# Hopf-Hopf bifurcation

Post-publication activity

Curator: John Guckenheimer

The Hopf-Hopf bifurcation is a bifurcation of an equilibrium point in a two-parameter family of autonomous ODEs at which the critical equilibrium has two pairs of purely imaginary eigenvalues. This phenomenon is also called the double-Hopf bifurcation.

The bifurcation point in the parameter plane lies at a transversal intersection of two curves of Andronov-Hopf bifurcations. Generically, two branches of torus bifurcations emanate from the Hopf-Hopf (HH) point. Depending on the system, other bifurcations occur for nearby parameter values, including bifurcations of Shilnikov's homoclinic orbits to the focus-focus equilibrium, and bifurcations of a heteroclinic structure connecting saddle limit cycles and equilibria.

This bifurcation, therefore, can imply a local birth of "chaos". Also quasi-periodicity is involved (Braaksma and Broer, 1982).

## 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 two pairs of purely imaginary eigenvalues $$\lambda_{1,2}=\pm i\omega_1(0), \lambda_{3,4}=\pm i\omega_2(0)$$ with $$\omega_1(0) > \omega_2(0) > 0\ .$$

This codimension two bifurcation is characterized by the conditions $${\rm Re}\ \lambda_{1,2}=0$$ and $${\rm Re}\ \lambda_{3,4}=0$$ and appears in open sets of two-parameter families of smooth ODEs. In such a family (1):

In a small fixed neighbourhood of $$x=0$$ for parameter values sufficiently close to $$\alpha=0\ ,$$ the system has at most one equilibrium, which can undergo the Andronov-Hopf bifurcations, producing limit cycles. Each torus bifurcation of these limit cycles generates an invariant two-dimensional torus with periodic or quasiperiodic orbits. The 2D invariant torus can be accompanied by an invariant set resembling a 3D torus, which can disappear via either a "heteroclinic destruction" or a "blow-up". In the former case, various homoclinic and heteroclinic orbits connecting the equilibrium and two cycles exist, while in the latter case, the invariant set hits the boundary of any small fixed neighbourhood of $$x=0 \ .$$ The complete bifurcation scenario is unknown.

## Four-dimensional case

To describe the Hopf-Hopf bifurcation analytically, consider the system (1) with $$n=4\ ,$$ $\tag{2} \dot{x} = f(x,\alpha), \ \ \ x \in {\mathbb R}^4 \ .$

If the following nondegeneracy conditions hold:

• $$k \omega_1(0) \neq l \omega_2(0)$$ for integer $$k,l >0, k+l \leq 3\ ;$$
• the map $$\alpha \mapsto ({\rm Re}\ \lambda_1(\alpha), {\rm Re}\ \lambda_3(\alpha))\ ,$$ where $$\lambda_{1,3}(\alpha)$$ are eigenvalues of the continuation of the critical equilibrium for small $$\|\alpha\|$$ such that $$\lambda_{1}(0)=i\omega_1(0), \lambda_3(0)=i\omega_2(0)\ ,$$ is regular at $$\alpha=0 \ ,$$

then this system is locally orbitally smoothly equivalent near the origin to the Poincare normal form $\dot{v}_1 = (\beta_1 + i\omega_1(\beta))v_1 + G_{2100}(\beta)v_1|v_1|^2 + G_{1011}(\beta) v_1|v_2|^2 + O(\|(v_1,\bar{v}_1,v_2,\bar{v}_2)\|^4) \ ,$ $\dot{v}_2 = (\beta_2 + i\omega_2(\beta))v_2 + H_{1110}(\beta)v_2|v_1|^2 + H_{0021}(\beta) v_2|v_2|^2 + O(\|(v_1,\bar{v}_1,v_2,\bar{v}_2)\|^4) \ ,$ where $$v_1,v_2 \in {\mathbb C},\ \beta \in {\mathbb R}^2\ ,$$ and $$G_{jklm}(\beta), H_{jklm}(\beta)$$ are complex-valued smooth functions. The formulas for $$G_{2100}(0), G_{1011}(0), H_{1110}(0),$$ and $$H_{0021}(0)$$ are given below.

The normal form is particularly simple in polar coordinates $$(r_k,\varphi_k), k=1,2\ ,$$ where it takes the form: $\dot{r}_1 = r_1(\beta_1 + a_{11}(\beta)r_1^2 + a_{12}(\beta)r_2^2) + O((r_1^2 + r_2^2)^2) \ ,$ $\dot{r}_2 = r_2(\beta_2 + a_{21}(\beta)r_1^2 + a_{22}(\beta)r_2^2) + O((r_1^2 + r_2^2)^3) \ ,$ $\dot{\varphi}_1 = \omega_1(\beta) + O(r_1^2 + r_2^2) \ ,$ $\dot{\varphi}_2 = \omega_2(\beta) + O(r_1^2 + r_2^2) \ ,$ where $a_{11}(\beta) = {\rm Re}\ G_{2100}(\beta),\ a_{12}(\beta)= {\rm Re}\ G_{1011}(\beta),\ a_{21}(\beta) = {\rm Re}\ H_{1110}(\beta),\ a_{22}(\beta)= {\rm Re}\ H_{0021}(\beta),$ and the $$O$$-terms are $$2\pi$$-periodic in $$\varphi_k\ .$$

In general, the bifurcation diagram of the normal form depends on the $$O$$-terms, although some of its features are determined by the truncated normal form: $\dot{r}_1 = r_1(\beta_1 + a_{11}(\beta)r_1^2 + a_{12}(\beta)r_2^2) \ ,$ $\dot{r}_2 = r_2(\beta_2 + a_{21}(\beta)r_1^2 + a_{22}(\beta)r_2^2) \ ,$ $\dot{\varphi}_1 = \omega_1(\beta) \ ,$ $\dot{\varphi}_2 = \omega_2(\beta) \ ,$ where the first two equations are independent of the last two defining monotone rotations. Local bifurcation diagrams of the planar amplitude system $\tag{3} \left\{ \begin{array}{rcl} \dot{r}_1 &=& r_1(\beta_1 + a_{11}(\beta)r_1^2 + a_{12}(\beta)r_2^2),\\ \dot{r}_2 &=& r_2(\beta_2 + a_{21}(\beta)r_1^2 + a_{22}(\beta)r_2^2), \end{array} \right.$

satisfying some extra genericity conditions can be found in Guckenheimer and Holmes (1983, Sec. 7.5). Here two cases should be distinguished:

• $$a_{11}(0)a_{22}(0) > 0$$ ("simple case", no periodic orbits in the amplitude system);
• $$a_{11}(0)a_{22}(0) < 0$$ ("difficult case", periodic and heteroclinic orbits in the amplitude system are possible).

Each case includes many subcases depending on $\theta = \frac{a_{12}(0)}{a_{22}(0)}\ ,\ \ \delta = \frac{a_{21}(0)}{a_{11}(0)}\ \ .$ The equilibrium $$(r_1,r_2)=(0,0)$$ of the amplitude system (3) corresponds to the equilibrium of the 4D-system (2). Nonzero equilibria $$(r_1,0)$$ and $$(0,r_2)$$ correspond to limit cycles, while positive equilibria $$(r_1,r_2); r_1 > 0, r_2 > 0$$ correspond to invariant 2D tori. Limit cycles of the amplitude system correspond to invariant 3D tori. The appearance of an equilibrium $$(r_1,0)$$ or $$(0,r_2)$$ in the amplitude system corresponds to Andronov-Hopf bifurcation in (2), while branching of a positive equilibrium from one of the above implies a torus bifurcation of the corresponding limit cycle. In the "difficult case", heteroclinic bifurcation in the amplitude system (3) suggests the breakdown of an invariant 3D torus and the appearance of chaotic invariant sets in the full 4D-system (2). Nearby, various homo- and heteroclinic orbits connecting the equilibrium and saddle limit cycles exist (Guckenheimer and Holmes, 1983; Broer, 1983; Broer and Vegter, 1984).

## Multidimensional case

In the $$n$$-dimensional case with $$n \geq 4\ ,$$ the Jacobian matrix $$A=f_x(0,0)$$ at the Hopf-Hopf bifurcation has

• two simple pairs of purely imaginary eigenvalues $$\lambda_{1,2}=\pm i \omega_1(0),\ \lambda_{3,4}=\pm i\omega_2(0)\ ,$$ as well as
• $$n_s$$ eigenvalues with $${\rm Re}\ \lambda_j < 0\ ,$$ and
• $$n_u$$ eigenvalues with $${\rm Re}\ \lambda_j > 0\ ,$$ with $$n_s+n_u+4=n\ .$$

According to the Center Manifold Theorem, there is a family of smooth four-dimensional invariant manifolds $$W^c_{\alpha}$$ near the origin. The $$n$$-dimensional system restricted on $$W^c_{\alpha}$$ is four-dimensional, hence has the normal form above. Also compare with (Broer, 2003).

## Cubic normal form coefficients

The cubic coefficients in the normal form can be computed for $$n \geq 4$$ as follows (Kuznetsov, 1999).

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 complex eigenvectors, $$q_{1,2} \in {\mathbb C}^n\ ,$$ $Aq_1=i\omega_1(0)q_1,\ \ Aq_2=i\omega_2(0) q_2 \ ,$ and two adjoint eigenvectors, $$p_{1,2} \in {\mathbb C}^n\ ,$$ $A^Tp_1=-i\omega_1(0)p_1,\ \ A^Tp_2=-i\omega_2(0) p_1.$ Normalize them such that $$\langle p_1,q_1 \rangle = \langle p_2,q_2 \rangle =1.$$ (The notation $$\langle v,w \rangle = \bar{v}^T w$$ denotes the inner product of two complex vectors.)

Compute $h_{1100}=-A^{-1}B({q_{1}},{\overline{q}_{1}}) \ ,$ $h_{2000}=(2i{\omega _{1}(0)} I_n - A)^{-1}B({q_{1}},{q_{1}}), \ ,$ $h_{1010}=[i(\omega _{1}(0)+\omega _{2}(0))I_n - A]^{-1}B({q_{1}},{q_{2}}) \ ,$ $h_{1001}=[i({\omega _{1}(0)}-{\omega _{2}(0)})I_n-A]^{-1}B({q_{1}}, \,{\overline{q}_{2}}), \ ,$ $h_{0020}=(2i{\omega _{2}(0)}I_n-A)^{-1}B({q_{2}},{q_{2}}), \ ,$ $h_{0011}=-A^{-1}B({q_{2}},{\overline{q}_{2}})$ and then evaluate $G_{2100}(0)=\frac{1}{2}\langle p_1, C(q_1,q_1,\overline{q}_1) + B(h_{2000},\overline{q}_1) + 2B(h_{1100},q_1)\rangle \ ,$ $G_{1011}(0)=\langle p_1, C(q_1,q_2,\overline{q}_2) + B(h_{1010},\overline{q}_2) + B(h_{1001},q_2) + B(h_{0011},q_1)\rangle \ ,$ $H_{1110}(0)=\langle p_2, C(q_1,\overline{q}_1,q_2) + B(h_{1100},q_2) + B(h_{1010},\overline{q}_1) + B(\overline{h}_{1001},q_1) \rangle \ ,$ $H_{0021}(0)=\frac{1}{2}\langle p_2, C(q_2,q_2,\overline{q}_2) + B(h_{0020},\overline{q}_2) + 2B(h_{0011},q_2) \rangle \ .$

The bifurcation software MATCONT computes these coefficients automatically.

## Gavrilov normal form

To analyze bifurcations of 2D tori, one has to normalize the fourth- and fifth-order terms. The resulting normal form is not unique. If the following nondegeneracy conditions hold:

• (HH.0) $$k \omega_1(0) \neq l \omega_2(0)$$ for integer $$k,l >0, k+l \leq 5\ ;$$
• (HH.1) $${\rm Re}\ G_{2100}(0) \neq 0\ ;$$
• (HH.2) $${\rm Re}\ G_{1011}(0) \neq 0\ ;$$
• (HH.3) $${\rm Re}\ H_{1110}(0) \neq 0\ ;$$
• (HH.4) $${\rm Re}\ H_{0021}(0) \neq 0\ ;$$
• (HH.5) the map $$\alpha \mapsto ({\rm Re}\ \lambda_1(\alpha), {\rm Re}\ \lambda_3(\alpha))\ ,$$ where $$\lambda_{1,3}(\alpha)$$ are eigenvalues of the continuation of the critical equilibrium for small $$\|\alpha\|$$ such that $$\lambda_{1}(0)=i\omega_1(0), \lambda_3(0)=i\omega_2(0)\ ,$$ is regular at $$\alpha=0 \ ,$$

then system (2) is locally orbitally smoothly equivalent near the origin to the complex normal form (Gavrilov, 1980) $\dot{v}_1 = (\beta_1 + i\omega_1(\beta))v_1 + P_{11}(\beta)v_1|v_1|^2 + P_{12}(\beta) v_1|v_2|^2 + i R_1(\beta)v_1|v_1|^4 + S_1(\beta) v_1|v_2|^4 + O(\|(v_1,\bar{v}_1,v_2,\bar{v}_2)\|^6) \ ,$ $\dot{v}_2 = (\beta_2 + i\omega_2(\beta))v_2 + P_{21}(\beta)v_2|v_1|^2 + P_{22}(\beta) v_2|v_2|^2 + S_2(\beta) v_2|v_1|^4 + i R_2(\beta)v_2|v_2|^4 + O(\|(v_1,\bar{v}_1,v_2,\bar{v}_2)\|^6) \ ,$ where $$v_1,v_2 \in {\mathbb C},\ \beta \in {\mathbb R}^2\ ;$$ $$S_k(\beta),\ P_{jk}(\beta)$$ are complex-valued smooth functions such that ${\rm Re}\ P_{11}(0) = {\rm Re}\ G_{2100}(0),\ \ {\rm Re}\ P_{12}(0) = {\rm Re}\ G_{1011}(0) \ ,$ ${\rm Re}\ P_{21}(0) = {\rm Re}\ H_{1110}(0),\ \ {\rm Re}\ P_{12}(0) = {\rm Re}\ H_{0021}(0) \ ,$ while $$R_{k}(\beta)$$ are real smooth functions. The formulas for $$S_1(0)$$ and $$S_2(0)$$ are lengthy and can be found in Kuznetsov (1999); MATCONT computes them automatically.

In the polar coordinates $$(r_k,\varphi_k), k=1,2\ ,$$ Gavrilov's normal form reads: $\tag{4} \left\{\begin{array}{rcl} \dot{r}_1 &=& r_1(\beta_1 + p_{11}(\beta)r_1^2 + p_{12}(\beta)r_2^2 + s_1(\beta)r_2^4) + O((r_1^2 + r_2^2)^3),\\ \dot{r}_2 &=& r_2(\beta_2 + p_{21}(\beta)r_1^2 + p_{22}(\beta)r_2^2 + s_2(\beta)r_1^4) + O((r_1^2 + r_2^2)^3),\\ \dot{\varphi}_1 &=& \omega_1(\beta) + O(r_1^2 + r_2^2),\\ \dot{\varphi}_2 &=& \omega_2(\beta) + O(r_1^2 + r_2^2), \end{array} \right.$

where $p_{kj}(\beta) = {\rm Re}\ P_{kj}(\beta),\ s_k(\beta)={\rm Re}\ S_k(\beta),\ \ \ k,j=1,2,$ and the $$O$$-terms are $$2\pi$$-periodic in $$\varphi_k\ .$$

The bifurcation diagram of this normal form also depends on the $$O$$-terms, but some of its important features are determined by the fifth-order amplitude system $\tag{5} \left\{ \begin{array}{rcl} \dot{r}_1 &=& r_1(\beta_1 + p_{11}(\beta)r_1^2 + p_{12}(\beta)r_2^2 + s_1(\beta)r_2^4),\\ \dot{r}_2 &=& r_2(\beta_2 + p_{21}(\beta)r_1^2 + p_{22}(\beta)r_2^2 + s_2(\beta)r_1^4). \end{array} \right.$

Local bifurcation diagrams of this system satisfying some extra genericity conditions can be found in Kuznetsov (2004, Sec. 8.6.2). In (5), the positive equilibrium exhibits the Andronov-Hopf bifurcation generating a limit cycle. This limit cycle corresponds to a 3D invariant torus in the truncated normal form (4). Taking into account the $$O$$-terms leads to the destruction of this torus, while a complicated invariant set close to it appears.

## Other cases

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