Consider the equation

for both positive and negative values of x. Find the equilibria as functions of a for values of a between -1 and 1. Draw a bifurcation diagram and describe in words what happens at a = 0. The change that occurs at a = 0 is called a saddle-node bifurcation.
When parameter values change, the number and stability of equilibria sometimes change. Such changes are called bifurcations and play a central role in the study of differential equations. The following illustrate several of the more important bifurcations. In each case, graph the value of equilibria as functions of the parameter value, using a solid line when an equilibrium is stable and a dashed line when an equilibrium is unstable. This picture is called a bifurcation diagram.

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dx dt