**Example:** Consider the autonomous equation

with parameter *a*.

**1.**- Draw the bifurcation diagram for this differential equation.
**2.**- Find the bifurcation values, and describe how the behavior of the solutions changes close to each bifurcation value.

**1.**- First, we need to find the equilibrium points (critical points). They are found by setting . We get
.

This a quadratic equation which solves into

,

we have

- if , then we have two equilibrium points;
- if , then we have one equilibrium point;
- and if , then we have no equilibrium points.

This clearly implies that the bifurcation occurs when , or equivalently , which gives . The bifurcation diagram is given below. The equilibrium points are pictured in white, red colored areas are areas with "up" arrows, and blue colored areas are areas with "down" arrows.

**2.**- The bifurcation values are
*a*= 4 and*a*= -4. Let us discuss what is happening around*a*=4 (similar conclusions hold for the other value):- Left of
*a*=4: no equilibrium; - At
*a*=4: we have a node (up), i.e. attractive from below and repelling from above (look at the bifurcation diagram); - Right of
*a*=4: we have two equilibria, the smaller one is a sink, the bigger one is a source, which explains the node behavior of .

- Left of

**
**

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