# Some Dynamic Aspects of a Sextic Galactic Potential in a Rotating Reference Frame
## 4 Stability Analysis
### Linear Stability
The two conditions \(M\geq 0\) and \(N\geq 0\) are satisfied together if  
\[1<\mu\leq\frac{3-7\Omega^{2}}{\Omega^{2}-1}. \tag{30}\]  
To guarantee the correctness of the last inequality, we find the possible values of \(\Omega\) such that \(1\ <\frac{3-7\Omega^{2}}{\Omega^{2}-1}\) is verified. Thus, we have \(\Omega\in]\frac{1}{\sqrt{2}},1[\). Let \(f(\mu)=M^{2}-4N\) and \(I=]1,\frac{3-7\Omega^{2}}{\Omega^{2}-1}]\). Direct calculations show that \(\frac{df}{d\mu}<0\) for all \(\mu\in I\), and  
\[f(1)=16(2\Omega^{2}-1)^{2}>0,\ \ f(\frac{3-7\Omega^{2}}{\Omega^{2}-1})=64(2 \Omega^{2}-1)(\Omega^{2}-1)<0.\]  
Thus, the polynomial \(f(\mu)\) has a unique root \(\mu_{1}=\frac{5-\Omega^{2}-8\Omega\sqrt{1-\Omega^{2}}}{1-\Omega^{2}}\) in the interval \(I\). Hence, the equilibrium position \(\mathcal{E}_{1,2}\) is a linearly stable maximum if  
\[d<b<0,\ \ \ \Omega\in]\frac{1}{\sqrt{2}},1[,\ \ \ 1<\frac{d}{b}\leq \frac{5-\Omega^{2}-8\Omega\sqrt{1-\Omega^{2}}}{1-\Omega^{2}}. \tag{31}\]Analogous procedures can be applied to investigate the linear stability of the equilibrium position \(\mathcal{E}_{3,4}\). The regions of linear stability of the equilibrium point \(\mathcal{E}_{1,2}\) are delimited by the two straight lines \(b=d\) and \(d=\mu_{1}(\Omega)b\), where \(\mu_{1}(\Omega)\) is the slope of the second line as well as the region in which the equilibrium positions \(\mathcal{E}_{1,2}\) are the maximums, as determined by the two lines \(d=0,b=d\). Hence, the size of the region of linear stability relies on the value of the angular velocity \(\Omega\). Figure 2 outlines the zone of linear stability in yellow, which grows as the value of the angular velocity \(\Omega\) increases, and it reaches the maximum size when \(\Omega\to 1\) (i.e., it covers all the regions of the maximum positions in gray for \(\mathcal{E}_{1,2}\)). This means the force resulting from the rotating frame controls the regions of stability and instability. On another side, if at least one condition in Equation (31) is not fulfilled, then the equilibrium position \(\mathcal{E}_{1,2}\) becomes Lyapunov unstable. The same discussion and similar graphs can be constructed for the equilibrium point \(\mathcal{E}_{3,4}\) using \(c,a\) instead of \(d,b\). Figure 2: Regions of stability and instability for the equilibrium position \(\mathcal{E}_{1,2}\) in the plan of the parameters \(b,d\)  
For each \(\mu\in]-\frac{1}{3},1[\), all the quantities \(1+\mu,1+3\mu\), and \(3+5\mu\) are positive, and the map \(g(\mu)=\frac{2(1+\mu)}{3+5\mu}\) is decreasing on \(\mu\in]-\frac{1}{3},1[\). For a fixed \(\mu\), the quantity \(M\geq 0\) if and only if \(\Omega^{2}\geq\frac{2(1+\mu)}{3+5\mu}=g(\mu)\). Direct computations show the following: * At the fixed value \(\mu_{1}\), we have \(\Omega^{2}<g(\mu_{1})\) for all \(\Omega\in[0,1[\). Therefore, for each \(\mu\in]-\frac{1}{3},\mu_{1}]\) and \(\Omega\in[0,1[\), we have \(\Omega^{2}<g(\mu_{1})\leq g(\mu)\), and then \(M<0\). Therefore, we must also exclude the interval \(]\frac{-1}{3},\mu_{1}]\) from the possible values for \(\mu\). * At the fixed value \(\mu_{2}\), we have \(\Omega^{2}\geq g(\mu_{2})\) for each \(\Omega\in]\frac{1}{\sqrt{2}},1[\). Therefore, for each \(\mu\in[\mu_{2},1[\) and \(\Omega\in]\frac{1}{\sqrt{2}},1[\), we have \(\Omega^{2}\geq g(\mu_{2})\geq g(\mu)\), and then \(M\geq 0\). The case where \(\Omega^{2}=\frac{1}{\sqrt{2}}\) yields \(\mu_{2}=1\) and must be excluded. * At the fixed point \(\mu=1\), we have \(\Omega^{2}<g(1)\) for \(\Omega\in]0,\frac{1}{\sqrt{2}}[\). Therefore, for each \(\mu\in]\mu_{2},1[\) and \(\Omega\in]0,\frac{1}{\sqrt{2}}[\), we have \(\Omega^{2}<g(1)<g(\mu)\), and then \(M<0\). Therefore, we must also exclude the interval \(\Omega\in]0,\frac{1}{\sqrt{2}}[\) from the possible values of \(\Omega\). Note that even though \(\mu=1\) is not included in the interval under consideration, \(g(\mu)\) is continuous at \(\mu=1\), so we can study the relation between \(\Omega\) and \(\mu\) when \(\mu\) tends toward \(1\).