Bautin bifurcation

Figure 1: Generalized Hopf (Bautin) bifurcation in the planar system \( \dot{r}=r(\beta_1 + \beta_2 r^2 - r^4),\ \dot{\varphi}=1 \ .\) The vertical axis corresponds to the Andronov-Hopf bifurcation (supercritical at \( H_{-}\) and subcritical at \( H_{+}\)); the curve \( LPC \) 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.

[edit] Definition

Consider an autonomous system of ordinary differential equations (ODEs) \[\tag{1} \dot{x}=f(x,\alpha),\ \ \ x \in {\mathbb R}^n \]

depending on two parameters \(\alpha \in {\mathbb R}^2\ ,\) where \(f\) is smooth.

Suppose that for all sufficiently small \(\|\alpha\|\) the system has an equilibrium \(x=0\ .\)
Further assume that its Jacobian matrix \(A(\alpha)=f_x(0,\alpha)\) has one pair of complex eigenvalues

\(\lambda_{1,2}(\alpha)=\mu(\alpha) \pm i\omega(\alpha)\) such that \(\mu(0)=0\) and \(\omega(0)=\omega_0>0\ .\)

Finally assume that the critical first Lyapunov coefficient for Andronov-Hopf bifucation \(l_1(0) = 0\ .\)

This bifurcation is characterized by two bifurcation conditions \({\rm Re}\ \lambda_{1,2}=0\) and \(l_1(0) = 0\) (has codimension two) and appears generically in two-parameter families of smooth ODEs.

Generically, \(\alpha=0\) is the origin in the parameter plane of

two branches of Andronov-Hopf bifurcation curve, corresponding to the super- and subcritical cases; and
a curve of saddle-node bifurcations of periodic orbits, where two limit cycles collide and disappear.

Moreover, these bifurcations are nondegenerate and no other bifurcation occur in a small fixed neighbourhood of \( x=0 \) for parameter values sufficiently close to \(\alpha=0\ .\) In this neighbourhood, the system has at most one equilibrium and two limit cycles.

[edit] Two-dimensional Case

To describe the Bautin bifurcation analytically, consider the system (1) with \(n=2\ ,\) \[ \dot{x} = f(x,\alpha), \ \ \ x \in {\mathbb R}^2 \ .\] If the following nondegeneracy conditions hold:

(GH.1) \(l_2(0)\neq 0\ ,\) where \( l_2(0)\) is the second Lyapunov coefficient (see below);
(GH.2) the map \( \alpha \mapsto (\mu(\alpha),l_1(\alpha))\) is regular at \( \alpha=0 \ ,\) where \( l_1(\alpha) \) is the parameter-dependent first Lyapunov coefficient (see below),

then this system is locally topologically equivalent near the origin to the normal form \[ \dot{y}_1 = \beta_1 y_1 - y_2 + \beta_2 y_1(y_1^2+y_2^2) + \sigma y_1(y_1^2+y_2^2)^2 \ ,\] \[ \dot{y}_2 = y_1 + \beta_1 y_2 + \beta_2 y_2(y_1^2+y_2^2) + \sigma y_2(y_1^2+y_2^2)^2 \ ,\] where \(y=(y_1,y_2)^T \in {\mathbb R}^2,\ \beta \in {\mathbb R}^2\ ,\) and \(\sigma= {\rm sign}\ l_2(0) = \pm 1\ .\) This normal form is particularly simple in polar coordinates \((r,\varphi),\) where it takes the form: \[ \dot{r} = r(\beta_1 r + \beta_2 r^2 + \sigma r^4) \ ,\] \[ \dot{\varphi} = 1 \]

The local bifurcation diagram of the normal form with \(\sigma=-1\) is presented in Figure 1. The point \( \beta=0 \) separates two branches of the Andronov-Hopf bifurcation curve: the half-line \[ H_{-}=\{(\beta_1,\beta_2): \beta_1=0,\ \beta_2<0 \} \] corresponds to the supercritical bifurcation that generates a stable limit cycle, while the half-line \[ H_{+}=\{(\beta_1,\beta_2): \beta_1=0,\ \beta_2>0 \} \] 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 \( H_{+} \) and the curve \[ LPC=\{(\beta_1,\beta_2): \beta_1= -\frac{1}{4}\beta_2^2 ,\ \beta_2 > 0 \} \ ,\] at which two cycles collide and disappear via a saddle-node bifurcation of periodic orbits. The abbreviation \( LPC \) stands for 'Limit Point of Cycles'.

Along the curve \( LPC \) the system has a unique nonhyperbolic limit cycle with the nontrivial Floquet multiplier \( +1\ .\)

The case \( \sigma=1 \) can be reduced to the one above by the substitution \( t \to -t, \ y_2 \to -y_2, \ \beta \to -\beta \ .\)

[edit] Multidimensional Case

In the \(n\)-dimensional case with \(n \geq 2\ ,\) the Jacobian matrix \(A_0=A(0)\) at the Bautin bifurcation has

a simple pair of purely imaginary eigenvalues \(\lambda_{1,2}=\pm i \omega_0\ ,\) as well as
\(n_s\) eigenvalues with \({\rm Re}\ \lambda_j < 0\ ,\) and
\(n_u\) eigenvalues with \({\rm Re}\ \lambda_j > 0\ ,\)

with \(n_s+n_u+2=n\ .\) According to the Center Manifold Theorem, there is a family of smooth two-dimensional invariant manifolds \(W^c_{\alpha}\) near the origin. The \(n\)-dimensional system restricted on \(W^c_{\alpha}\) is two-dimensional, hence has the normal form above.

Moreover, under the non-degeneracy conditions (GH.1) and (GH.2), the \(n\)-dimensional system is locally topologically equivalent near the origin to the suspension of the normal form by the standard saddle, i.e. \[ \dot{y}_1 = \beta_1 y_1 - y_2 + \beta_2 y_1(y_1^2+y_2^2) + \sigma y_1(y_1^2+y_2^2)^2 \ ,\] \[ \dot{y}_2 = y_1 + \beta_1 y_2 + \beta_2 y_2(y_1^2+y_2^2) + \sigma y_2(y_1^2+y_2^2)^2 \ ,\] \[ \dot{y}^s = -y^s \ ,\] \[ \dot{y}^u = +y^u \ ,\] where \(y \in {\mathbb R}^2\ ,\) \(y^s \in {\mathbb R}^{n_s}, \ y^u \in {\mathbb R}^{n_u}\ .\)

[edit] Lyapunov Coefficients

The Lyapunov coefficients \(l_1(\alpha)\) and \(l_2(0)\ ,\) which are involved in the nondegeneracy conditions (GH.1) and (GH.2), can be computed for \(n \geq 2\) as follows.

Write the Taylor expansion of \(f(x,\alpha)\) at \(x=0\) as \[ f(x,\alpha)=A(\alpha)x + \frac{1}{2}B(x,x,\alpha) + \frac{1}{6}C(x,x,x,\alpha) + O(\|x\|^4), \] where \(B(x,y,\alpha)\) and \(C(x,y,z,\alpha)\) are the multilinear functions with components \[ \ \ B_j(x,y,\alpha) =\sum_{k,l=1}^n \left. \frac{\partial^2 f_j(\xi,\alpha)}{\partial \xi_k \partial \xi_l}\right|_{\xi=0} x_k y_l \ ,\] \[ C_j(x,y,z,\alpha) =\sum_{k,l,m=1}^n \left. \frac{\partial^3 f_j(\xi,\alpha)}{\partial \xi_k \partial \xi_l \partial \xi_m}\right|_{\xi=0} x_k y_l z_m \ ,\] for \(j=1,2,\ldots,n\ .\) Let \(q_{\alpha}\in {\mathbb C}^n\) be a complex eigenvector of \(A(\alpha)\) corresponding to the eigenvalue \(\lambda(\alpha)=\mu(\alpha) + i\omega(\alpha)\ :\) \(A(\alpha)q_{\alpha}=\lambda(\alpha) q_{\alpha}\ ,\) \( \langle q_{\alpha}, q_{\alpha} \rangle =1\ .\) Introduce also the adjoint eigenvector \(p_{\alpha} \in {\mathbb C}^n\ :\) \(A^T(\alpha) p_{\alpha} = \bar{\lambda}(\alpha) p_{\alpha}\ ,\) \( \langle p_{\alpha}, q_{\alpha} \rangle =1\ .\) Here \(\langle p_{\alpha}, q_{\alpha} \rangle = \bar{p}_{\alpha}^Tq_{\alpha}\) is the inner product in \({\mathbb C}^n\) and the vectors \( q_{\alpha} \) and \( p_{\alpha} \) can be assumed to depend smoothly on the parameters.

Then \[ l_1(\alpha) = \frac{{\rm Re}\; c_1(\alpha)}{\omega(\alpha)} - \mu(\alpha) \frac{{\rm Im}\; c_1(\alpha)}{\omega^2(\alpha)} \ ,\] where \[ \begin{array}{rcl} c_1(\alpha) &=& \frac{1}{2} \left[\langle p_{\alpha},C(q_{\alpha},q_{\alpha},\bar{q}_{\alpha},\alpha) \rangle + 2 \langle p_{\alpha}, B(q_{\alpha},((\lambda(\alpha)+\bar{\lambda}(\alpha))I_n-A(\alpha))^{-1}B(q_{\alpha},\bar{q}_{\alpha},\alpha),\alpha)\rangle + \right. \\ &&~~~\left. \langle p_{\alpha}, B(\bar{q}_{\alpha},(2\lambda(\alpha) I_n-A(\alpha))^{-1} B(q_{\alpha},q_{\alpha},\alpha),\alpha)\rangle \right]. \end{array} \] Here \(I_n\) is the unit \(n \times n\) matrix.

To compute the second Lyapunov coefficient \( l_2(0) \ ,\) write the Taylor expansion of \(f(x,0)\) at \(x=0\) as \[ f(x,0)=A_0x + \frac{1}{2}B_0(x,x) + \frac{1}{6}C_0(x,x,x) + \frac{1}{24} D_0(x,x,x,x) + \frac{1}{120} E_0(x,x,x,x,x) + O(\|x\|^6), \] where \(B_0(x,y)=B(x,y,0),\ C_0(x,y,z)=C(x,y,z,0)\ ,\) and \(D_0(x,y,z,v)\) and \(E_0(x,y,z,v,w)\) are the multilinear functions with components \[ D_{0,j}(x,y,z,v) =\sum_{k,l,m,p=1}^n \left. \frac{\partial^4 f_j(\xi,0)} {\partial \xi_k \partial \xi_l \partial \xi_m}\right|_{\xi=0} x_k y_l z_m v_p \ ,\] \[ E_{0,j}(x,y,z,v,w) =\sum_{k,l,m,p,q=1}^n \left. \frac{\partial^5 f_j(\xi,0)} {\partial \xi_k \partial \xi_l \partial \xi_m \partial \xi_p \partial \xi_q}\right|_{\xi=0} x_k y_l z_m v_p w_q \ ,\] for \(j=1,2,\ldots,n\ .\)

Then the critical second Lyapunov coefficient is given by \[ l_2(0)=\frac{{\rm Re\ }c_2(0)}{\omega(0)} \ ,\] with \[ \begin{array}{rcl} c_2(0)&=&\frac{1}{12}\langle p_0,E_0(q_0,q_0,q_0,\overline{q}_0,\overline{q}_0) + D_0(q_0,q_0,q_0,\overline{h}_{20}) + 3D_0(q_0,\overline{q}_0,\overline{q}_0,h_{20}) +6D_0(q_0,q_0,\overline{q}_0,h_{11}) \\ &&~~~+ C_0(\overline{q}_0,\overline{q}_0,h_{30}) +3C_0(q_0,q_0,\overline{h}_{21})+6C_0(q_0,\overline{q}_0,h_{21}) +3C_0(q_0,\overline{h}_{20},h_{20}) \\ &&~~~+6 C_0(q_0,h_{11},h_{11}) +6C_0(\overline{q}_0,h_{20},h_{11}) + 2B_0(\overline{q}_0,h_{31}) + 3B_0(q_0,h_{22}) \\ &&~~~+B_0(\overline{h}_{20},h_{30})+3B_0(\overline{h}_{21},h_{20}) + 6B_0(h_{11},h_{21}) \rangle , \end{array} \] where \[ h_{20} = (2i\omega_0 I_n - A_0)^{-1}B_0(q_0,q_0) \ ,\] \[ h_{11}=-A_0^{-1}B_0(q_0,\overline{q}_0) \ .\] The complex vector \( h_{21} \) is found by solving the nonsingular \( (n+1)\)-dimensional complex system \[ \left(\begin{array}{cc} i\omega_0 I_n-A_0 & q_0\\ \overline{p}^{T} & 0 \end{array} \right) \left(\begin{array}{c} h_{21}\\s\end{array}\right)= \left(\begin{array}{c} C_0(q_0,q_0,\overline{q}_0)+B_0(\overline{q}_0,h_{20})+2B_0(q_0,h_{11}) -2c_1(0)q_0\\0\end{array}\right), \] while \[ \begin{array}{rcl} h_{30}&=&(3i\omega_0 I_n - A_0)^{-1}[C_0(q_0,q_0,q_0)+3B_0(q_0,h_{20})],\\ h_{31}&=&(2i\omega_0 I_n -A_0)^{-1} [D_0(q_0,q_0,q_0,\overline{q}_0)+3C_0(q_0,q_0,h_{11})+3C_0(q_0,\overline{q}_0,h_{20})\\ &&~~~~~~~~~~~~~~~~~ + 3B_0(h_{20},h_{11}) + B_0(\overline{q}_0,h_{30})+3B_0(q_0,h_{21})-6c_1(0)h_{20}],\\ h_{22}&=&-A_0^{-1}[D_0(q_0,q_0,\overline{q}_0,\overline{q}_0)+4C_0(q_0,\overline{q}_0,h_{11}) +C_0(\overline{q}_0,\overline{q}_0,h_{20}) +C_0(q_0,q_0,\overline{h}_{20}) \\ &&~~~~~~ + 2B_0(h_{11},h_{11})+2B_0(q_0,\overline{h}_{21})+2B_0(\overline{q}_0,h_{21}) + B_0(\overline{h}_{20},h_{20})]. \end{array} \]

Standard bifurcation software MATCONT computes \(l_2(0)\) automatically.

[edit] Other Cases

Bautin (GH) bifurcation occurs also in infinitely-dimensional ODEs generated by PDEs and DDEs, to which the Center Manifold Theorem applies.

[edit] References

V.I. Arnold (1983) Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren Math. Wiss., 250, Springer
J. Guckenheimer and P. Holmes (1983) Nonlinear Oscillations, Dynamical systems and Bifurcations of Vector Fields. Springer
Yu.A. Kuznetsov (2004) Elements of Applied Bifurcation Theory, Springer, 3rd edition.

Internal references

Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
Willy Govaerts, Yuri A. Kuznetsov, Bart Sautois (2006) MATCONT. Scholarpedia, 1(9):1375.
James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
Yuri A. Kuznetsov (2006) Saddle-node bifurcation. Scholarpedia, 1(10):1859.
Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
Emmanuil E. Shnol (2007) Stability of equilibria. Scholarpedia, 2(3):2770.
Bard Ermentrout (2007) XPPAUT. Scholarpedia, 2(1):1399.