Example 1.7.8.
Now consider the one-parameter family
\begin{equation*}
\frac{dy}{dt} = y^3 - \alpha y = y (y^2 - \alpha).
\end{equation*}
We will have an equilibrium solution at zero for all values of \(\alpha\) and two additional equilibrium solutions at \(\pm \sqrt{\alpha}\) for \(\alpha \gt 0\text{.}\) This type of bifurcation is a pitch fork bifurcation (Figure 1.7.9).
a parabola with a vertex at the origin and opeing to the right and a horizontal line on the horizontal axis