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?