## Fixed Points of Autonomous Systems

In one dimension an autonomous system obeys an equationIf at a pointthe the pointis said to be a fixed point, labelledandfor any motion which starts at a fixed point.

One example is shown above,This motion has three fixed points, at These points may be classified as stable, or unstable.

The functionmay be expanded in a Taylor series about a fixed pointto givewith solution

whereIfthentends tofor any particle with initial position sinceand iftends tosince