1. van der Pol Oscillator Appendix J
Triode oscillator (limit cycle):  < 0 for small q
Normalized version: →
Fixed point:

2. At fixed point (0,0): → →
(0,0) is a spiral repellor for R < 2
a repellor for R > 2
Energy:
Set
= average input power - average power dissipation
= 0 for steady state

3. Small R, sinusoidal oscillation:→
Steady state:
Circle of radius Q0 in state space
R = 3 relaxation oscillation
R = .1 Sinusoidal oscillation

4. Stability of the Limit Cycle
Method of slowly varying amplitude & phase
(KrylovBogoliubovMitropusky averaging method)
Ansatz for the case of small RHS:
and →
(1) →
(2)
(1),(2):

5. As trajectory nears limit cycle, a,Φ~ const during one cycle.
Replace RHS’s by their averages over a cycle:
where
Limit cycle is reached when
i.e.,
Rate of approach to the l.c.
Set →
Set Poincare section = positive Q axis →
Setting
gives a Poincare map
near a*

6. For R < 1
Slope at fixed point a* ,i.e., Q = 2 √R , is e-2πR < 1.
Fixed point (Q,U) = (0,0) is repulsive → slope at Q = 0 is > 1.