, where F denotes the positive
branch of the hyperbola at which the flip bifurcation occurs, H denotes the positive
branch of the hyperbola at which the Neimark-Hopf bifurcation occurs, and the
shaded area represents the learning region). If
c
1
=2 < c
2
<2c
1
;
(5.111)
then the two hyperbolas do not intersect, and the learning region is bounded only by
the flip bifurcation curve (Fig. 5.6a), whereas if
2c
2
<c
1
<2c
2
C
A or 2c
1
<c
2
<2c
1
C
A
(5.112)
then the two hyperbolas intersect in the positive orthant of the plane .a
1
=B;a
2
=B/,
so that the learning region is bounded by an arc of the Neimark-Hopf bifurcation
curve and by two arcs of the flip bifurcation curve
1
(Fig. 5.6b).
If the parameters a
1
=B and/or a
2
=B are varied, so that they cross the bound-
ary of the stability region along the portion of curve F , then the equilibrium point
changes from a stable node to a saddle point via a supercritical flip bifurcation.
2
1
For c
1
D 2c
2
the curve F degenerates into the pair of straight lines a
1
=B D 6=.3c
2
C A/ and
a
2
=B D 6=B.Forc
2
D 2c
1
the curve F degenerates into the pair of straight lines a
1
B D 6=A
and a
2
=B
A/.
2
A rigorous proof of the supercritical nature of the flip bifurcation requires a center manifold
reduction and the evaluation of higher order derivatives, up to the third order (see for example
Guckenheimer and Holmes (1983)). This is a rather tedious calculation for a two-dimensional map,
D
6=.3c
1
C
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