Bautin bifurcation

• Dr. John Guckenheimer, Cornell University, Ithaca, NY, USA

• Prof. Yuri A. Kuznetsov, Department of Mathematics, Utrecht University, The Netherlands

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.

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.

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 $t \to -t, \ y_2 \to -y_2, \ \beta \to -\beta \ .$

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 $y \in {\mathbb R}^2\ ,$ $y^s \in {\mathbb R}^{n_s}, \ y^u \in {\mathbb R}^{n_u}\ .$

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 $$D_{0,j}(x,y,z,v) =\sum_{k,l,m,p=1}^n \left. \frac{\partial^4 f_j(\xi,0)} {\partial \xi_k \partial \xi_l \partial \xi_m}\right|_{\xi=0} x_k y_l z_m v_p \ ,$$ $$C_{0,j}(x,y,z,v,w) =\sum_{k,l,m,p,q=1}^n \left. \frac{\partial^5 f_j(\xi,0)} {\partial \xi_k \partial \xi_l \partial \xi_m \partial \xi_p \partial \xi_q}\right|_{\xi=0} x_k y_l z_m v_p w_q \ ,$$ 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.

Other Cases

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

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.

See Also

Andronov-Hopf Bifurcation, Saddle-node Bifurcation, Saddle-node Bifurcation of Periodic Orbits, Bifurcations, Center Manifold Theorem, Dynamical Systems, Equilibria, MATCONT, Ordinary Differential Equations, XPPAUT

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Reviewed by: Anonymous

Accepted on: 2007-04-26 11:49:51 GMT