I am currently working on analyzing the Flip bifurcation using Jury's condition. I derived the following condition for the bifurcation to not occur:
\begin{equation} \frac{1}{2(-4+b^2)}\left(-2(b^2(-2+c)+4(2+c))+a(8\alpha+2b^2\alpha+b^3(-1+\alpha)\alpha+4b(1-3\alpha+2\alpha^2))\right) < 0 \end{equation}
Since this inequality is quite complex, I differentiated it with respect to α to obtain:
\begin{equation} a(8-12b+2b^2-b^3)+a(16b+2b^3)\alpha < 0 \end{equation}
However, when I tried to find the range of α, I obtained a negative result:
\begin{equation} \alpha < -\frac{a(8-12b+2b^2-b^3)}{a(16b+2b^3)} \end{equation}
I have never seen such a result in other papers, so I think I might have made a big mistake. Could you please help me figure out what I did wrong and how to proceed?