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

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

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

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):

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*

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.