Summary of non-linear systems

Flows on the line

1.   General form : plot in phase-plane (  vs  ).

2.   Equilibria occur when  a  they may be stable (  ), unstable (  ) or half-stable (  ).

3.   Define Potential, .

4.   Consider polynomial potentials up to 4^th order:

a.      harmonic

b.     limit point instability

c.      transcritical bifurcation

d.     pitchfork bifurcation

e.      cusp catastrophe

5.   Abstract to  control-space diagram and bifurcation set.

6.   No oscillation possible

Flows on the circle

7.   “To and fro” oscillations (librations) vs “over the top” oscillations (rotations).

8.   Overdamped limit:  

9.   Two equilibria  one stable and one unstable.