of the disk surface density using the polar coordinates $(R,\phi)$: $$\begin{aligned}
\frac{\Sigma(R,\phi)}{\Sigma_{\rm 0}(R)} 
    =  \sum_{m=0}^{\infty} A_m(R)\cos\{m[\phi-\phi_m(R)]\}.\end{aligned}$$ Here, $A_m$ and $\phi_m(R)$ denote the Fourier amplitude and phase angle for the $m$-th mode, respectively [e.g, @RixZaritsky1995]. Upon execution of the simulation, spiral patterns initially developed from the noise with an $e$-folding time of about 4 galactic rotations ($T_{\rm rot} \sim 4$), and subsequently, the patterns settled to a nearly constant level, consistent with the report by @Fujii+2011. While the global features were in the quasi-steady state, the local spiral features were not static. The most prominent mode always changed on a rotational time scale, and it also showed radial dependence.

![image](fig2.eps){width=".9\textwidth"}

An important dynamical feature of the time-dependent multi-arm spirals is that the spirals “co-rotate” with the galactic rotation. Figure \[fig:PatternSpeed\] shows the angular pattern speed of a dominant stellar spiral ($m=4$). The angular frequency is analyzed by the rate of change of phase $\phi_{\rm m=4}(R)$ during two galactic rotation periods. The pattern speed is not constant. In fact, the pattern speed decreases with radius in a manner similar to galactic rotation for any radii. This result shows that there is no “single” co-rotating point in the disk, in contrast to stationary 