Figure \[fig:gh-t-n\](a-1) shows that the spin $n$ changes its sign from positive to negative at $t_{ru}\approx 112\,\tilde{t}$, and Fig. \[fig:gh-t-n\](b-1) shows the spin $n$ changes its sign from negative to positive at $t_{rs} \approx 810\,\tilde{t}$. Garcia and Hubbard’s solutions $\overline{n}(t)$ of Eqs. (\[eq:gh-solution-p\]) and (\[eq:gh-solution-m\]) are shown by the dashed lines in Figs. \[fig:gh-t-n\](a-1) and (b-1), respectively; they are in good agreement with the numerical simulations.

The angular velocities $\omega_{x}$ and $\omega_{y}$ oscillate in much shorter time scale, and their amplitudes evolve differently depending on the spin direction. In the case of Fig. \[fig:gh-t-n\](a), where the positive initial spin reverses to negative, the amplitude of $\omega_{x}$ becomes large and reaches its maximum around $t_{ru}$; the amplitude of $\omega_{y}$ also becomes large around both sides of $t_{ru}$ but shows the local minimum at $t_{ru}$. Both $\omega_{x}$ and $\omega_{y}$ oscillate at the pitch frequency $\omega_{p} \approx
1.44\,\tilde{\omega}$. In the case of Fig. \[fig:gh-t-n\](b) where the negative spin reverses to positive, the situation is similar but the amplitude of $\omega_{y}$ reaches its maximum around $t_{rs}$, and $\omega_{x}$ and $\omega_{y}$ oscillate at the roll frequency $\omega_{r} \approx 0.602\,\tilde{\omega}$.

These features can be understood based on the analysis in the previous section as follows. The 