1. The problem of automatic synchronization of triode oscillators was studied by Appleton† and van der Pol‡; it gives rise to the differential equation
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100025196/resource/name/S0305004100025196_eqn001.gif?pub-status=live)
where α, γ, ω, E, ω1 are positive constants such that α/ω, γ/ω, (ω − ω1)/ω are small and dots denote differentiations with respect to t. When these conditions are satisfied, it is easy to see that
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100025196/resource/name/S0305004100025196_eqn002.gif?pub-status=live)
is an approximate solution over a limited time for any b1, b2 chosen to fit initial conditions on v and ṿ provided that v and ṿ are not too large. If it is assumed that b1 and b2 vary slowly compared with ω1t, so that
can be neglected, and ḃ1, ḃ2 are comparatively small, the equations
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100025196/resource/name/S0305004100025196_eqn003.gif?pub-status=live)
where
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100025196/resource/name/S0305004100025196_eqnU001.gif?pub-status=live)
are obtained. These are sufficiently accurate for the discussion of most of the physical phenomena, and have been used in this form (or in the polar-coordinate form obtained by putting b1 = b cos ø, b2 = b sin ø) by various authors §. Solutions of (1) with period 2π/ω1 are obtained approximately by putting ḃ1 = ḃ2 = 0 in (3). The steady-state solutions of (1) other than those with period 2π/ω1 correspond to periodic solutions of (3). All other solutions converge to one or other of these types of solution.