Here is a Pitchfork bifurcation animation.  The equation is (dy/dt)=a y -(y^3)  for a∈[-1,1].  At left is a graph of many solutions for that particular value of the parameter.  At right is the graph of the function f(y)=a y -(y^3) .  Notice that the graph of f(y) crosses the horizontal axis at only one point for a≤0 and at three points for a>0.  Thus, 0 is a bifurcation value for the parameter and one can see at left that the equilibrium at 0 is asymptotically stable for a≤0 and becomes unstable for a>0, while the two new equilibria are asymptotically stable.

Manipulate[{StreamPlot[{.1,a*y-y^3},{x,-3,3},{y,-3,3}],Plot[a*y-y^3,{y,-2,2},PlotRange->{{-2,2},{-5,5}}]},{a,-2,2, .01,Appearance->"Open"},FrameLabel->Text[Style["Phase Portrait and graph of dy/dt = ay-y^3",FontSize->18,FontFamily->"Times",Bold]]]