# Some Dynamic Aspects of a Sextic Galactic Potential in a Rotating Reference Frame
## 5 Periodic Motion
\begin{split}& x(\zeta)=x_{0}+n\Big{[}k_{1}V_{xy}(x_{0},y_{0}) \cos\zeta+((\gamma_{0}^{2}-V_{xx}(x_{0},y_{0}))k_{1}-2\Omega\gamma_{0}k_{2}) \sin\zeta\Big{]},\\ & y(\zeta)=y_{0}+n\Big{[}k_{1}((\gamma_{0}^{2}-V_{yy}(x_{0},y_{0} ))-2k_{2}\Omega\gamma_{0})\cos\zeta+k_{2}V_{xy}(x_{0},y_{0})\sin\zeta\Big{]},\\ & E=E_{0}+n^{2}\Bigg{[}\frac{k_{1}^{2}+k_{2}^{2}}{4}\gamma_{0}^{6 }-2\Omega k_{1}k_{2}\gamma_{0}^{5}+\frac{\gamma_{0}^{4}}{4}(k_{1}^{2}(4\Omega ^{2}+V_{xx}(x_{0},y_{0})-2V_{yy}(x_{0},y_{0})\\ &\quad\quad+k_{2}^{2}(4\Omega^{2}-2V_{xx}(x_{0},y_{0})+2V_{yy}(x _{0},y_{0}))-\frac{1}{4}\gamma_{0}^{2}(-k_{2}^{2}V_{xx}(x_{0},y_{0})+2((k_{1}^ {2}+k_{2}^{2})V_{yy}(x_{0},y_{0})\\ &\quad\quad-2k_{2}^{2}\Omega^{2})V_{xx}-k_{1}^{2}V_{yy}^{2}(x_{0 },y_{0})-4k_{1}^{2}\Omega^{2}V_{yy}(x_{0},y_{0})-3(k_{1}^{2}+k_{2}^{2})V_{xy}^{ 2}(x_{0},y_{0})))\\ &\quad\quad+\frac{(V_{xx}(x_{0},y_{0})V_{yy}^{2}(x_{0},y_{0})}{4}( k_{1}^{2}V_{yy}(x_{0},y_{0})+k_{2}^{2}V_{xx}(x_{0},y_{0})\Bigg{]},\end{split}  \tag{45}\]  
where \(k_{1}\) and \(k_{2}\) are arbitrary constants introduced instead of the old ones for the suitability, while \(\gamma_{0}\) is the frequency, and it takes the value  
\[\gamma_{0}=\sqrt{\frac{V_{yy}(x_{0},y_{0})+V_{xx}(x_{0},y_{0})+4\Omega^{2}\pm \sqrt{(V_{yy}(x_{0},y_{0})+V_{xx}(x_{0},y_{0})+4\Omega^{2})^{2}-4\mathcal{R}}}{ 2}}, \tag{46}\]where \(\mathcal{R}\) is given by \[\mathcal{R}=V_{xx}(x_{0},y_{0})V_{yy}(x_{0},y_{0})-V_{xy}^{2}(x_{0},y_{0}).\] (47) Based on the values \(V_{xx}(x_{0},y_{0}),V_{yy}(x_{0},y_{0}),V_{xy}(x_{0},y_{0})\), and \(\Omega\), there are single or double values of the frequency \(\gamma_{0}\). Let us clarify this point: If \(\mathcal{R}=V_{xx}(x_{0},y_{0})V_{yy}(x_{0},y_{0})-V_{xy}^{2}(x_{0},y_{0})>0\), this means \((x_{0},y_{0})\) is an extreme point for the effective potential in Equation (19). Thus, we have two frequencies if the condition \[\Omega^{2}>\frac{1}{4}[2\sqrt{\mathcal{R}}-V_{yy}(x_{0},y_{0})-V_{xx}(x_{0},y_{ 0})],\] (48) is verified. This condition is always satisfied when \((x_{0},y_{0})\) is the minimum value of the effective potential. Thus, there are two values for the frequency. Consequently, there are two periodic families of orbits depending on the parameter \(n\) about the stable minimum point of the effective potential. 2. If \(\mathcal{R}=V_{xx}(x_{0},y_{0})V_{yy}(x_{0},y_{0})-V_{xy}^{2}(x_{0},y_{0})<0\), then the point \((x_{0},y_{0})\) is a saddle point for the effective potential in Equation (19). There is a unique value for the frequency \(\gamma_{0}\) which is obtained from Equation (46), which takes the positive sign. Hence, there is exactly one periodic orbit around the saddle point for the equilibrium point. We outlined the obtained results graphically by introducing certain values for the included parameters. The equilibrium point \(\mathcal{E}_{0}=(0,0)\) is a stable minimum critical point for the effective potential in Equation (19) if we select \(\Omega=0.5,n=10^{-5},k_{1}=0.5\), and \(k_{2}=1.5\). For these values, Equation (46) gives two real frequencies: \(\gamma_{0}=1.5\) and \(\gamma_{0}=0.5\). Thus, there are two periodic orbits close to the equilibrium point \(\mathcal{E}_{0}\), as outlined in Figure 3(a). For the equilibrium point \(\mathcal{E}_{1,2}\), we choose \(\Omega=1.5,b=0.2\), which guarantees the existence of this equilibrium alongside \(n=10^{-5},k_{1}=0.5\). There are two possible choices for the parameter \(d\):  
* If \(d=0.1\), then the equilibrium point \(\mathcal{E}_{1,2}\) is a saddle point for the effective potential since \(\frac{d}{\mathbb{p}}=0.5<1\). Equation (46) gives a single frequency \(\gamma_{0}=3.688455\). Thus, one periodic orbit is close to \(\mathcal{E}_{1,2}\), as outlined in Figure 3(b). * If \(d=0.4\), then \(\frac{d}{\mathbb{p}}=2>1\), and so the equilibrium point \(\mathcal{E}_{1,2}\) is a stable minimum critical point for the effective potential in Equation (19). Equation (46) provides two real values of the frequency: \(\gamma_{0}=3.85078\) and \(\gamma_{0}=0.649219\).