Ring

Other configurations

A ring of n point vortices

Region of Stability

for $\theta\in[0,\pi]$ the co-latitude, the ring is stable if $$\begin{array}{r:l}n=3 & \forall \theta\cr n=4 & \cos^2\theta > 1/3 \cr n=5 & \cos^2\theta > 1/2 \cr n=6 & \cos^2\theta > 4/5 \end{array}$$

For $n\geq 7$ the ring is always unstable (there are real eigenvalues)

Bifurcations

Where do bifurcations occur?

The eigenvalues of the $\theta$-part of the reduced Hessian are (for $\mu\neq0$ when $\ell=1$ as reduced space is smaller) $$\lambda^{(\ell)} = \frac{n}{2\sin^2\theta_0}[-(n-\ell-1)(\ell-1) + (n-1)\cos^2\theta_0].$$ The $\phi$-part is positive definite. Therefore a bifurcation occurs when $$\cos^2\theta_0 = \frac{1}{n-1}(n-\ell-1)(\ell-1).$$ (Recall $\mu=n\cos\theta_0$.)

On the $\ell=1$ mode the Hessian is positive definite.

Since $2\leq\ell\leq[n/2]$, and $\cos^2\theta_0<1$, we have that bifurcations with an eigenvalues passing through 0 only occur for

$\ell=2:\; n\geq 4$
$\ell=3:\; n=6$
$\ell\geq4$ never

Note: For $\ell\geq4$ (so $n\geq8$), the $\theta$-part of the Hessian is always negative definite and so the eigenvalues of the linear system are all real (and non-zero).
Also, for $n=7,\,\ell=3$ the expression simplifies to $\lambda^(3) = -21$ (independent of $\theta$), and again the eigenvalues of the linear system are all real.

Bifurcation types:

For $\ell=n/2$ and n even, a 2-dimensional mode, the action has kernel $\mathbb{Z}_{n/2}$ so is effectively a rep of $\mathbb{Z}_2\times \mathbb{Z}_2$ ($=D_n/\mathbb{Z}_{n/2}$). The quadratic invariants are therefore $(\delta\phi)^2+\lambda (\delta\theta)^2$ which has imaginary or real eigenvalues (it's 2-dimensional so there are no other possibilities anyhow). This gives rise (generically) to a $\mathbb{Z}_2$-pitchfork bifurcation in the $\theta$-direction. (The genericity condition is a non-zero coefficient in $(\delta\theta)^4$.)
When the mode is 4-dimensional (in particular $\ell=2$), the $\phi$-space and the $\theta$-space are invariant and Lagrangian (ie the symplectic rep is of the from $V\oplus V$, with V abs. irred.). The quadratic invariant is therefore $|p|^2+\lambda|q|^2$, which is stable when $\lambda\gt0$ and has all eigenvalues real when $\lambda\lt0$. The bifurcation is a $D_{n'}$-pitchfork, where $n'=n$ if n is odd, and $=n/2$ if n is even. What is the genericity condition? (And which type of pitchfork occurs when $n'=4$?) See here for pictures

Summary:

$n=3$: no bifurcations (only have $\ell=1$ mode, and eigenvalues are always imaginary).

$n=4,\; \ell=2$: The ring loses stability at $\cos^2\theta=1/3$ through this mode, which is 2-dimensional. On this mode, the D₄ action factors through one of $\mathbb{Z_2\times Z_2}$ (as rotation by $\pi$ acts trivially on this mode), so it's a reducible representation, and one pair of the eigenvalues passes through zero, becoming real. This suggests a $\mathbb{Z}_2$-pitchfork bifurcation. The broken symmetry is the rotation by $\pi/2$.

$n=5,\; \ell=2$: a 4 dimensional mode, with positive definite Hessian for $\cos^2\theta_0 \gt 1/2$. When $\cos^2\theta_0$ passes through 1/2, the system undergoes a $D_5$-pitchfork bifurcation.

$n=6$:

$\ell=3$: a 2 dimensional mode, with positive definite Hessian for $\cos^2\theta_0>4/5$ and the linearization has real eigenvalues if $\cos^2\theta<4/5$. is it sub- or super-critical?
$\ell=2$: a 4 dimensional mode, with positive definite Hessian for $\cos^2\theta_0>3/5$. When $\cos^2\theta_0$ passes through 3/5, the system undergoes a $D_3$-pitchfork bifurcation (such bifurcations are generically transcritical).

$n=7,\;\ell=2$: a 4 dimensional mode, positive definite if $\cos^2\theta_0>2/3$ When $\cos^2\theta_0$ passes through 2/3, the system undergoes a $D_7$-pitchfork bifurcation. What type?

$n=8,\;\ell=2$: a 4 dimensional mode, positive definite if $\cos^2\theta_0>5/7$. When $\cos^2\theta_0$ passes through 5/7, the system undergoes a $D_4$-pitchfork bifurcation. Which type? Transcritical or sub-/super-critical?

$n=9,\; \ell=2$: a 4 dimensional mode, positive definite if $\cos^2\theta_0>3/4$ When $\cos^2\theta_0$ passes through 3/4, the system undergoes a $D_9$-pitchfork bifurcation. What type?