712 Dr. B. van der Pol on Oscillation Hysteresis in a 



We shall now proceed to investigate the conditions of 

 stability of our four solutions i., ii., iii., iv. of (11) 

 separately. 



i. "/ = 0, b/ = Q. 



After substitution of these values in (14) we find as the 

 condition for which both amplitudes remain zero : 



-a >E I -io i Bii>'0 



and a 2 b 2 > 0, 



or, as Ei and En are both positive, 



a 2 <0 

 and V < 0. 



These inequalities are expressions for the fact that only 

 when the circuit conditions (resistances, retroaction, etc.) 

 are such that no oscillations are " possible " at all, can 

 the system be kept in the non-oscillatory state, from which 

 it may be concluded that, when oscillations are " possible " 

 at all, some form of oscillation (either ii., iii., or iv.) will 

 build up automatically. 



ii. a s 2 = J(2V-« 2 ), 



b*=i{W-K 2 ). 



This represents the case in which both coupling frequencies 

 would be present simultaneously. But the conditions of 

 stability here are from (14) easily found to be 



E^ + EiA^O 



| ..." . (15 a, b) 



Now for a s and b s to be possible at all we must obviously 



a, 2 > 



which relations are incompatible with (15b). We thus see 

 that the simultaneous occurrence of finite stationary oscilla- 

 tions of both the coupling frequencies represents an unstable 

 condition and can therefore not be realized in practice. 



