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.

## Contents

* 1 Definition
* 2 Two-dimensional Case
* 3 Multidimensional Case
* 4 Lyapunov Coefficients
* 5 Other Cases
* 6 References
* 7 External Links
* 8 See Also

## [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.

## [edit] External Links

## [edit] 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