# Bautin bifurcation

Post-publication activity

Curator: John Guckenheimer

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 $$\omega(0)=\omega_0>0\ .$$

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

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