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



Whether b 2 has the tendency to build up when once 

 a/ = a 2 and b 2 = 0, is seen from (9 b), which can be 

 written 



d\ogb 2 



— j t — =E) n (6o 2 ~20, 



which shows that log b 2 or b 2 itself will only increase when 

 cd 2 2 has been given such a value that 



b 2 ~2a 2 >0. 

 This is the case when 



co 2 2 = OW (fig. 5). 



We can therefore bring co 2 2 from a value o> 2 2 < ^i 2 through 

 resonance (co 2 2 = a) 1 2 ) to a value a> 2 2 = OW>G)i 2 while all 

 the time the system continues to vibrate in the first mode 

 only. But as soon as w 2 has reached the value OW where 

 the square, of the amplitude which would obtain if the system 

 vibrated in the second mode only equals twice the square of 

 the amplitude of the vibrations in the first mode actualty 

 present (b 2 — 2a 2 or TV = 2TU), then the oscillation sud- 

 denly jumps from the first mode to the second. A further 

 increase of co 2 2 to any bigger value will leave the second 

 mode only present. In the same way bringing (o 2 back 

 from a big value through resonance to a smaller value 

 results in the second mode being present up to the point P 

 where 2b 2 = a 2 . It is therefore clear that the mode of 

 vibration once obtained persists up to the point where it is 

 no longer stable (not, as occasionally stated in a linear treat- 

 ment, where it is no longer possible) though a region may 

 have been traversed where the other mode would separately 

 be stable, and thus the hysteresis effect found experimentally 

 can be explained by theory. 



Another experimental fact can further be found theo- 

 retically. When one branch of the primary circuit, such 

 e. g. as the grid circuit, is first open and thereupon the 

 circuit is closed, then it is found that in general when 

 a> 2 2 < w{ 2 only vibrations of frequency &>i build up, while, 

 when co 2 2 > ft)! 2 , the system starts vibrating in the second 

 mode. 



The initial conditions here are therefore 



for t = Q, a 2 = b 2 ^Q. 



Now (9) may be integrated graphically for these initial 

 conditions, and fig. 6 is the result for a special case where 

 G> 2 2 < ft?! 2 and therefore a 2 > b$ 2 . This figure shows clearly 



