Here the bifurcations concern singularities with co-dimension greater than 1. These are only unfolded by local families that depend on more than 1 parameter.
Given
a smooth vector field with a cusp singularity at the
origin. Then
but
=
=
and
has normal form (with
and
).
The
local
family
is
a versal unfolding of the cusp singularity above. (Note that the cusp
singularity requires at least two parameters to completely unfold it).
Considering this local family
,
under some conditions (not shown here) in particular about parameters, it is
possible to generate three bifurcation types: (1) Saddle
node, (2) Hopf bifurcation, (3)
Saddle connection.
Generalized Hopf bifurcations. (Takens 1973b, Guckenheimer
& Holmes 1983, Arrowsmith & Place 1990). Here, we describe the
generalized Hopf singularity of type
.
The vector field in the previous co-dimension 1 Hopf bifurcation was obtained
by posing
in the normal
form
If
a degeneracy is introduced by posing
=
=
=
=
with
,
then the system above is said to have a generalized Hopf
singularity of type
at
=
.
(Note that a co-dimension
Hopf singularity is said to be of type
).
The framework of the generalized Hopf singularity highlights
the relationship between the degeneracy of a singularity and the number of
parameters contained in its versal unfolding.
If a Hopf singularity of type
occurs, it is possible to reduce the system above to a simpler system.
Given
that
can also be
written:
with
=
.
Because
>
,
we have
>
for all
enough close to the origin. Thus there exists a neighborhood of
=
on which the system is topologically equivalent to the (simpler)
vector
field:
While
preserving the topological equivalence, we can replace
by
in the vector field above. Two cases are then possible according to a
singularity of the type:
or
.
A versal unfolding for the vector field above can be obtained through the
following Takens theorem (1973b):
The versal unfolding of a generalized Hopf singularity of type
given in the simpler vector field above is written (where
,
=
):
According to the sign of
,
the unfoldings of in this theorem gives two cases
and
.
Here, we only describe the first case
,
and the latter case can be related to the other by a time reversal (so the
stabilities of fixed points and limit cycles are inverted). The formula in
this theorem, through the polar form
(
with here
=
),
gives for
the bifurcational behavior by using the equivalent equation:
The behavior is independent of the terms of order
(as described for the co-dimension 1 Hopf bifurcation), so it suffices to
consider
.
This equation in particular provides limit cycles. (1) For
=
,
by posing
=
,
we obtain
,
then a limit cycle occurs whose radius is
only when
<
.
(2) For
=
,
by posing
=
,
we obtain
and non-trivial zeros result from
.
Then we obtain:
If
>
and
>
(or
<
):
no limit cycle but a repelling spiral.
If
<
and
>
:
two limit cycles.
If
<
:
one limit cycle. (see
Fig.
below).
This
figure shows the bifurcation diagram for the type-(2,+) Hopf bifurcation. It
shows that a supercritical (subcritical) Hopf bifurcation occurs when
increases through zero with
<
(subcritical with
>
),
and a double limit cycle bifurcation occurs on the semi-parabola. If
decreases through the semi-parabola, a non-hyperbolic limit cycle occurs and
then divides into two hyperbolic cycles.