Consider an
autonomous system
of ordinary differential equations
depending on a parameter
where
is a smooth function. Suppose that at
,
the system
(a1)
has an
equilibrium position
with a simple pair of purely imaginary eigenvalues
,
,
of its Jacobian matrix
.
Then, generically, a unique
limit cycle
bifurcates from the equilibrium while it changes stability, as
passes through
.
This phenomenon is called the
Hopf
(or
Andronov–Hopf)
bifurcation
[a1],
[a7],
[a2],
[a3].
It is characterized by a single bifurcation
condition
(has codimension one) and appears generically in one-parameter families.
First, consider a smooth planar system
that has for all sufficiently small
the equilibrium
with eigenvalues
,
,
.
If the following non-degeneracy (genericity) conditions hold:
1)
,
where
is the first Lyapunov coefficient (see below);
2)
,
then
(a2)
is
locally topologically equivalent (cf.
Equivalence of dynamical systems)
near the origin to the
normal form
where
,
,
(see
[a2],
[a6]).
Figure: h110260a
Supercritical Hopf bifurcation on the plane
Consider the case
.
Then the system
(a3)
has an equilibrium at the origin
,
which is stable for
(weakly at
)
and unstable for
.
Moreover, there is a unique and stable circular limit cycle that exists for
and has radius
(see
Fig.a1).
This is a
supercritical Hopf bifurcation.
Figure: h110260b
Subcritical Hopf bifurcation on the plane
For
,
the origin in
(a3)
is stable for
and unstable for
(weakly at
),
while a unique and unstable limit cycle exists for
(see
Fig.a2).
This is a
subcritical Hopf bifurcation.
In the
-dimensional
case, the Jacobian matrix
evaluated at the equilibrium
has a simple pair of purely imaginary eigenvalues
,
,
as well as
eigenvalues with
,
and
eigenvalues with
(
).
According to the
centre manifold theorem
(cf.
Centre manifold)
[a5],
[a7],
[a2],
there is an invariant two-dimensional
centre manifold
near the origin, the restriction of
(a1)
to which has the form
(a2).
Moreover,
[a2],
under the non-degeneracy conditions
1) and 2), the system
(a1)
is
locally topologically equivalent (cf.
Equivalence of dynamical systems)
near the origin to the suspension of the normal form
(a3)
by the
standard saddle:
where
,
,
,
.
Fig.a3
shows the phase portraits of the system
(a4)
in the three-dimensional
case, when
,
,
,
and
.
Figure: h110260c
Hopf bifurcation in
The first Lyapunov coefficient
can be computed (to within a scalar multiple) in terms of the right-hand
side of
(a1)
at
.
Represent
as
where
and
are multilinear functions (cf. also
Multilinear mapping).
In coordinates one has
where
.
Let
be a complex eigenvector of
corresponding to the eigenvalue
:
Introduce also the
adjoint eigenvector
:
where
is the
inner product
in
.
Then (see, for example,
[a6])
There is an analogue of the Hopf bifurcation for discrete-time dynamical
systems, called the
Neimark–Sacker bifurcation
[a7],
[a4],
[a2],
[a8],
[a6].
Under certain non-degeneracy conditions, it generates a closed invariant
curve around a fixed point which changes stability due to the transition of a pair
of its complex eigenvalues through the unit circle.