in turn changes the structure of the spirals. During this process, the random energy of individual stars in the system does [*not*]{} increase monotonically. In other words, local interactions between the non-steady arms and stars increase or decrease the total energy of individual stars locally; however, the energy remains around its value for circular motion with the occurrence of a small dispersion. This is because the interaction causes the migration of the guiding centers of the stars without increasing their eccentricity or random energy. This “dynamical cooling” mechanism is essential to prevent heating of the stellar disk and erasure of the spiral arms, and the mechanism produces “swarms” of stars moving between non-steady spirals. The non-linear epicycle motion of the stars and their non-linear coupling with the density perturbation is the fundamental physics of the recurrently formed, non-steady spiral arms in a stellar disk.

![image](fig9.eps){width=".90\textwidth"}

![image](fig10.eps){width=".90\textwidth"}

![image](fig11.eps){width=".8\textwidth"}

![image](fig12.eps){width=".8\textwidth"}

Discussion {#sec:Discussion}
==========

Figure \[fig:spiralproperties\] compares the morphological properties (the pitch angle $i$ and the amplitude $|A_{m}|$) of the stellar spiral arms between simulations and observations. For the comparison, we used a 2D fast Fourier transform (2D FFT) analysis with logarithmic spirals [for details, see for e.g., @PuerariDottori1992]. The plotted points in 