Bifurcation with symmetry

The most thoroughly studied bifurcation problems in physics tend to have symmetries. This should be no surprise: The analysis of a linear eigenvalue problem for a partial differential equation is greatly simplified if separation of variables is possible, and what makes separation of variables possible is a symmetry. Also, as we shall see, the study of bifurcation problems with symmetry offers new mathematical features: rather than finding one branch of bifurcating solutions as in the examples studied above, we shall see that there are several possible ``patterns," and the study of the stability of bifurcating solutions offers an insight into the question which pattern is preferred by nature.

On the one hand, symmetry complicates the study of bifurcation. This is because the assumption of a simple eigenvalue as in the preceding two sections typically does not hold if symmetries are present: If there is one eigenvector, then there will be others obtained from it by applying symmetry operations, and the more symmetries there are, the larger the multiplicity of eigenvalues will typically be. So the center manifold has a larger dimension. On the other hand, the equations on the center manifold inherit the symmetries of the original problem, which makes them simpler to study than a generic set of equations would be.

The concrete example we choose to illustrate the main ideas is the Hopf bifurcation with O(2)-symmetry. Let us consider the Bénard problem of Example 13. As before, we confine attention to solutions which are periodic in x with a given period $2\pi/\alpha$. The problem is invariant under translations in the x-direction, that is, if

$$\Psi(x,t)=(u(x,t),v(x,t),p(x,t),T(x,t))$$ (131)

is a solution, then

$$T_\phi\Psi(x,t)=(u(x+\phi/\alpha,t),v(x+\phi/\alpha,t),p(x+\phi/\alpha,t), T(x+\phi/\alpha,t))$$ (132)

is also a solution. Note that $T_{2\pi}$ is the identity because of the imposed periodicity. Also, the equations are invariant under reflection $x\mapsto -x$:

$$R\Psi(x,t)=(-u(-x,t),v(-x,t),p(-x,t),T(-x,t)).$$ (133)

Abstractly, we say that the system

$$\dot u=A(\lambda)u+f(u,\lambda)$$ (134)

has O(2)-symmetry if there exist linear mappings $T_\phi$ and R such that  

$$T_{\phi+\psi}=T_\phi T_\psi,\quad T_0=T_{2\pi}=I,\quad R^2=I,\quad RT_\phi =T_{-\phi}R,$$ (135)

and

   $$\begin{eqnarray} A(\lambda)T_\phi&=&T_\phi A(\lambda),\quad A(\lambda)R=RA(\lamb... ...,\lambda)&=&T_\phi f(u,\lambda),\quad f(Ru,\lambda)=Rf(u,\lambda).\end{eqnarray}$$

We consider a Hopf bifurcation for such a system. Thus, we assume that at $\lambda=0$, there is an eigenvalue $i\omega_0$. Moreover, we assume there are two eigenvectors $a_1(\lambda)$ and $a_2(\lambda)$ related by reflection: $a_2(\lambda)=Ra_1(\lambda)$. Finally, we assume that  

$$T_\phi a_1(\lambda)=e^{i\phi}a_1(\lambda),$$ (136)

and consequently  

$$T_\phi a_2(\lambda)=e^{-i\phi}a_2(\lambda).$$ (137)

In the context of a problem like the Bénard problem, $a_1(\lambda)$ would involve an eigenfunction proportional to $\exp(i\alpha x)$, and $a_2(\lambda)$would involve an eigenfunction proportional to $\exp(-i\alpha x)$. The Bénard problem as discussed in Example 13 does not have Hopf bifurcations, but if additional physical effects are considered, Hopf bifurcations are possible. For example, the Bénard problem for a viscoelastic fluid is known to allow Hopf bifurcation [14,16].

Thus the eigenvalue $i\omega_0$ is double rather than simple, and we have two complex amplitudes z₁ and z₂:

$$u=z_1a_1(\lambda)+z_2a_2(\lambda)+\overline{z_1a_1(\lambda)}+\overline{z_2a_2 (\lambda)}+y.$$ (138)

We can go through center manifold reduction and Birkhoff normal form as before; we omit the details. The result is a coupled system for the complex amplitudes z₁ and z₂. This system must reflect the symmetries of the original problem. We note that $T_\phi$ corresponds to the transformation $(z_1,z_2)\mapsto (\exp(i\phi)z_1,\exp(-i\phi)z_2)$, while R corresponds to the transformation $(z_1,z_2)\mapsto (z_2,z_1)$. At cubic order, the most general equation which respects these symmetries and includes only resonant terms is

   $$\begin{eqnarray} \dot z_1&=&\sigma(\lambda)z_1+\beta_1(\lambda)\vert z_1\vert^2z... ...(\lambda)\vert z_2\vert^2z_2+\beta_2(\lambda)\vert z_1\vert^2 z_2.\end{eqnarray}$$

We can immediately identify two types of solutions for which (3.92) takes on exactly the same form as the standard Hopf bifurcation. First, we can set z₂=0 (or z₁=0). This satisfies the second equation identically and the first becomes  

$$\dot z_1=\sigma(\lambda)z_1+\beta_1(\lambda)\vert z_1\vert^2z_1.$$ (139)

From the previous section we know that there is a branch of periodic solutions, which is supercritical if ${\rm Re}\,\beta_1(0)<0$ and subcritical if ${\rm Re}\,\beta_1(0)\gt$. It can be shown that the full system has a branch of periodic solutions approximated by the solution to (3.93). This solution is a traveling wave, it has the property that

$$T_\phi u(t)=u(t+\phi/\omega).$$ (140)

If, instead of setting z₂=0, we set z₁=0, we find a traveling wave in the opposite direction, i.e.

$$T_\phi u(t)=u(t-\phi/\omega).$$ (141)

The second type of solutions is a standing wave which satisfies u=Ru, i.e, z₁=z₂. For this case, (3.92) simplifies to

$$\dot z_1=\sigma(\lambda)z_1+(\beta_1(\lambda)+\beta_2(\lambda))\vert z_1\vert^2z_1.$$ (142)

THis equation has a branch of periodic solutions which is supercritical if ${\rm Re}(\beta_1(0)+\beta_2(0))<0$ and subcritical if ${\rm Re}(\beta_1(0)+\beta_2(0))\gt$.

The strategy employed here is typical for bifurcation problems with symmetry. What we have done is restrict the type of solution we consider by imposing symmetries on it, in such a way that the restricted problem is of the same type as the bifurcation at a simple eigenvalue. We can then use the results for bifurcation at a simple eigenvalue to show the existence of a branch of periodic solutions. In general, of course, we have no guarantee that all bifurcating periodic solutions can be found in this fashion (for the specific situation of Hopf bifurcation with O(2) symmetry, this is the case, but for more complicated situations it may not be).

Not everything is as it would be without symmetries, however. We still have to consider the stability of the bifurcating solutions. Let us first consider the traveling waves. In (3.92), let $z_1=(R+v_1)\exp(i\omega t)$, $z_2= v_2\exp(i\omega t)$, where $R\exp(i\omega t)$ is the periodic solution, i.e. $\sigma(\lambda)+R^2\beta_1(\lambda)=i\omega$. The linearized system reads

   $$\begin{eqnarray} \dot v_1+i\omega v_1&=&\sigma(\lambda)v_1+\beta_1(\lambda)R^2(2... ... \dot v_2+i\omega v_2&=&\sigma(\lambda)v_2+\beta_2(\lambda)R^2v_2.\end{eqnarray}$$

The first equation is exactly as in the previous section, yielding a stable eigenvalue if the traveling wave is supercritical and an unstable eigenvalue if it is subcritical. However, the second equation yields the eigenvalue  

$$\sigma(\lambda)-i\omega+\beta_2(\lambda)R^2=R^2(\beta_2(\lambda)-\beta_1 (\lambda)).$$ (143)

For the traveling wave to be stable, it must therefore be supercritical, and, in addition, the real part of $\beta_2(\lambda)-\beta_1(\lambda)$ must be negative. Thus the traveling wave is stable if both of the following conditions hold

   $$\begin{eqnarray} {\rm Re}(\beta_1(\lambda))<0,\quad {\rm Re}(\beta_2(\lambda)-\beta_1(\lambda))<0.\end{eqnarray}$$

A consequence of these inequalities is that ${\rm Re}(\beta_2(\lambda)+\beta_1 (\lambda))<0$, which, as we saw above, means that the standing wave is also supercritical.

To study stability of the standing wave, we set

$$z_1=(R+v_1)\exp(i\omega t),\quad z_2=(R+v_2)\exp(i\omega t),$$ (144)

where now

$$i\omega=\sigma(\lambda)+R^2(\beta_1(\lambda)+\beta_2(\lambda)).$$ (145)

The linearized system is now

   $$\begin{eqnarray} \dot v_1+i\omega v_1&=&\sigma(\lambda)v_1+R^2\beta_1(\lambda)(2... ..._2+\bar v_2) +R^2\beta_2(\lambda)(v_2+v_1+\bar v_1).\nonumber\ &&\end{eqnarray}$$

With v=v₁+v₂, w=v₁-v₂, we can decouple this system into

   $$\begin{eqnarray} \dot v+i\omega v&=&\sigma(\lambda)v+R^2(\beta_1(\lambda)+\beta_... ...ambda)w+R^2\beta_1(\lambda)(2w+\bar w)-R^2\beta_2 (\lambda)\bar w.\end{eqnarray}$$

The first equation is just as before; it yields a stable eigenvalue if the bifurcation is supercritical and an unstable eigenvalue if it is subcritical. The second equation can be further simplified to

$$\dot w=R^2(\beta_1(\lambda)-\beta_2(\lambda))(w+\bar w).$$ (146)

This yields a zero eigenvalue associated with the translation symmetry of the problem, and in addition the eigenvalue $R^2(\beta_1(\lambda)-\beta_2(\lambda))$.For the stability of the standing wave, we therefore find the conditions  

$${\rm Re}(\beta_1(\lambda)+\beta_2(\lambda))<0,\quad {\rm Re}(\beta_1(\lambda)- \beta_2(\lambda))<0.$$ (147)

Note that these conditions imply that ${\rm Re}\,\beta_1(\lambda)<0$, i.e. that the traveling wave is supercritical.

We see from this that the simple criterion that supercritical branches are stable is valid only for ``simple" bifurcations and loses its applicability in bifurcations with symmetry. Supercriticality remains necessary for stability but is no longer sufficient. Indeed, in the example discussed here, both the traveling and standing wave had to be supercritical in order for either one to be stable. If both are supercritical, then one of them is stable, and the other is unstable, since (3.99) and (3.105) are mutually exclusive.

An example of Hopf bifurcation with O(2) symmetry arises in the viscoelastic Taylor-Couette flow problem (flow between concentric rotating cylinders), see [1]. This problem is mathematically very similar to the Newtonian Taylor-Couette flow with counterrotaing cylinders, which is discussed in depth in [2]. If the cylinders are long, they can be modeled as infinite, and we have symmetry under translation along the axial direction of the cylinder, and also symmetry under reflection in this direction. In flow visualizations, the traveling waves have the appearance of ``spirals," while the standing waves appear as ``ribbons."