# Fold-Hopf Bifurcation

(Redirected from Zero-Hopf bifurcation)
Post-publication activity

Curator: John Guckenheimer

The fold-Hopf bifurcation is a bifurcation of an equilibrium point in a two-parameter family of autonomous ODEs at which the critical equilibrium has a zero eigenvalue and a pair of purely imaginary eigenvalues. This phenomenon is also called the zero-Hopf (ZH) bifurcation, saddle-node Hopf bifurcation or Gavrilov-Guckenheimer bifurcation.

The bifurcation point in the parameter plane lies at a tangential intersection of curves of saddle-node bifurcations and Andronov-Hopf bifurcations. Depending on the system, a branch of torus bifurcations can emanate from the ZH-point. In such cases, other bifurcations occur for nearby parameter values, including saddle-node bifurcations of periodic orbits on the invariant torus, torus breakdown, and bifurcations of Shil'nikov homoclinic orbits to saddle-foci and heteroclinic orbits connecting equilibria.

This bifurcation, therefore, can imply a local birth of "chaos".

## 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 at $$\alpha=0$$ the system has an equilibrium $$x=0\ .$$
• Assume that its Jacobian matrix $$A=f_x(0,0)$$ has a zero eigenvalue $$\lambda_1=0$$ and a pair of purely imaginary eigenvalues $$\lambda_{2,3}=\pm i\omega$$ with $$\omega>0\ .$$

This codimension two bifurcation is characterized by the conditions $$\lambda_{1}=0$$ and $${\rm Re}\ \lambda_{2,3}=0$$ and appears in open sets of two-parameter families of smooth ODEs.

Generically, $$\alpha=0$$ lies at a tangential intersection of curves of

An early example of this bifurcation in a specific system is provided by the Brusselator reaction-diffusion system in one spatial dimension (Guckenheimer 1980, Wittenberg and Holmes 1997).

In a small fixed neighbourhood of $$x=0$$ for parameter values sufficiently close to $$\alpha=0\ ,$$ the system has at most two equilibria, which can collide and disappear via a saddle-node bifurcation or undergo an Andronov-Hopf bifurcation producing a limit cycle. Additional curves of codimension one bifurcations accumulate at $$\alpha=0$$ in the parameter plane. Which codimension one bifurcations appear depends largely upon the quadratic Taylor coefficients of $$f(x,0) \ .$$ The most complicated case is associated with the appearance of a branch of torus bifurcations (Neimark-Sacker bifurcations) of the limit cycles generated by the Hopf bifurcations. This curve of torus bifurcations is transversal to the saddle-node and Andronov-Hopf bifurcation curves. The torus bifurcation generates an invariant two-dimensional torus, i.e. "interaction of the saddle-node and Andronov-Hopf bifurcations can lead to tori". The invariant torus disappears via either a "heteroclinic destruction" or a "blow-up". In the former case, homoclinic and heteroclinic orbits connecting the two equilibria appear and disappear (Champneys and Kirk 2004), while in the latter case, the torus hits the boundary of any small fixed neighbourhood of $$x=0 \ .$$ The dynamics on the torus can either be periodic or quasiperiodic, and the torus can lose its smoothness before disappearance. The complete bifurcation scenario is unknown.

## Three-dimensional Case

To describe the fold-Hopf bifurcation analytically, consider the system (1) with $$n=3\ ,$$ $\dot{x} = f(x,\alpha), \ \ \ x \in {\mathbb R}^3 \ .$ If the following nondegeneracy conditions hold:

• (ZH.1) $$B(0)C(0)E(0) \neq 0\ ;$$
• (ZH.2) the map $$(x,\alpha) \mapsto (f(x,\alpha),{\rm Tr}(f_x(x,\alpha)),\det(f_x(x,\alpha)))$$ is regular at $$(x,\alpha)=(0,0) \ ,$$

then this system is locally orbitally smoothly equivalent near the origin to the complex normal form $\dot{\xi} = \beta_1 + \xi^2 + s |\zeta|^2 + O(\|(\xi,\zeta)\|^4) \ ,$ $\dot{\zeta}= (\beta_2 + i\omega)\zeta +(\theta(\beta)+i\theta_1(\beta))\xi \zeta + \xi^2 \zeta + O(\|(\xi,\zeta)\|^4) \ ,$ where $$\xi \in {\mathbb R},\ \zeta \in {\mathbb C},\ \beta \in {\mathbb R}^2\ ,$$ and $s = {\rm sign}\ B(0)C(0) = \pm 1,\ \theta(0)=\frac{{\rm Re}\ H_{110}}{B(0)} \ .$ When $$E(0)<0\ ,$$ the orbital equivalence includes reversal of time.

The formulas for $$B(0), C(0), E(0),$$ and $$H_{110}$$ are given below. This normal form is particularly simple in real cylindrical coordinates $$(\rho,\varphi,\xi),$$ where it takes the form: $\dot{\xi} = \beta_1 + \xi^2 + s \rho^2 + O((\xi^2 + \rho^2)^2) \ ,$ $\dot{\rho} = \rho(\beta_2 + \theta(\beta) \xi + \xi^2) + O((\xi^2 + \rho^2)^2) \ ,$ $\dot{\varphi} = \omega + \theta_1(\beta)\xi + O(\xi^2 + \rho^2),$ where the $$O$$-terms are $$2\pi$$-periodic in $$\varphi\ .$$

In general, the bifurcation diagram of the normal form depends on the $$O$$-terms, although its essential features are determined by the "truncated normal form": $\dot{\xi} = \beta_1 + \xi^2 + s \rho^2 \ ,$ $\dot{\rho} = \rho(\beta_2 + \theta(\beta) \xi + \xi^2) \ ,$ $\dot{\varphi} = \omega + \theta_1(\beta)\xi,$ where the first two equations are independent of the third one, which describes a monotone rotation. Local bifurcation diagrams of the planar system $\dot{\xi} = \beta_1 + \xi^2 + s \rho^2 \ ,$ $\dot{\rho} = \rho(\beta_2 + \theta(\beta) \xi + \xi^2)$ with

• (ZH.3) $$\theta(0) \neq 0$$

are presented in Kuznetsov (2004, Sec. 8.5.2) and Guckenheimer and Holmes (1983, Sec. 7.4). Here four cases should be distinguished:

• $$s=1,\ \theta(0) > 0$$ (subcritical Hopf bifurcations and no tori);
• $$s=-1,\ \theta(0) < 0$$ (subcritical Hopf bifurcations and no tori);
• $$s=1,\ \theta(0) < 0$$ (sub- and supercritical Hopf bifurcations and torus "heteroclinic destruction");
• $$s=-1,\ \theta(0) > 0$$ (sub- and supercritical Hopf bifurcation and torus "blow-up").

Normal forms for bifurcations are not unique. In the present case, the stability of tori is determined by cubic terms in the normal form and the equivalence of different choices is discussed in Guckenheimer and Holmes (1983, Sec. 7.4). Whether the heteroclinic destruction of tori gives rise to chaotic invariant sets is not determined by properties of finite degree normal form expansions.

## Multidimensional Case

In the $$n$$-dimensional case with $$n \geq 3\ ,$$ the Jacobian matrix $$A=f_x(0,0)$$ at the fold-Hopf bifurcation has

• a simple zero eigenvalue $$\lambda_{1}=0$$ and a simple pair of purely imaginary eigenvalues $$\lambda_{2,3}=\pm i \omega\ ,$$ as well as
• $$n_s$$ eigenvalues with $${\rm Re}\ \lambda_j < 0\ ,$$ and
• $$n_u$$ eigenvalues with $${\rm Re}\ \lambda_j > 0\ ,$$ with $$n_s+n_u+3=n\ .$$

According to the Center Manifold Theorem, there is a family of smooth three-dimensional invariant manifolds $$W^c_{\alpha}$$ near the origin. The $$n$$-dimensional system restricted on $$W^c_{\alpha}$$ is three-dimensional, hence has the normal form above.

## Normal Form Coefficients

The normal form coefficients which are involved in the nondegeneracy conditions (ZH.1) and (ZH.3), can be computed for $$n \geq 3$$ as follows.

Write the Taylor expansion of $$f(x,0)$$ at $$x=0$$ as $f(x,\alpha)=Ax + \frac{1}{2}B(x,x) + \frac{1}{6}C(x,x,x) + O(\|x\|^4),$ where $$B(x,y)$$ and $$C(x,y,z)$$ are the multilinear functions with components $\ \ B_j(x,y) =\sum_{k,l=1}^n \left. \frac{\partial^2 f_j(\xi,0)}{\partial \xi_k \partial \xi_l}\right|_{\xi=0} x_k y_l \ ,$ $C_j(x,y,z) =\sum_{k,l,m=1}^n \left. \frac{\partial^3 f_j(\xi,0)}{\partial \xi_k \partial \xi_l \partial \xi_m}\right|_{\xi=0} x_k y_l z_m \ ,$ where $$j=1,2,\ldots,n\ .$$

Introduce two eigenvectors, $$q_0 \in {\mathbb R}^n$$ and $$q_1 \in {\mathbb C}^n\ ,$$ $$Aq_0=0,\ \ Aq_1=i\omega q_1,$$ and two adjoint eigenvectors, $$p_0 \in {\mathbb R}^n$$ and $$p_1 \in {\mathbb C}^n\ ,$$ $$A^Tp_0=0,\ \ A^Tp_1=-i\omega p_1.$$ Normalize them such that $$\langle p_0,q_0 \rangle = \langle p_1,q_1 \rangle =1.$$ (The notation $$\langle v,w \rangle$$ denotes the inner product of two vectors.)

Compute $G_{200} = \frac{1}{2}\langle p_0,B(q_0,q_0)\rangle,\ H_{110} = \langle p_1,B(q_0,q_1)\rangle,\ G_{011} = \langle p_0,B(q_1,\overline{q}_1)\rangle,$ and $G_{300}=\frac{1}{6}\langle p_0, C(q_0,q_0,q_0)+3B(q_0,h_{200})\rangle \ ,$ $G_{111}=\langle p_0, C(q_0,q_1,\overline{q}_1)+B(q_1,\overline{h}_{110}) + B(\overline{q}_1,h_{110}) + B(q_0,h_{011})\rangle \ ,$ $H_{210}=\frac{1}{2}\langle p_1, C(q_0,q_0,q_1) + 2B(q_0,h_{110}) + B(q_1,h_{200})\rangle \ ,$ $H_{021}=\frac{1}{2}\langle p_1, C(q_1,q_1,\overline{q}_1) + 2B(q_1,h_{011}) + B(\overline{q}_1,h_{020}) \rangle \ ,$ where $h_{020}=(2i\omega I_n - A)^{-1}B(q_1,q_1) \ ,$ while the vectors $$h_{200},\ h_{011},$$ and $$h_{110}$$ are the solutions of the following nonsingular systems $\left( \begin{array}{cc} A& ~~q_0\\ p_0^{T}&~~0 \end{array} \right) \left( \begin{array}{c} h_{200} \\ s \end{array} \right)= \left( \begin{array}{c} -B(q_0,q_0) + \langle p_0,B(q_0,q_0)\rangle q_0\\ 0 \end{array}\right) \ ,$ $\left( \begin{array}{cc} A&~~q_0\\ p_0^{T}&~~0 \end{array} \right) \left( \begin{array}{c} h_{011} \\ s \end{array} \right)= \left( \begin{array}{c} -B(q_1,\overline{q}_1) + \langle p_0,B(q_1,\overline{q}_1)\rangle q_0\\ 0 \end{array}\right)$ and $\left(\begin{array}{cc} i\omega I_n-A & ~~q_1\\ \overline{p}_1^T & ~~0 \end{array} \right) \left(\begin{array}{c} h_{110}\\s\end{array}\right)= \left(\begin{array}{c} B(q_0,q_1) - \langle p_1,B(q_0,q_1)\rangle q_1\\0\end{array}\right).$

Finally, $$B(0)=G_{200},\ C(0)=G_{011},$$ while $E(0)={\rm Re}\left[H_{210}(0)+H_{110}(0) \left(\frac{{\rm Re\ }H_{021}(0)}{G_{011}(0)}-\frac{3G_{300}(0)}{2G_{200}(0)}+ \frac{G_{111}(0)}{2G_{011}(0)}\right) - \frac{H_{021}(0)G_{200}(0)}{G_{011}(0)}\right].$

The bifurcation software MATCONT computes $$s, \theta(0)$$ and $$E(0)$$ automatically.

## Other Cases

Fold-Hopf (ZH) bifurcation occurs also in infinite-dimensional ODEs generated by PDEs and DDEs to which the Center Manifold Theorem applies.