# 4.5. Oscillations and Limit Circle¶

Vacuum oscillations are observed because the phase portrait of states forms neutrally stable limit circles.

To visualize this we use the flavor isospin picture. The equation of motion for flavor isospin is

$\dot{\mathbf s} = \mathbf s \times \mathbf H.$

Suppose we could plot out $$\dot{\mathbf s}$$ vs $$\mathbf s$$, we could see that these vectors are periodic. The reason is that given a $$\mathbf s$$ we know $$\dot{\mathbf s}$$ is perpendicular to $$\mathbf s$$. To discretize the process, we move our $$\mathbf s$$ to next step according to $$\mathbf s \to \mathbf s + \dot{\mathbf s}$$.

Fig. 4.16 Limit circle in flavor isospin space for vacuum oscillations. Such a limit circle tells us that the state is going circular in this space thus oscillations.

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