Generalized Hopf (Bautin) bifurcation in the planar system
<math> \dot{r}=r(\beta_1 + \beta_2 r^2 - r^4),\ \dot{\varphi}=1 \ .</math> The vertical axis corresponds to the Andronov-Hopf
bifurcation (supercritical at <math> H_{-}</math> and subcritical at <math> H_{+}</math>); the curve <math> LPC </math>
corresponds to the saddle-node bifurcation of periodic orbits.
The Bautin bifurcation is a bifurcation of an equilibrium in
a two-parameter family of autonomous ODEs at which the critical equilibrium
has a pair of purely imaginary eigenvalues and the first Lyapunov coefficient for
the Andronov-Hopf bifucation vanishes.
This phenomenon is also called the generalized Hopf (GH) bifurcation.
The bifurcation point separates branches of sub- and supercritical
Andronov-Hopf bifurcations in the parameter plain. For nearby parameter values, the system has
two limit cycles which collide and disappear via a
saddle-node bifurcation of periodic orbits.
Consider an autonomous system of ordinary differential equations (ODEs)
- <math>\label{ode1}
\dot{x}=f(x,\alpha),\ \ \ x \in {\mathbb R}^n
</math>
depending on two parameters <math>\alpha \in {\mathbb R}^2\ ,</math> where <math>f</math>
is smooth.
- Suppose that for all sufficiently small <math>\|\alpha\|</math> the system has an equilibrium <math>x=0\ .</math>
- Further assume that its Jacobian matrix <math>A(\alpha)=f_x(0,\alpha)</math> has one pair of complex eigenvalues
<math>\lambda_{1,2}(\alpha)=\mu(\alpha) \pm i\omega(\alpha)</math>
such that <math>\mu(0)=0</math> and <math>\omega(0)=\omega_0>0\ .</math>
This bifurcation is
characterized by two bifurcation conditions <math>{\rm Re}\ \lambda_{1,2}=0</math> and
<math>l_1(0) = 0</math> (has codimension two)
and appears generically in two-parameter families of smooth ODEs.
Generically, <math>\alpha=0</math> is the origin
in the parameter plane of
Moreover, these bifurcations are nondegenerate and
no other bifurcation occur in a small fixed neighbourhood of
<math> x=0 </math> for parameter values sufficiently close to <math>\alpha=0\ .</math>
In this neighbourhood, the system has at most one equilibrium and two limit cycles.
To describe the Bautin bifurcation analytically, consider the system \eqref{ode1}
with <math>n=2\ ,</math>
- <math>
\dot{x} = f(x,\alpha), \ \ \ x \in {\mathbb R}^2
\ .</math>
If the following
nondegeneracy conditions hold:
-
(GH.1) <math>l_2(0)\neq 0\ ,</math> where <math> l_2(0)</math> is the second Lyapunov coefficient (see below);
-
(GH.2) the map <math> \alpha \mapsto (\mu(\alpha),l_1(\alpha))</math> is regular at <math> \alpha=0 \ ,</math> where <math> l_1(\alpha) </math> is the parameter-dependent first Lyapunov coefficient (see below),
then this system is locally
topologically equivalent
near the origin to the
normal form
- <math>
\dot{y}_1 = \beta_1 y_1 - y_2 + \beta_2 y_1(y_1^2+y_2^2) + \sigma y_1(y_1^2+y_2^2)^2
\ ,</math>
- <math>
\dot{y}_2 = y_1 + \beta_1 y_2 + \beta_2 y_2(y_1^2+y_2^2) + \sigma y_2(y_1^2+y_2^2)^2
\ ,</math>
where <math>y=(y_1,y_2)^T \in {\mathbb R}^2,\ \beta \in {\mathbb R}^2\ ,</math> and
<math>\sigma= {\rm sign}\ l_2(0) = \pm 1\ .</math> This normal form is particularly simple in polar coordinates <math>(r,\varphi),</math> where it takes the form:
- <math>
\dot{r} = r(\beta_1 r + \beta_2 r^2 + \sigma r^4)
\ ,</math>
- <math>
\dot{\varphi} = 1
</math>
The local bifurcation diagram of the normal form with <math>\sigma=-1</math> is presented in <figref>Bautin.gif</figref>. The point <math> \beta=0 </math> separates two branches of the
Andronov-Hopf bifurcation curve: the half-line
- <math>
H_{-}=\{(\beta_1,\beta_2): \beta_1=0,\ \beta_2<0 \}
</math>
corresponds to the supercritical bifurcation that generates a stable limit cycle, while
the half-line
- <math>
H_{+}=\{(\beta_1,\beta_2): \beta_1=0,\ \beta_2>0 \}
</math>
corresponds to the subcritical bifurcation that generates an unstable limit cycle.
Two hyperbolic limit cycles (one stable and one unstable) exist in the region between
the line <math> H_{+} </math> and the curve
- <math>
LPC=\{(\beta_1,\beta_2): \beta_1= -\frac{1}{4}\beta_2^2 ,\ \beta_2 > 0 \}
\ ,</math>
at which two cycles collide and disappear via a
saddle-node bifurcation of periodic orbits. The abbreviation <math> LPC </math> stands for 'Limit Point of Cycles'.
Along the curve <math> LPC </math> the system has a unique nonhyperbolic limit cycle
with the nontrivial Floquet multiplier <math> +1\ .</math>
The case <math> \sigma=1 </math> can be reduced to the one above by the
substitution <math> t \to -t, \ y_2 \to -y_2, \ \beta \to -\beta \ .</math>
In the <math>n</math>-dimensional case with <math>n \geq 2\ ,</math> the Jacobian
matrix <math>A_0=A(0)</math> at the Bautin bifurcation has
- a simple pair of purely imaginary eigenvalues <math>\lambda_{1,2}=\pm i \omega_0\ ,</math> as well as
- <math>n_s</math> eigenvalues with <math>{\rm Re}\ \lambda_j < 0\ ,</math> and
- <math>n_u</math> eigenvalues with <math>{\rm Re}\ \lambda_j > 0\ ,</math>
with <math>n_s+n_u+2=n\ .</math>
According to the
Center Manifold Theorem, there is a family of smooth
two-dimensional
invariant manifolds <math>W^c_{\alpha}</math> near the origin.
The <math>n</math>-dimensional system restricted on <math>W^c_{\alpha}</math> is
two-dimensional, hence has the normal form above.
Moreover, under the non-degeneracy conditions (GH.1) and (GH.2),
the <math>n</math>-dimensional system is
locally topologically equivalent near the origin
to the suspension of the normal form by the standard saddle, i.e.
- <math>
\dot{y}_1 = \beta_1 y_1 - y_2 + \beta_2 y_1(y_1^2+y_2^2) + \sigma y_1(y_1^2+y_2^2)^2
\ ,</math>
- <math>
\dot{y}_2 = y_1 + \beta_1 y_2 + \beta_2 y_2(y_1^2+y_2^2) + \sigma y_2(y_1^2+y_2^2)^2
\ ,</math>
- <math>
\dot{y}^s = -y^s
\ ,</math>
- <math>
\dot{y}^u = +y^u
\ ,</math>
where <math>y \in {\mathbb R}^2\ ,</math> <math>y^s \in {\mathbb R}^{n_s}, \ y^u
\in {\mathbb R}^{n_u}\ .</math>
The Lyapunov coefficients <math>l_1(\alpha)</math> and <math>l_2(0)\ ,</math>
which are involved in the nondegeneracy
conditions (GH.1) and (GH.2), can be computed for <math>n \geq 2</math> as follows.
Write the Taylor expansion of <math>f(x,\alpha)</math> at <math>x=0</math> as
- <math>
f(x,\alpha)=A(\alpha)x + \frac{1}{2}B(x,x,\alpha) + \frac{1}{6}C(x,x,x,\alpha) + O(\|x\|^4),
</math>
where <math>B(x,y,\alpha)</math> and <math>C(x,y,z,\alpha)</math> are the multilinear functions with components
- <math>
\ \ B_j(x,y,\alpha) =\sum_{k,l=1}^n \left. \frac{\partial^2 f_j(\xi,\alpha)}{\partial \xi_k \partial
\xi_l}\right|_{\xi=0} x_k y_l
\ ,</math>
- <math>
C_j(x,y,z,\alpha) =\sum_{k,l,m=1}^n \left. \frac{\partial^3 f_j(\xi,\alpha)}{\partial \xi_k \partial
\xi_l \partial \xi_m}\right|_{\xi=0} x_k y_l z_m
\ ,</math>
for <math>j=1,2,\ldots,n\ .</math> Let <math>q_{\alpha}\in {\mathbb C}^n</math> be a
complex eigenvector of <math>A(\alpha)</math> corresponding to the eigenvalue <math>\lambda(\alpha)=\mu(\alpha)
+ i\omega(\alpha)\ :</math> <math>A(\alpha)q_{\alpha}=\lambda(\alpha) q_{\alpha}\ ,</math>
<math> \langle q_{\alpha}, q_{\alpha} \rangle =1\ .</math>
Introduce also the
adjoint eigenvector <math>p_{\alpha} \in {\mathbb C}^n\ :</math>
<math>A^T(\alpha) p_{\alpha} = \bar{\lambda}(\alpha) p_{\alpha}\ ,</math>
<math> \langle p_{\alpha}, q_{\alpha} \rangle =1\ .</math>
Here <math>\langle p_{\alpha}, q_{\alpha} \rangle = \bar{p}_{\alpha}^Tq_{\alpha}</math> is the inner product
in <math>{\mathbb C}^n</math> and the vectors <math> q_{\alpha} </math> and
<math> p_{\alpha} </math> can be assumed to depend smoothly on the parameters.
Then
- <math></math>