Bautin bifurcation
| John Guckenheimer and Yuri A. Kuznetsov (2007), Scholarpedia, 2(5):1853. | doi:10.4249/scholarpedia.1853 | revision #91035 [link to/cite this article] |
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.
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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.
External Links
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
