Fixed Points of Autonomous Systems
![](/university-physics/non-linear-dynamics/fixed-points-of-autonomous-systems-html-m5bb0fe1d.gif)
![](/university-physics/non-linear-dynamics/fixed-points-of-autonomous-systems-html-71d11c60.gif)
![](/university-physics/non-linear-dynamics/fixed-points-of-autonomous-systems-html-73d2e1c0.gif)
![](/university-physics/non-linear-dynamics/fixed-points-of-autonomous-systems-html-22b6ae22.gif)
![](/university-physics/non-linear-dynamics/fixed-points-of-autonomous-systems-html-7db520a3.gif)
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 point
to give
with solution
whereIf
then
tends to
for any particle with initial position
since
and if
tends to
since