spiral waves (or bar) with a rigid rotation. Therefore, no single spiral can last for more than one rotational period (see Section \[sec:StarAroundSpiralArm\]). @Wada+2011 and @Grand+2012 have also reported that the spiral arms in their simulations are co-rotating, winding and short-lived. Observationally, the phenomenon of co-rotating spiral arms has been confirmed in at least a few of the nearby spiral galaxies such as M51, NGC 1068, M101, IC 342, NGC 3938, and NGC 3344 using the Tremaine-Weinberg (TWR) analysis [@Merrifield+2006; @Meidt+2008; @Meidt+2009].

It is noteworthy that the pattern speed curve seems slightly flatter (particularly when $R \sim 10-15$ kpc) than the circular rotation curve. In order to investigate the cause of the flattening, we divided this period into five periods and performed the same analysis for each period (Figure \[fig:PatternSpeedAppendix\]). The pattern speeds in the all periods decrease with increasing radius in a manner similar to galactic rotation for any radii; however, they show slightly flatter distributions at certain instants (e.g., $T_{\rm rot}= 12.4-12.8$). This flatter distribution lasts for less than one galactic rotation period ($\Delta T_{\rm rot}<1$). Thus, this structure is not a long-lasting structure.

![image](fig3.eps){width=".9\textwidth"}

![image](fig4.eps){width=".9\textwidth"}

Further, it is significant that the gravitational scattering of stars by the 