Summary of non-linear systems
Flows on the line
1.
General form :
plot in phase-plane (
ˆàvs
ˆà).
2.
Equilibria occur when ˆàa
ˆàthey may be stable (
ˆà), unstable (
ˆà) or half-stable (
ˆà).
3.
Define Potential, .
4. Consider polynomial potentials up to 4th order:
a.
ˆà
ˆàharmonic
b.
ˆà
ˆàlimit point instability
c.
ˆà
ˆàtranscritical bifurcation
d.
ˆà
ˆàpitchfork bifurcation
e.
ˆà
ˆàcusp catastrophe
5. Abstract to ˆàcontrol-space diagram and bifurcation set.
6.
No oscillation possible
Flows on the circle
7. òÀÜTo and froòÀÝ oscillations (librations) vs òÀÜover the topòÀÝ oscillations (rotations).
8.
Overdamped limit:
9.
Two equilibria ˆàone stable and one unstable.