Ring2poles

Other configurations

Ring and two poles $$C_{nv}(R,\,2p)$$

In the graphs, the horizontal axis is $$\kappa_N$$, the vertical is $$\kappa_S$$.

$$\theta_0$$ is the colatitude of the ring (so $$\theta_0=\pi/2$$ lies on the equator); all are for the ring lying in the Northern hemisphere.

Lyapounov stable points in red elliptic ones in blue

The grey line in the figures for $$n\geq4$$ is the instability boundary given by the higher modes.

Contents:

n=2

θ0 = π/2 (= 1.57)

θ0 = 1.45

θ0 = 1.3

θ0 = 1.15

θ0 = 1.0

θ0 = 0.5

n=3

With $$|\kappa_j| \leq 10$$ (zoomed in)

θ0 = π/2 (= 1.57)

θ0 = 1.45

θ0 = 1.3

θ0 = 1.0

θ0 = 0.5

n = 3 with $$|\kappa_j| \leq 50$$ (zoomed out)

θ0 = π/2 ( = 1.57)

θ0 = 1.5

θ0 = 1.3

θ0 = 1.0

θ0 = 0.5

θ0=π/2 (= 1.57)

θ0=1.45

θ0=1.3

θ0=1.0

θ0=0.75

θ0=0.5

θ0=π/2 (= 1.57)

θ0=1.45

θ0=1.3

θ0=1.0

θ0=0.75

θ0=0.5

n=5

$$|\kappa_j|\leq 10$$

θ0 = π/2 (= 1.57)

θ0 = 1.45

θ0 = 1.3

θ0 = 1.0

$$|\kappa_j|\leq 50$$

θ0 = π/2 (= 1.57)

θ0 = 1.45

θ0 = 1.3

θ0 = 1.0

n=12

$$|\kappa_j|\leq 50$$

θ0 = π/2 (= 1.57)

θ0 = 1.4

$$|\kappa_j|\leq 100$$

θ0 = π/2 (= 1.57)

θ0 = 1.4