significantly faster than the roll frequency. Consequently, the time for reversal is much shorter for the unsteady direction $n_{i}>0$, where the pitching is induced, than for the steady direction $n_{i}<0$, where the rolling is induced. We denote the time for reversal for the unsteady direction as $t_{ru}$ and that for the steady direction as $t_{rs}$ when we consider a specific spinning direction.

Results
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### General behavior for the parameter set GH

![\[fig:gh-t-n\] A typical spin evolution and the corresponding $\omega_{x}$ and $\omega_{y}$ for GH (Table \[tab:parameters\]). (a) The case of the initial spin in the unsteady direction. The initial condition is specified by Eqs. (\[eq:sim-ic-1\])–(\[eq:sim-ic-3\]) with $n_{i} = 0.1\,\tilde{\omega}$ and $|\omega_{xy0}|=0.01\,\tilde{\omega}$. (b) The case of the initial spin in the steady direction with $n_{i} = -0.1\,\tilde{\omega}$ and $|\omega_{xy0}|=0.01\,\tilde{\omega}$. The dashed lines in (a-1) and (b-1) show Garcia and Hubbard’s solution $\overline{n}(t)$ given by Eqs. (\[eq:gh-solution-p\]) and (\[eq:gh-solution-m\]), respectively.](5r.pdf){width="8cm"}

In Fig. \[fig:gh-t-n\] we show a typical simulation result of the time evolution of the spin $n(t)$ along with the angular velocities $\omega_{x}(t)$ and $\omega_{y}(t)$ for the parameter set GH (Table \[tab:parameters\]) in the case of the unsteady direction $n_i > 0$ (a), and the steady direction $n_i < 0$ (b).

