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Energy fluctuations in a van der Pol oscillator
E N E R G Y FLUCTUATIONS I N A VAN DER POL OSCILLATOR BY
N. MINORSKY, Ph.D.1 1. INTRODUCTION
B y analogy with t h e differential equation of t h e harmonic oscillator + x = 0
(1.1)
- x~)~ + x = 0
(1.2)
o n e c a n c o n s i d e r t h e equation -
~(1
as representing a special, v a n der Pol oscillator (abbreviation V D P ) . The important feature o f t h e h a r m o n i c oscillator is t h e fact t h a t (1.1) has t h e first integral ½z~ + ½x~ = h, (1.3) w h i c h expresses t h e law of conservation of energy (we assume here for t h e sake o f simplicity m -- c = w0 -- 1 in t h e u s u a l form of t h e equation m:~ + cx = 0 o f t h e harmonic oscillator). A l t h o u g h t h e V D P oscillator does n o t possess this feature, t h e introduction o f e n e r g y as a d e p e n d e n t variable is of a certain i n t e r e s t in applications a n d this will form t h e s u b j e c t of this note. If one uses t h e phase-plane representation (5 = y), (1.3) is x 2q_y2 = r2 = p = 2h.
(1.4)
That is, r~ = p represents twice t h e t o t a l energy o f t h e harmonic oscillator a n d , moreover, this energy is c o n s t a n t t h r o u g h o u t t h e cycle. I n o r d e r t o simplify t h e writing we will assume t h a t r2 = p = 2h is t h e t o t a l energy of t h e oscillator t h u s disregarding t h e factor 2. Equation 1.1 written as a s y s t e m o f two e q u a t i o n s is 2 =y;
9 -
--x.
Transforming t h e s e e q u a t i o n s into p, 0 coordinates one g e t s : do 0; dt =
dO dt -
1.
(1.5)
Eliminating time between t h e s e e q u a t i o n s o n e o b t a i n s t h e integral curve (or "trajectory") do dO
0.
(1.6)
J Department of Electrical Engineering, Stanford University, Stanford, Calif.
205
206
N . MINORSKY
[J. F. 1.
The first equation (1.5) expresses t h e law o f conservation of energy ; (1.6) represents a circle in t h e p, 0 plane a n d t h e s e c o n d equation (1.5) gives 0 = - t, as t h e c o n s t a n t of integration c a n obviously b e a s s u m e d t o b e zero b y a p r o p e r choice of t h e origin of either 0 o r t. The sign m i n u s appearing in this equation is o f n o special significance a n d arises m e r e l y from t h e fact t h a t t h e angles are c o u n t e d as positive in t h e trigonometric sense (counterclockwise) w h i l e positive direction o n t h e integral curve is clockwise. If o n e applies this procedure t o (1.2) one o b t a i n s similar e q u a t i o n s d__o# = 2~(1 - p Cos2 0)0 Sin2 O,
(1.7)
dt
dO
d~ =
dp dO
- 1 + ~(1 - p Cos2 O) Sin 0 Cos O,
(1.8)
2~(1 - p Cos2 O) p Sin2 0
(1.9)
~(1 - p C o s 20) S i n 0 C o s 0 - 1
T h e s e e q u a t i o n s reduce t o E q s . 1.5 a n d 1.6, respectively, when ~ = 0. Since it is k n o w n t h a t t h e V D P oscillator is periodic, its integral curve is closed, t h a t is
f o 2" dp dO = f o2~ do = 0.
(1.10)
If, moreover, e << 1 t h e V D P oscillator differs b u t l i t t l e from a h a r m o n i c oscillator so t h a t o(0) ~ p0 = C o n s t . In this case neglecting t h e s m a l l term with e in t h e denominator of (1.9) o n e c a n w r i t e
dp
d--0 -~ - 2e(1 - p0 Cos~ 0)o0 Sin s 0, a n d t h e condition (1.10) o f periodicity gives
f:-
do = - 2e po
Sins OdO
--
po ~
f:-
Cos20 Sins OdO
]
= 0,
(I.11)
whence p0 = r0~ = 4, which is well k n o w n (1).~ If, however, e = 0, condition (1.10) is satisfied identically in (1.11) a n d o n e has no r i g h t t o e q u a t e t h e square b r a c k e t in (1.11) t o zero. In this case o0 c a n n o t b e determined from (1.11) which is also o b v i o u s since in this case t h e V D P oscillator degenerates into a h a r m o n i c oscillator w h o s e energy c o n t e n t is a r b i t r a r y d e p e n d i n g o n t h e initial c o n d i t i o n s : F o r a V D P oscillator, o n t h e contrary, t h e initial conditions u l t i m a t e l y d o n o t play a n y role a n d t h e oscillator "selects," so t o speak, a definite energy cont e n t o0 = 4 as long as ~ << 1 b u t n o t zero. So far we were able m e r e l y t o retrace some well-known f a c t s , 2 The boldface numbers in parentheses refer to the references appended to this paper.
Sept., I949.]
FLUCTUATIONS
I N A VAN DER POL
OSCILLATOR
starting with a n e w d e p e n d e n t variable p, t h e e n e r g y . we shall e n d e a v o r t o elaborate this line of argument.
20 7
In what follows
2. PHASE TRAJECTORY
W e will c o n s i d e r first t h e differential equation (1.9) of t h e p h a s e trajectory assuming e << 1. Expanding t h e r i g h t h a n d of (1.9) into a series o n e gets do
dO
-
2pl-e(1 - p Cos2 0) Sin"- 0 + d(1 - p Cos"- 0)2 X X S i n 3 0 C o s 0 + d(1 - p Cos"- 0)3Sin4 0 Cos'-' 0 + . . . ] ,
(2.1)
w h e r e p = p(O) is a n u n k n o w n periodic function of 0 w h i c h we will e n d e a v o r t o determine b y t h e perturbation method. If one s e t s o = p0 + ep~ + do2 + ".- in t h e preceding e q u a t i o n , o n e o b t a i n s a series of successive approximations arranged according t o t h e o r d e r of e.
(a) Approximation of order zero dpo -0; dO
oo = K0 = C o n s t .
(b) First approximation From (2.1) one has
dpl dO
2(1 - po Cos"- O)po Sin s 0,
whence p~ = K1 - po(1 - -~po)O + ,~po Sin 20 - -~p0"- Sin 40,
(2.2)
K1 being a n integration c o n s t a n t . The s e c o n d term o n t h e r i g h t of (2.2) is clearly a secular term a n d c a n n o t exist in a stationary s t a t e . Determining t h e c o n s t a n t p0 so as t o eliminate this term o n e o b t a i n s a g a i n p0 = 4 so t h a t (2.2) becomes m = K I + 2 Sin 20 - Sin 40.
(2.3)
The c o n s t a n t K1 is determined in tile next approximation.
(c) Second approximation Equating t h e t e r m s with ~2 in (2.1) one o b t a i n s dp.z
dO
-
-
2pt Sin2 0 + 4pip0 Sin2 0 Cos" 0 - 2p0 Sin a 0 Cos 0 + + 4p02 Sin~ 0 Cos~ 0 - 2pd Sin3 0 Cos~ 0.
Replacing pt a n d p0 b y t h e i r v a l u e s , p a s s i n g t o t h e m u l t i p l e arguments, integrating, a n d eliminating t h e s e c u l a r term (which yields K1 = 0)
208
N. MINORSKY
[J. F. I.
one gets 3 1 .~ 1 p2 = K 2 - ~ C o s 2 0 + ~ C o s 4 0 + l ~ C ° s 6 0 - ~ C ° s 8 0 '
(2.4)
where K~ is a constant of integration.
(d) Third approximation One obtains K2 = ~ and
1 5 Sin 40 + 23 37 Sin 80 p3 = - - - S i n 2 0 - 192 ~ Sin 60 - 192 12 -485Sin100+~4Sin120+K3"
(2.5)
If one limits the approximation up to the terms of the order ,4, the periodic function p(0) is then given by the following expression arranged by the orders of ~: p( O) = 4 +
+ ,(2 Sin 20 - Sin 40)+ +~
~-~Cos20+~Cos40+
+,3 K 3 -
'
)
Cos 60 - ~ Cos 80 +
1 5 23 37 ~ S i n 2 0 - 192 Sin 40 + ~-~ Sin 60 - --192 X
(2.6)
X Sin 80 - 5 Sin 100 + 5Sin 120) + ~
-
•
•
..
If the terms of this array are arranged according to the order of harmonics one gets p(O) = (4+-~2+ .. . ) + + [ ( 2 ~ - - 1 e3+ ...
) Sin 2 0 + ( - ~3 ~2+. • . ) Cos 2O]+
+[(-~-1-~3+'")Sin40+(1~+""
) Cos 40]+
(2.7)
) Sin 6 0 + ( 5 , 2 + ... ) Cos 60]+
+ Jr-
[ ( ,_~_@~3+... 7 •
•
) S i n S 0 + ( - ~1, ~ + . . . ) Cos 80]+
..
The non-written terms in brackets are ascending powers of e. Unfortunately, the calculations beyond the third approximation are so
Sept., I949.]
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I N A VAN DER POL
OSCILLATOR
209
long t h a t i t b e c o m e s i m p r a c t i c a b l e t o c o n t i n u e t h e m u n l e s s , p e r h a p s , with the aid of s o m e m e c h a n i c a l c o m p u t e r s . I t is p o s s i b l e , h o w e v e r , t o form c e r t a i n g e n e r a l c o n c l u s i o n s r e g a r d i n g the s e r i e s (2.7). 1. T h e coefficients ao, al, bl, as, b2 . . • of the t r i g o n o m e t r i c t e r m s in (2.7) a r e p o w e r s e r i e s in ~ a p p r o a c h i n g zero w h e n E --* 0, e x c e p t a0 w h i c h a p p r o a c h e s the v a l u e 4. T h e a m p l i t u d e s ~/al" + b~2 . . . of the v a r i o u s h a r m o n i c s also a p p r o a c h zero w h e n ~--+ 0. A s t o t h e i r p h a s e s , they a p p r o a c h a t the l i m i t ~ --~ 0 the p h a s e of the t e r m w h o s e coefficient cont a i n s a l o w e r p o w e r of ~. 2. I t s e e m s l i k e l y ( a l t h o u g h the a p p r o x i m a t i o n s in o u r case d i d n o t p r o g r e s s far e n o u g h t o be able t o a s s e r t t h i s ) t h a t for a sufficiently s m a l l ~, the coefficients a , , b, of the t r i g o n o m e t r i c s e r i e s (2.7) a p p r o a c h zero w h e n n --+ oo. If s u c h is the case the t r i g o n o m e t r i c s e r i e s (2.7) is actually a Fourier series. 3. T h e c o n s t a n t t e r m (4 + }~" -i- " " ) m a y be r e g a r d e d a s a " c o m p o n e n t " of a h a r m o n i c oscillator w h i l e the t r i g o n o m e t r i c t e r m s r e p r e s e n t m e r e l y the energy fluctuations b e t w e e n the V D P oscillator a n d the e x t e r n a l s o u r c e . I t s h o u l d be n o t e d t h a t the p r e c e d i n g d i s c u s s i o n r e l a t e s only t o the i n t e g r a l c u r v e p(O) of the V D P o s c i l l a t o r i n a s m u c h as time does n o t e n t e r h e r e . I n the f o l l o w i n g s e c t i o n we s h a l l i n v e s t i g a t e the d e p e n d e n c e on t i m e , t h a t is, the a c t u a l f l u c t u a t i o n s of e n e r g y . It m a y be u s e f u l t o s a y a few w o r d s r e g a r d i n g the c o n v e r g e n c e of the exp a n s i o n (2.1). I n a s m u c h a s this e x p a n s i o n was o b t a i n e d from the f u n c t i o n a p p e a r i n g on the r i g h t side of (1.9) it is c l e a r t h a t the only case w h e n i t is d i v e r g e n t is w h e n dO/dt = 0. This case is o b v i o u s l y r u l e d out for ~ << 1 in w h i c h we are i n t e r e s t e d h e r e . It can be s h o w n , howe v e r , (section 5 b e l o w ) t h a t , even in the case w h e n ~ is l a r g e , dO/dt ~ 0 a l t h o u g h it m a y b e c o m e of the o r d e r of 1/~2 for some s p e c i a l i n t e r v a l s of 0. In t h e s e i n t e r v a l s the c o n v e r g e n c e b e c o m e s so slow t h a t t h e pert u r b a t i o n m e t h o d b e c o m e s i m p r a c t i c a b l e in v i e w of the i m p o s s i b i l i t y of c a r r y i n g the s u b s e q u e n t a p p r o x i m a t i o n s indefinitely on a c c o u n t of a r a p i d l y i n c r e a s i n g c o m p l e x i t y of c a l c u l a t i o n s . F o r this r e a s o n the p r e c e d i n g p r o c e d u r e is a p p l i c a b l e only for ~ < 1 a n d , p r e f e r a b l y , for E << 1. In m a n y a p p l i c a t i o n s , h o w e v e r , one is i n t e r e s t e d p r e c i s e l y in the r a n g e of s m a l l E so t h a t the p e r t u r b a t i o n p r o c e d u r e c a n be u s e d t h e n . A s r e g a r d s the so-called " r e l a x a t i o n r a n g e " (e >> 1), a l t h o u g h the p e r t u r b a t i o n m e t h o d c e a s e s t o be a p p l i c a b l e , the e n e r g y f l u c t u a t i o n s in a V D P o s c i l l a t o r can be s t i l l e x p l o r e d in c o n n e c t i o n with the d a t a o b t a i n e d by the i s o c l i n e m e t h o d a s will be s h o w n in s e c t i o n 5. 3. I~..NERGY FLUCTUATIONS
T h e p r e c e d i n g s t u d y c o n c e r n s the f u n c t i o n p(0) in w h i c h the time does n o t a p p e a r . In o r d e r t o i n v e s t i g a t e the e n e r g y f l u c t u a t i o n s o(t) in
2IO
N. MINORSKY
[J, F. I.
a V D P oscillator it is necessary t o d e a l directly with E q s . 1.7 a n d 1.8 containing t i m e . In o t h e r w o r d s , it is necessary t o determine t h e m o t i o n p(t) o n t h e trajectory o(0). The perturbation procedure remains t h e same as before with t h e exception t h a t it is t o b e carried o u t simultaneously in b o t h variables p a n d 0 considered as functions of t i m e . Equations 1.7 a n d 1.8 c a n b e w r i t t e n
do
d-)- = el0(1 - Cos 20) - 102(1 - Cos 4 0 ) ] ,
(3.1)
dO d-t
=
e[(½ -- ¼o) Sin 20 - -~o Sin 403 - 1.
(3.2)
Since t h e time does n o t a p p e a r explicitly in t h e s e e q u a t i o n s , t h e r e is n o difference between t a n d t + to, to being arbitrary. This m e a n s t h a t t o a given trajectory (or integral curve) o(0) c o r r e s p o n d s a n infinity of motions o(t), O(t) differing b y a n a r b i t r a r y p h a s e to d e p e n d i n g o n t h e initial c o n d i t i o n s . The perturbation procedure is carried o u t b y s e t t i n g = po + col + e202 -t- " " , 0 = 00 -4- e01 -I- d02 -k- " " .
p
Since 0 appears only u n d e r t h e sign of t h e trigonometric functions we h a v e t o e x p a n d Sin 20, Cos 20, • • • into a Taylor's series a r o u n d 0 = 0o, w h i c h gives Cos 20 = Cos 200 - 2(~01(t) + ~202(t) + . . . ) Sin 200 + . . . , Sin 20 = Sin 200 + 2(e01(t) + ¢~02(t) + . . . ) Cos 200 + . . . . Rearranging t h e results according t o t h e powers of e one finds d-2 = el-p0(1 - Cos 200) - 1oo2(1
dt
Cos 400)] -4-
+ ~2~2po01 Sin 20o + o1(1 - Cos 200) - 0o201 Sin 400 - ½ p o m [ 1 - Cos 400)3 + < . . . 3 + . . .
(3.3)
dO = _ 1 + d-(½ - Ip0) Sin 200 - kp0 Sin 40o] +
dt
+ #[-(1 - ½oo)01 Cos 200 - lol Sin 200 -
10001 Cos 400 - -~plSin 40o-] + Of-...-1 + " " .
The approximation of t h e zero o r d e r yields a g a i n
do--2 = O; dt
__dO° = _ 1, dt
t h a t is
00 =
- t,
(3.4)
Sept., ~949.]
FLUCTUATIONS
I N A VAN DER P O L
OSCILLATOR
211
w h i c h c o i n c i d e s with Eqs. 1.5 for the h a r m o n i c oscillators. The e l i m i n a t i o n of the s e c u l a r t e r m in the f i r s t a p p r o x i m a t i o n g i v e s a g a i n p0 = 4. T h e r e a p p e a r s , h o w e v e r , a n o t h e r c o n s t a n t of i n t e g r a t i o n in the s e c o n d e q u a t i o n w h i c h m u s t be d e t e r m i n e d by the initial conditions, the i n d e p e n d e n t v a r i a b l e b e i n g now t. T h e s i m p l e s t w a y of i n t r o d u c i n g the i n i t i a l c o n d i t i o n is t o a s s u m e t h a t for t -- 0, 0(0) = 01(0) = 02(0) =
" ° •
~-
0,
T h e f i r s t a p p r o x i m a t i o n b e c o m e s then
pl(t) = KI - 2 S i n 2t + Sin 4 t ; 01(t) = ~- - } Cos 2t - { - C o s 4t,
(3.5)
w h e r e K1 is a n i n t e g r a t i o n c o n s t a n t d e t e r m i n e d in the s e c o n d a p p r o x i m a t i o n . T h e p r o c e d u r e from now on b e c o m e s o b v i o u s , viz: for t h e s e c o n d a p p r o x i m a t i o n we c o l l e c t the t e r m s with e2 in (3.3) a n d (3.4) a n d it is o b s e r v e d t h a t do2/dt a n d dO2/dt a r e e x p r e s s i b l e now in t e r m s of po, 0o, t)~ a n d 01. I n t e g r a t i n g t h e s e e q u a t i o n s , e l i m i n a t i n g the s e c u l a r t e r m in 02 ( w h i c h l e a d s t o the d e t e r m i n a t i o n of K1) a n d d e t e r m i n i n g the c o n s t a n t of i n t e g r a t i o n in 02 by the a s s u m e d i n i t i a l c o n d i t i o n s , one o b t a i n s the s e c o n d a p p r o x i m a t i o n p2 and 02; one f i n d s 7 2 02 = K 2 + C o s 2 t - ~ C o s 4 t + ~ C o s 6 t , w h e r e K1 = 0. I t is t o be n o t e d t h a t the e x p r e s s i o n for o2(t) is n o t the s a m e as t h a t for p~(0) a n d l i k e w i s e for o t h e r a p p r o x i m a t i o n s so t h a t , in g e n e r a l , the t r i g o n o m e t r i c s e r i e s for p(t) is n o t the s a m e a s t h a t for p(O). T h e r e a s o n for t h a t is due t o the n o n - u n i f o r m i t y of r o t a t i o n of the r a d i u s v e c t o r as is seen from the e x p r e s s i o n for O~(t) w h i c h g i v e s in this a p p r o x i m a t i o n 0 =
- t +
~(~- ¼ C o s 2 t - ~Cos4t).
T h i s s h o w s t h a t t h e r e e x i s t s a kind of a " p h a s e m o d u l a t i o n " a f f e c t i n g all t r i g o n o m e t r i c t e r m s . T h e f l u c t u a t i o n of e n e r g y in time up t o the t e r m s of the o r d e r e'~ is then
p(t) = 4 + ~ ( - 2 S i n 2 t + S i n 4 t ) + + ~2
(
Cos2t-~Cos4t+~Cos6t
)
+K2,
w h e r e the c o n s t a n t of i n t e g r a t i o n K2 is d e t e r m i n e d in the next a p p r o x i m a t i o n a n d so on. 4. E F F E C T O F T H E
FORM O F T H E V A N DER POL E Q U A T I O N O N T H E
FREQUENCY SPECTRUM
So far we h a v e b e e n c o n c e r n e d with Eq. 1.2 o r i g i n a l l y f o r m u l a t e d by v a n d e r Pol (1). L a t e r on, this e q u a t i o n was g e n e r a l i z e d by A.
N. MINORSKY
212
[J. F . I.
L i 6 n a r d (2), E. & H. C a r t a n (3), N. L e v i n s o n a n d O. K. S m i t h (4) a n d o t h e r s . T h e g e n e r a l form of the v a n d e r Pol e q u a t i o n is 2 -- ef(x, 2)2 + g(x) = O,
(4.1)
w h e r e the f u n c t i o n s f a n d g m u s t s a t i s f y c e r t a i n c o n d i t i o n s (4). W e s h a l l give a few e x a m p l e s r e l a t i v e t o the e l e c t r o n t u b e c i r c u i t s a s s u m i n g t h a t they do n o t i n v o l v e a n y n o n - l i n e a r c a p a c i t i e s in w h i c h case g(x) = x, a n d c o n s i d e r f ( x , 2) = f ( x ) in the L i 6 n a r d s e n s e , viz: f ( x ) is a n even f u n c t i o n of x so t h a t F(x) = f f ( x ) d x is o d d a n d p o s i t i v e for s m a l l v a l u e s of x a n d b e c o m e s n e g a t i v e b e g i n n i n g with x = x0, d e c r e a s i n g m o n o t o n i c a l l y for x > x0. T h e p o l y n o m i a l s of the form F(x) = a x - b x a ; F(x) = ax + bxa - cx5 e t c . , c l e a r l y , s a t i s f y this c o n d i t i o n a n d are f r e q u e n t l y u s e d in a p p l i c a t i o n s . T h u s , for i n s t a n c e , w h e n F(x) is a p o l y n o m i a l of the fifth d e g r e e , the v a n d e r Pol e q u a t i o n h a s the form 2 - ~(1 + ~ x2 - 5x4)2 + x = 0.
(4.2)
U s i n g t h e p r e c e d i n g p r o c e d u r e one gets dp -dO
2p[-¢( ) Sin 2 0 + ~2( )= S i na 0 Cos 0 + ca( )a Sin 4 0 C o s~ 0 +
. . . ],
w h e r e ( ) = (1 + ap Cos 2 0 - /~p2 Cos 4 0). T h e p e r t u r b a t i o n m e t h o d l i m i t e d t o the z e r o , a n d the f i r s t a p p r o x i m a t i o n g i v e s p0 = K0 = C o n s t ; pl = -- 2 [ ' p o f S i n s OdO 4- o~po2 f Cos 2 0 Sin 2 OdO -- 3po3 f Cos 40 S i ns OdO] + K1 a n d the e l i m i n a t i o n of the s e c u l a r t e r m g i v e s oo =
+
+
so t h a t , up t o the t e r m s of the o r d e r e~ one gets
'
o =p0+~pl =p0- ~
+
p0+~-~/~00'
)
Sin 2 0 +
1
l po~ ( a - -~ [3 ) Sin 40 -
l ~po' Sin 60 ] + K2.
T h e r e e x i s t s o b v i o u s l y n o p0 for j3 = 0 s i n c e the d a m p i n g t e r m is t h e n n e g a t i v e for all v a l u e s of x. F o r a = 0, B = 1, (4.2) b e c o m e s -
~(1 - x')2 + x = 0,
(4.5)
in w h i c h case o0 = 2V2. I n t e r e s t i n g c a s e s a r i s e w h e n the form of the v a n d e r Pol e q u a t i o n for a n e l e c t r o n t u b e oscillator is m o d i f i e d by the p r e s e n c e of c e r t a i n
Sept., 1949.]
FLUCTUATIONS
I N A VAN DER POL
OSCILLATOR
2I 3
variable conductors in the oscillating circuit. We shall consider one such case in connection with the circuit shown in Fig. 1. If all p a r a m eters (L, C and R) are constant, it is well known (5) that the van der Pol equation in this case is of the form d2v dr2
dv (fl + 2~v - 3~v2) dr + v = O,
(4.6)
where v = vu/v~ is a dimensionless variable (v~ is the grid voltage ; v~ the
t_ R
FIG. 1.
constant "saturation voltage"), r = ~0t ; c00 = ~ / - f / L C ; ~ = ( ~ 1 -- RC)o~o '*/ ----- ~k'Y1600;
~ ----- ~k~l~O0;
81 = S 1 ;
"~'1 = ,~2Ps;
~1 ~" - - S 3 p s 2,
and the non-linear function, the a n o d e current, is assumed to be represented by the polynomial /~ = ¢ ( ~ ) = I0 + $ 1 ~ + $2~o~ + $ 3 ~ / . Let us modify the problem by replacing the constant resistor R by a variable one R ( I i l ), depending on the absolute v a l u e of the current and
214
N. MINORSKY
[J. 1~. I.
a s s u m e , m o r e o v e r , t h a t the non-linearity of R ( ] i ] ) a p p e a r s before the non-linearity of the e l e c t r o n t u b e is felt. In o t h e r w o r d s , we wish t o c o n s i d e r the case w h e n the e l e c t r o n t u b e o p e r a t e s s t i l l on the r e c t i l i n e a r p a r t of its c h a r a c t e r i s t i c ( t h a t is, "r = ~ --~ 0). E q u a t i o n 4.6 in s u c h a case b e c o m e s d2v
d~ 2
Ex~l -
R ( [ i l ) C ] d~ dr + v = 0.
(4.7)
It is necessary to make a certain assumption regarding the function R([i]). Assume, for example, that the resistance of the non-linear resistor is of the form R(Ii[)
= m +niil,
where m and n are positive constants, that is, the resistance increases as a linear function of the absolute v a l u e of the current which is generally the case for conductors having a positive temperature coefficient. On the other hand vo =
whence
{f
idt,
dp (t v
so that
Ill = Cv,cOo d~ and (4.7) becomes dr ~
~
dr ] dr + v = O,
where = (X~I - . m C ) ;
b = nC2v,~Oo.
The preceding equation becomes d2v
-tiT2 ---
~
(
1 - K ddr~ ] d~ ~r
+ v =0,
K = -b"
The equivalent system is dv
--Y;
dy ~(1 -- K l y ] ) y - v. d-; =
Assuming that e and r are small, Eq. 1.9 in this case is do 2e( )o Sin2 0 dO ~( ) S i n 0 C o s 0 - 1 -
-
=
=
- - 2 o [ - e ( ) S i n 2 0 + e2( )2 S i n a 0 C o s 0 + ~3( )3 S i n ~ 0 C o s2 0 + . . - ~ ,
where ( ) = (1 -- K r l S i n 0[) = (1 -- Ko~[Sin 01).
Sept., I949.]
ITLUCTUATIONS I N A VAN DER I'OL
OSCILLATOR
215
T h e perturbation method gives dP-2° = 0 ; dO
p0 = K0 = C o n s t .
d m = _ 2(1 - K p 0 ~ ] S i n 0])p0 S i n2 0 = dO = - p0 + o0 Cos 20 + K p 0 ~ ( I S i n 01 - Cos 2 0 1 S i n 01). Replacing ISin 0[
. . . .7
C o s 2 0 ] S i n 0] =
7r
~ C o s 2 0 +
Cos 4 0 +
Cos 6 0 + . . .
76 Cos 40 . . . . - 3~r -2- + 28 ~ Cos 20 - 105---~
a n d e l i m i n a t i n g the s e c u l a r term one o b t a i n s p0
=
r02
=
8 K /
'
w h i c h s t a r t s the p e r t u r b a t i o n p r o c e d u r e w h i c h we will n o t c o n t i n u e h e r e . I t is w o r t h m e n t i o n i n g m e r e l y t h a t in this case we e n c o u n t e r a s i t u a t i o n in w h i c h a l r e a d y in the f i r s t a p p r o x i m a t i o n t h e r e a p p e a r s the w h o l e t r i g o n o m e t r i c s e r i e s i n s t e a d of t r i g o n o m e t r i c p o l y n o m i a l s cont a i n i n g b u t a few t e r m s a s this was the case for Eqs. 1.2 a n d 4.5. T h e f r e q u e n c y s p e c t r u m of e n e r g y f l u c t u a t i o n s in a n oscillator of this kind is, therefore, far r i c h e r in h a r m o n i c s t h a n in the c a s e s j u s t m e n t i o n e d . T h e s e e x a m p l e s show t h a t , f o l l o w i n g this p r o c e d u r e , it is a l w a y s p o s s i b l e t o form a n idea a s t o t h e f r e q u e n c y s p e c t r u m of e n e r g y f l u c t u a t i o n s in a V D P o s c i l l a t o r for a g i v e n n o n - l i n e a r f u n c t i o n F(x) p r o v i d e d it satisfies the L i ~ n a r d c o n d i t i o n a n d p r o v i d e d ~ is sufficiently s m a l l . 5.
cAs~
~
~ is
L~GE
T h e p e r t u r b a t i o n m e t h o d c e a s e s t o be a p p l i c a b l e w h e n ~ is n o t s m a l l . C e r t a i n a d d i t i o n a l c o n c l u s i o n s c a n be s t i l l o b t a i n e d from (1.7), (1.8) a n d (1.9) a l s o , in this c a s e , w i t h the aid of the d a t a y i e l d e d by the m e t h o d of isoclines. I t is t o be n o t e d t h a t the m a x i m a a n d m i n i m a of both p o l a r c u r v e s 0(0) a n d r(O) are s i t u a t e d on t h e s a m e r a d i i - v e c t o r s s i n c e the 0(0) c u r v e is o b t a i n e d m e r e l y by s q u a r i n g the r a d i i v e c t o r s of the r(O) c u r v e for each 0 a n d , c o n v e r s e l y if p(0) c u r v e is g i v e n , the r(O) c u r v e can be o b t a i n e d b y a n i n v e r s e o p e r a t i o n . \,Ve s h a l l s p e a k p r e f e r a b l y of the e n e r g y c u r v e p(0) i n a s m u c h a s in this m a n n e r it will be p o s s i b l e t o d e r i v e certain p h y s i c a l c o n c l u s i o n s r e g a r d i n g the V D P oscillator b u t o c c a s i o n a l l y we s h a l l r e f e r also t o the i n t e g r a l c u r v e r(O) in view of the a b o v e r e l a t i o n b e t w e e n t h e s e two c l o s e d c u r v e s . F r o m (1.7) it f o l l o w s t h a t the m a x i m a a n d m i n i m a of e n e r g y o c c u r w h e n p(0) = l / C o s20 w h i c h in c a r t e s i a n c o o r d i n a t e s is o b v i o u s l y x = =t= 1.
216
N. 1V[INORSKY
[J. F. I.
For these particular values of 0 the term with i in (1.2) vanishes and the VDP oscillator coincides instantaneously with the harmonic oscillator of this particular (Om~x or pmi,,) energy content. One can also verify this circumstance by replacing p = l/Cos" 0 in (1.8) w h i c h gives 0 = - 1 w h i c h in conjunction with the condition ~ = 0 gives conditions (1.5) for a harmonic oscillator. In o r d e r to distinguish m a x i m a from m i n i m a it is sufficient to calculate )5 which gives = 2d-~(Sin2 0 - 20 Sin2 0 Cos2 0) + 200(Sin 0 Cos 0 + p Sin3 0 Cos 0 - 0 Sin 0 Cos3 0)7. Replacing in this, expressions ~ = 0, 0 = - 1, p = 1 / C o s~ o, one finds )5 = - 4 ~ T a n 30, w h i c h shows that the m a x i m a of energy are located in the first and the t h i r d quadrants of the p o l a r curve 0(0) and m i n i m a in the second and in the fourth quadrants w h i c h also agrees with the data yielded by the m e t h o d of isoclines. Similarly, when ~ increases the isocline m e t h o d shows that the m a x i m a increase and the m i n i m a decrease but since the condition p Cos2 0 = 1 is independent of ~, it is clear that the values of 0 corresponding to Pm~ approach the y axis and those for pmi, recede to the x axis while still being in t h e i r respective quadrants. This also agrees with the data furnished by the isocline method. T h e r e exists thus a definite relation between the magnitude of the m a x i m a and m i n i m a of energy and t h e i r phase in the p o l a r curve. For a sufficiently large , and in the neighborhood of m a x i m a of the energy content, the rate of change of energy ~ varies very r a p i d l y even for a small variation 50 around the value 0 = 0 . . . . Thus, for instance, from the isocline curve for ~ = 30, pm~ ~ 1300 and 0 .... ~ 88.4°. For A0 = 4- 0.1°, # changes from about - 1 3 0 0 0 (for A0 = -- 0.1 °) to a b o u t + 1 1 0 0 0 (for A0 = + 0.1 °) in the dimensionless units used here, whereas in the neighborhood of m i n i m a the same variation A0 accounts for a small part of one per cent of the above variation. If one wishes to give a physical analogue of the behavior of the VDP oscillator in the neighborhood of its m a x i m a energy points, at least as far as energy fluctuations are concerned, perhaps a pneumatic hammer or a similar device may give an adequate analogy. In fact in a device of this kind a r a p i d acceleration of the hammer with the incident increase in its energy content (which is the kinetic energy here) is followed by a not less r a p i d deceleration when the hammer strikes an energy absorbing m e d i u m and thus loses its energy. In addition to one maximum and one minimum of energy per one half cycle (the o t h e r half cycle being identical) the VDP oscillator has also one stationary point for 0 = 0 (for the other half-cycle this stationa r t point occurs for 0 = 7r) a t w h i c h the energy passes through a stat i o n a r y v a l u e without being either maximum or minimum. One sees
~el)t., I949.]
FLUCTUATIONS
I N A VAN DER
PoL O,gCILLATOI{
2I 7
this from (1.7) w h i c h s h o w s t h a t # p a s s e s t h r o u g h zero a t t h e s e p o i n t s w i t h o u t c h a n g i n g its s i g n . O n e n o t e s also t h a t ? = 0 a t this p o i n t . W h e n e is l a r g e t h e s e s t a t i o n a r y p o i n t s are c l o s e t o the m i n i m a p o i n t s of e n e r g y . T h e i n v e s t i g a t i o n of the b e h a v i o r of a n i n t e g r a l c u r v e in the n e i g h b o r h o o d of t h e s e s t a t i o n a r y p o i n t s is far from b e i n g s i m p l e , if E
5q ~2
50 28 2a 22
20 18
-i6
J - i s -x.'o -o:5
-
lO a?
zo
z3
2.o
FIG. 2.
is l a r g e e n o u g h a n d this is p r o b a b l y one of the p r i n c i p a l difficulties of the v a n d e r Pol e q u a t i o n . T h e m e t h o d of i s o c l i n e s s h o w s , in f a c t , t h a t t h e r e is a s h a r p b e n d of i n t e g r a l c u r v e s in this " r e l a x a t i o n r e g i o n " a s c a n be seen from Fig. 2, r e p r e s e n t i n g the i n t e g r a l c u r v e r(O) for E = 30 cons t r u c t e d b y the m e t h o d of isoclines. ~ If one f o l l o w s this i n t e g r a l c u r v e The author is indebted to Miss E. Yost for this construction.
218
N. MINORSKY
[J. F. I.
in the positive direction ( t h a t is, clockwise) one finds that it cuts the x-axis at r i g h t angles (point A) but immediately a f t e r that the integral curve has an a l m o s t 90° bend and follows practically the x-axis t o w a r d the origin. The accuracy of the graphical construction is obviously insufficient to be able to specify the details of what happens in this critical region and it is necessary to make a closer analysis. We shall indicate first an approximation based on the results of the isocline data and shall analyze the situation l a t e r independently of the m e t h o d of isoclines. Let us assume, as an approximation, that in this "relaxation region" (say between x ~ 1.9 and x ----- 1.2 in Fig. 2) the integral curve is parallel to the x-axis, that is, coincides with the zero isocline whose equation is y
= x/
(1
-
If one replaces this value of y into the expression for 0 in the x, y coordinates, viz: 0 = - 1 + ~(1 - x~)xy/o, a f t e r a few transformations, one finds the following approximate expression for 0 in this region 0=
1
d(1
-
x ~ ) 2'
which shows that 0 is small of the order 1 / d and negative as everywhere else on the integral curve. In view of this, from (1.8) we get ,(1 - 0 Cos~ o) ~ 1/Sin 0 Cos O, so that (1.7) becomes dpp ~ 20 Tan 0 --~ 200.
do--
(5.1)
Since 0 < 0 in this region is small and changes very slowly, if one t a k e s its average v a l u e 0 in this interval, (5.1) yields an approximate formula p = poe~°~
(5.2)
w h e r e p0 corresponds to the beginning of this "relaxation" region. This approximate relation shows that the energy is slowly drained away .from the system in a m a n n e r in which the electrostatic energy of a charged condenser disappears gradually owing to an imperfect dielectric. This somewhat crude analysis is merely a physical interpretation of the results yielded by the m e t h o d of isoclines. For that reason it is preferable to investigate the behavior of the functions b(0) and 0(0) in a small interval (01, -01); 01 > 0 a r o u n d 0 = 0 (the a r g u m e n t applies also to 0 = ~r). This will permit obtaining certain conclusions directly without relying on the m e t h o d of isoclines.
Sept., 1949.]
F L U C T U A T I O N S IN A VAN D E R P O L
OSCILLATOR
2I 9
Differentiating (1.8) with respect to 0 and setting 0 = 0 a f t e r the differentiation one gets
(
(dO)d_O o = - ~ ( p o - 1)" =
d20)o = O;
=
dO~ / o so that 02
0(0) -- 00 + 0
0 ÷
( " ) 0
03
= -- 1 -- ~(p0-- 1)0 + 4 5 p 0 - - 2)-~ + . . . .
(5.3)
Since 0 is small and p0 is of the o r d e r 4 one can limit this expansion to the l i n e a r term in 0. U n d e r this assumption, 0 becomes very small for 0<0 such that ~(p0-1)] 01 is of the o r d e r of unity, that is [al ~0(1/3~). In such a case the term with ~03 can be neglected as we did. If, therefore, the interval 01, -01 in which we propose to investigate the behavior of ~(0) and 0(0) is fixed on this basis, 0 < 0 is a monotonically increasing function. In fact 0 -- - 1 - K 0 has the values 10101 = = 1 + K 0 1 ; ]0Io = l a n d [ 0 [ - o , = l - K 0 1 . In a similar m a n n e r differentiating (1.7) with respect to 0 and setting 0 = 0 a f t e r the differentiation, one gets =
=
-
-
As, moreover, ~0 = ~(0) = 0, one gets ~(0) - ~(0) + 0
~
0 + ~
~-~/0 +
"
~ - 2~p0(p0 - 1)¢-.
(5.4)
Since ~I 0] is assumed to be of the o r d e r one, the preceding expression shows that ~(0) undergoes a small variation of the o r d e r 0 in the interval (01, -01). As first approximation we can assume, therefore that p ~ p0 in this interval. We shall indicate l a t e r how a second approxim a t i o n can be obtained. It is useful now to specify the interval (01, - 0 1 ) , 01 > 0 in which the preceding assumptions apply. In the construction of the curve of Fig. 2, ~ = 30 was assumed which is a purely a r b i t r a r y value selected with a view to go a little beyond the v a l u e (~ = 10) w h i c h van der Pol used in his construction. This permits obtaining a more pronounced "relaxation region" between the points A and B. On the o t h e r hand, van der Pol, in his analysis of a relaxation oscillation circuit with constants commonly encountered in applications, finds ~ ~ 3.10L that is K ~ 1 0 6. This gives the value of 0 < 0, for w h i c h 0 becomes very
:220
N. MINORSKY
[J. F. 1,
s m a l l , of the o r d e r of 1 sec. of arc. T h i s m e a n s t h a t the m o t i o n of the r e p r e s e n t a t i v e p o i n t w h i c h is p r a c t i c a l l y t a n g e n t i a l ( w i t h r e s p e c t t o the r a d i u s v e c t o r ) for 0 = 0, b e c o m e s r a d i a l for 0 ~ - 1 s e c . , w h i c h g i v e s a n i d e a a s t o the o r d e r of m a g n i t u d e of the i n t e r v a l (01, - 0~) in q u e s t i o n for the v a r i o u s v a l u e s of ~. A s r e g a r d s the f u n c t i o n ~(0), (5.4) s h o w s t h a t , u n d e r the a s s u m e d a p p r o x i m a t i o n , it is a n even f u n c t i o n of 0; m o r e o v e r ~(0) is n e g a t i v e in this i n t e r v a l , t h a t is, the e n e r g y d e c r e a s e s b e t w e e n 01, a n d - 0 1 . W e p r o p o s e now t o c a l c u l a t e the loss of e n e r g y - A p l , a n d - A p 2 in t h e h a l f - i n t e r v a l s (0~, 0) a n d (0, - 0 1 ) , respectively, w h e n the r a d i u s v e c t o r t r a v e r s e s the s m a l l arc (01, - 0 1 ) . O n e h a s -
-
Apl ----
f0 i
i~dt;
-- 5p2 =
ixtt.
F r o m (5.3) a n d (5.4) we h a v e o =
w h e r e K = e(p0 -- 1).
= _ 2poK92,
- (1 + Ko);
Thus 02
[Jdt = 2poK 1 nt- K~O dO a n d , therefore, -- Apl = 2poK fo° O2KodO; ~1+
-- Ap2 = 2poK fo -°~
dO. 0~ I +KO
U n d e r this a s s u m p t i o n --Am=
2p0 Ks[½(I+K01) 2 - 2 ( 1 + K 0 , ) + l n ( 1 + K 0 1 ) +~],
(5.5)
--Ap2 =
~ [ ½ ( 1 - - K 0 1 )2 - 2 ( 1 - K 0 , ) + l n ( 1 - K O 1 ) + ~ ] .
(5.6)
S i n c e in t h e half-interval (0, - 0 1 ) the q u a n t i t y 1 - KOt > 0 b e c o m e s v e r y s m a l l , In (1 ~ KO~) is a v e r y l a r g e n e g a t i v e n u m b e r so t h a t Ap2 >> Apl. On the o t h e r hand, since --Apl and --Ap2 represent the decrease of the r a d i u s vector in (01, 0) and (0, - 0 1 ) , respectively, one obtains a situation depicted in Fig. 3 which explains the origin of a s h a r p bend in the curve of energy p(0) (and also the integral curve r(O)) in the negative neighborhood (0 < 0) of the stationary point A. Once the dissymmetry of 0(0) has been ascertained starting from the assumption that p ~ p0 in this angular interval (0t, - 0 1 ) , it is possible to c a r r y out the second approximation using, for example, the values p0 + Am/2, po - A p 2 / 2 instead of p0 (and the corresponding values Kt and K2) in both half-intervals w h i c h gives a more correct estimate for Apl and Ap2 and thus calculate the shape of the p(0) curves in this neighborhood.
Sept., 1949.]
I~LUCTUATIONS I N A VAN DER POL
OSCILLATOR
22I
T h e s h a r p b e n d of the c u r v e s 0(0) a n d r(O) is t h u s due t o a v e r y m a r k e d a s s y m m e t r y of the f u n c t i o n 0(0) in this r e g i o n w h i l e the f u n c tion #(0) r e m a i n s s y m m e t r i c a l a b o u t t h e s e s t a t i o n a r y p o i n t s . F r o m a p h y s i c a l s t a n d p o i n t this c i r c u m s t a n c e m e a n s t h a t , on a c c o u n t of a s m a l l e r v e l o c i t y 0 in the (0, - 01) half i n t e r v a l the s a m e d r a i n of e n e r g y (for the s a m e 0) p e r s i s t s for a l o n g e r time in the n e g a t i v e half i n t e r v a l than in the p o s i t i v e one w h i c h is t r a v e r s e d r a p i d l y , so t h a t the a d t u a l d e c r e a s e of e n e r g y is g r e a t e r in (0, - 0 1 ) t h a n in (01, 0). It m u s t be n o t e d t h a t for v e r y l a r g e v a l u e s of E, like t h o s e m e n t i o n e d by v a n d e r Pol, the b e n d of i n t e g r a l c u r v e s b e c o m e s so s h a r p t h a t the a n a l y t i c i t y of the c u r v e in this r e g i o n is bad. F o r t h a t r e a s o n the T a y l o r e x p a n s i o n in this r e g i o n b e c o m e s r a t h e r c o m p l i c a t e d a s c o m p a r e d t o the r e m a i n i n g p o r t i o n s of the c u r v e . I t is l i k e l y t h a t t h e s e local c o m p l i c a t i o n s in the
I
Fro. 3.
a n a l y t i c a l s t r u c t u r e of t h e i n t e g r a l c u r v e a t the b e g i n n i n g of this " r e l a x a t i o n " r e g i o n , is the m a i n r e a s o n w h y n o e x p l i c i t s o l u t i o n ~of] the v a n d e r Pol e q u a t i o n (for e x a m p l e , in the form of a s e r i e s ) c o u l d be f o u n d a s y e t , in s p i t e of n e a r l y t w e n t y y e a r s of effort, a l t h o u g h c o n s i d e r a b l e p r o g r e s s h a s r e c e n t l y b e e n a c c o m p l i s h e d in c o n n e c t i o n with a q u a l i t a tive a n a l y s i s of this d i f f i c u l t p r o b l e m (6). CONCLUDING REMARKS
T h e m o s t i n t e r e s t i n g f e a t u r e of the V D P oscillator is the fact t h a t its b e h a v i o r is w i d e l y d i f f e r e n t for d i f f e r e n t v a l u e s of the p a r a m e t e r ~. A t one end of the r a n g e (~ << 1) the V D P oscillator d i f f e r s v e r y little from the h a r m o n i c oscillator a n d t h u s offers p r a c t i c a l l y a p e r f e c t p h y s i c a l i m a g e of the l a t t e r . P r o b a b l y it is n o t a n e x a g g e r a t i o n t o s a y t h a t , s i n c e the time of G a l i l e o w h o i n t r o d u c e d the c o n c e p t of the
222
N. MINORSKY
[J. F. I.
h a r m o n i c oscillator, we w e r e a b l e t o a c t u a l l y o b s e r v e a p r a c t i c a l l y p e r f e c t p h y s i c a l i m a g e of the l a t t e r only w h e n the V D P o s c i l l a t o r b e c a m e a v a i l a b l e r e l a t i v e l y a s h o r t time ago. In f a c t , v e r y likely, t h e r e e x i s t s no b e t t e r p h y s i c a l i m a g e of a s i m p l e h a r m o n i c m o t i o n t h a n t h a t •w h i c h is p r o d u c e d by m o d e r n h i g h q u a l i t y e l e c t r o n t u b e oscillators, p a r t i c u l a r l y t h o s e w h o s e f r e q u e n c y is s t a b i l i z e d b y q u a r t z u n i t s . B u t this a l m o s t p e r f e c t p h y s i c a l i m a g e c o n t a i n s a g e r m of a n i m m e n s e comp l e x i t y due t o the p r e s e n c e of a n infinite s p e c t r u m of f r e q u e n c i e s with w h i c h the e n e r g y f l u c t u a t e s b e t w e e n the oscillator a n d t h e s o u r c e of e n e r g y . T h e s e f l u c t u a t i o n s e s c a p e o u r o b s e r v a t i o n , h o w e v e r , if ~ is very s m a l l a n d , as the r e s u l t of t h i s , w h a t is a c t u a l l y o b s e r v e d is only the c o n s t a n t t e r m in the infinite a r r a y of t e r m s (2.7). H o w e v e r , i t is p r e c i s e l y this h i d d e n c o m p l e x i t y of v a n i s h i n g l y s m a l l e n e r g y f l u c t u a t i o n s w h i c h p e r m i t s o b t a i n i n g a s i m p l e h a r m o n i c oscillation in its apparently pure form. On the o t h e r end of the r a n g e (~ >> 1) the b e h a v i o r of t h e V D P o s c i l l a t o r is r a d i c a l l y d i f f e r e n t from t h a t of a h a r m o n i c oscillator. If one h a s t o look for a n a n a l o g y , a s far as e n e r g y f l u c t u a t i o n s are conc e r n e d , in this r a n g e the V D P o s c i l l a t o r r e s e m b l e s a p n e u m a t i c h a m m e r or a s i m i l a r d e v i c e in w h i c h the e n e r g y f l u c t u a t i o n s ( f r o m a b s o r p t i o n t o d i s s i p a t i o n a n d vice v e r s a ) o c c u r in a q u a s i - d i s c o n t i n u o u s m a n n e r . On the o t h e r h a n d , a c l o s e r i n v e s t i g a t i o n of the b e h a v i o r of the o s c i l l a t o r r e v e a l s t h a t in s p i t e of this a p p a r e n t difference of its b e h a v i o r for d i f f e r e n t v a l u e s of ~, t h e r e e x i s t s a f e a t u r e c o m m o n t o all c a s e s , n a m e l y the e n e r g y r e s u m e s exactly the s a m e v a l u e a f t e r one p e r i o d a l t h o u g h d u r i n g the p e r i o d it f l u c t u a t e s b e t w e e n the o s c i l l a t o r a n d the s o u r c e with infinitely m a n y f r e q u e n c i e s of e v e n h a r m o n i c s . A t t i m e s the e n e r g y is a b s o r b e d , a t t i m e s it is d i s s i p a t e d b u t the time i n t e g r a l s of all t h e s e f l u c t u a t i o n s w i t h d i f f e r e n t f r e q u e n c i e s over the p e r i o d are zero a n d t h e r e r e m a i n s only the a v e r a g e c o n s t a n t t e r m a r o u n d w h i c h t h e s e f l u c t u a t i o n s o c c u r . If t h e s e f l u c t u a t i o n s are v a n i s h i n g l y s m a l l in comp a r i s o n with the c o n s t a n t e n e r g y c o n t e n t of the V D P oscillator, it b e h a v e s a p p r o x i m a t e l y a s a c o n s e r v a t i v e s y s t e m . If ~>> 1, o n the c o n t r a r y , the f l u c t u a t i o n s of e n e r g y d o m i n a t e e v e r y t h i n g else t o such a n e x t e n t t h a t it b e c o m e s even i m p o s s i b l e t o a n s w e r the q u e s t i o n : a r o u n d w h i c h a v e r a g e c o n s t a n t v a l u e of e n e r g y do t h e s e f l u c t u a t i o n s o c c u r ? In a d d i t i o n t o t h e s e f l u c t u a t i o n s of e n e r g y , t h e r e e x i s t also f l u c t u a t i o n s in the a n g u l a r v e l o c i t y of the r a d i u s v e c t o r of t h e r e p r e s e n t a t i v e p o i n t m o v i n g on the i n t e g r a l c u r v e . If ~ is s m a l l e n o u g h t h e s e f l u c t u a t i o n s are also s m a l l so t h a t , from this s t a n d p o i n t , the m o t i o n on i n t e g r a l c u r v e s d o e s n o t d i f f e r m u c h from t h e c o r r e s p o n d i n g m o t i o n for t h e h a r m o n i c o s c i l l a t o r in w h i c h case t h e s e f l u c t u a t i o n s a r e r i g o r o u s l y z e r o . F o r a n i n c r e a s i n g ~ t h e s e f l u c t u a t i o n s in a n g u l a r v e l o c i t y b e c o m e m o r e a n d m o r e p r o n o u n c e d a n d , for v e r y l a r g e v a l u e s of ~, the a n g u l a r v e l o c i t y b e c o m e s of the o r d e r 1 / , 2, t h a t is v e r y s m a l l , in the n e i g h b o r h o o d of the
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p o i n t s a t w h i c h the e n e r g y p a s s e s t h r o u g h a s t a t i o n a r y v a l u e . In this n e i g h b o r h o o d one o b s e r v e s the i n t e r e s t i n g p h e n o m e n o n of a " r e l a x a t i o n a l " d r a i n of e n e r g y from the s y s t e m a n d r e s u l t s in a s h a r p b e n d in the i n t e g r a l c u r v e . In s p i t e of a n u m b e r of c o m m o n f e a t u r e s b e t w e e n a h a r m o n i c , a n d a V D P o s c i l l a t o r for ~ << 1, t h e r e e x i s t s a f u n d a m e n t a l difference bet w e e n the two oscillators, n a m e l y : a h a r m o n i c o s c i l l a t o r can oscillate with any e n e r g y c o n t e n t p r e s c r i b e d b y the initial c o n d i t i o n s , w h e r e a s the e n e r g y c o n t e n t of a V D P oscillator does n o t d e p e n d a t all on t h e s e c o n d i t i o n s . In fact, the o s c i l l a t o r " s e l e c t s " so t o s p e a k its own e n e r g y c o n t e n t c o n s i s t e n t with the form of the n o n - l i n e a r f u n c t i o n F(x) as w e l l as with the p a r a m e t e r s of the s y s t e m . This r e m a r k a b l e p e c u l i a r i t y of the V D P oscillator, a s we saw, is due t o the fact t h a t a s t a t i o n a r y p e r i o d i c m o t i o n in this case is p o s s i b l e only in the case w h e n all s e c u l a r t e r m s are r e d u c e d t o z e r o , if we wish t o t h i n k in t e r m s of the p e r t u r b a tion t h e o r y . T h e r e q u i r e m e n t for the e l i m i n a t i o n of s e c u l a r t e r m s in the infinite f r e q u e n c y s p e c t r u m of e n e r g y f l u c t u a t i o n s t h u s i m p o s e s a r e q u i s i t e n u m b e r of a d d i t i o n a l c o n d i t i o n s o w i n g t o w h i c h only one p e r i o d i c m o t i o n is p o s s i b l e , i n s t e a d of infinity of s u c h m o t i o n s a s in the case of a h a r m o n i c oscillator. Acknowledgments T h e a u t h o r is i n d e b t e d t o P r o f . M . S c h i f f e r for v a l u a b l e d i s c u s s i o n s of this m a t t e r . This w o r k was c a r r i e d out u n d e r a r e s e a r c h p r o g r a m of the Office of N a v a l R e s e a r c h , N a v y D e p a r t m e n t . REFERENCES
(1) BALTH, VAN DER POL, "On Relaxation Oscillations," Phil. ,1fag., Vol. 27 (1926). - - - - - "Oscillations Sinusoidales et de Relaxation," l'Onde Eleclrique, Vol. 9 (1930). - "The Non-Linear Theory of Electrical Oscillations," Proc. I R E , Vol. 22 (1934). (2) A. LIft;NARD, "Etude des Oscillations Entretenues," Revue G~ndrale de l'Electric#d, Vol. 23 (1928). (3) E . ANDH. CARTAN, "Note sur la G6n~ratlon des Oscillations Entretenues," A n n des P. T. T., Vol. 12 (1925). (4) N . LEVINSON AND O. K. SMITH, " A General Equation for Relaxation Oscillations," Duke Math..Tour., Vol. 9 (1942). (.5) A. ANDRONOW AND S. CHAIKIN, "Theory of Oscillations," Chapter VII, Moscow (1937); English language edition of this book will a p p e a r shortly in Princeton Univ. Press. (6) D. I'LANDERS AND J. STOKER, "The L i m i t Case of Relaxation Oscillations," New York University (1946). J. LA SALLE, "Relaxation Oscillations" (to appear shortly).
N. MINORSKY, Ph.D.1 1. INTRODUCTION
B y analogy with t h e differential equation of t h e harmonic oscillator + x = 0
(1.1)
- x~)~ + x = 0
(1.2)
o n e c a n c o n s i d e r t h e equation -
~(1
as representing a special, v a n der Pol oscillator (abbreviation V D P ) . The important feature o f t h e h a r m o n i c oscillator is t h e fact t h a t (1.1) has t h e first integral ½z~ + ½x~ = h, (1.3) w h i c h expresses t h e law of conservation of energy (we assume here for t h e sake o f simplicity m -- c = w0 -- 1 in t h e u s u a l form of t h e equation m:~ + cx = 0 o f t h e harmonic oscillator). A l t h o u g h t h e V D P oscillator does n o t possess this feature, t h e introduction o f e n e r g y as a d e p e n d e n t variable is of a certain i n t e r e s t in applications a n d this will form t h e s u b j e c t of this note. If one uses t h e phase-plane representation (5 = y), (1.3) is x 2q_y2 = r2 = p = 2h.
(1.4)
That is, r~ = p represents twice t h e t o t a l energy o f t h e harmonic oscillator a n d , moreover, this energy is c o n s t a n t t h r o u g h o u t t h e cycle. I n o r d e r t o simplify t h e writing we will assume t h a t r2 = p = 2h is t h e t o t a l energy of t h e oscillator t h u s disregarding t h e factor 2. Equation 1.1 written as a s y s t e m o f two e q u a t i o n s is 2 =y;
9 -
--x.
Transforming t h e s e e q u a t i o n s into p, 0 coordinates one g e t s : do 0; dt =
dO dt -
1.
(1.5)
Eliminating time between t h e s e e q u a t i o n s o n e o b t a i n s t h e integral curve (or "trajectory") do dO
0.
(1.6)
J Department of Electrical Engineering, Stanford University, Stanford, Calif.
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The first equation (1.5) expresses t h e law o f conservation of energy ; (1.6) represents a circle in t h e p, 0 plane a n d t h e s e c o n d equation (1.5) gives 0 = - t, as t h e c o n s t a n t of integration c a n obviously b e a s s u m e d t o b e zero b y a p r o p e r choice of t h e origin of either 0 o r t. The sign m i n u s appearing in this equation is o f n o special significance a n d arises m e r e l y from t h e fact t h a t t h e angles are c o u n t e d as positive in t h e trigonometric sense (counterclockwise) w h i l e positive direction o n t h e integral curve is clockwise. If o n e applies this procedure t o (1.2) one o b t a i n s similar e q u a t i o n s d__o# = 2~(1 - p Cos2 0)0 Sin2 O,
(1.7)
dt
dO
d~ =
dp dO
- 1 + ~(1 - p Cos2 O) Sin 0 Cos O,
(1.8)
2~(1 - p Cos2 O) p Sin2 0
(1.9)
~(1 - p C o s 20) S i n 0 C o s 0 - 1
T h e s e e q u a t i o n s reduce t o E q s . 1.5 a n d 1.6, respectively, when ~ = 0. Since it is k n o w n t h a t t h e V D P oscillator is periodic, its integral curve is closed, t h a t is
f o 2" dp dO = f o2~ do = 0.
(1.10)
If, moreover, e << 1 t h e V D P oscillator differs b u t l i t t l e from a h a r m o n i c oscillator so t h a t o(0) ~ p0 = C o n s t . In this case neglecting t h e s m a l l term with e in t h e denominator of (1.9) o n e c a n w r i t e
dp
d--0 -~ - 2e(1 - p0 Cos~ 0)o0 Sin s 0, a n d t h e condition (1.10) o f periodicity gives
f:-
do = - 2e po
Sins OdO
--
po ~
f:-
Cos20 Sins OdO
]
= 0,
(I.11)
whence p0 = r0~ = 4, which is well k n o w n (1).~ If, however, e = 0, condition (1.10) is satisfied identically in (1.11) a n d o n e has no r i g h t t o e q u a t e t h e square b r a c k e t in (1.11) t o zero. In this case o0 c a n n o t b e determined from (1.11) which is also o b v i o u s since in this case t h e V D P oscillator degenerates into a h a r m o n i c oscillator w h o s e energy c o n t e n t is a r b i t r a r y d e p e n d i n g o n t h e initial c o n d i t i o n s : F o r a V D P oscillator, o n t h e contrary, t h e initial conditions u l t i m a t e l y d o n o t play a n y role a n d t h e oscillator "selects," so t o speak, a definite energy cont e n t o0 = 4 as long as ~ << 1 b u t n o t zero. So far we were able m e r e l y t o retrace some well-known f a c t s , 2 The boldface numbers in parentheses refer to the references appended to this paper.
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starting with a n e w d e p e n d e n t variable p, t h e e n e r g y . we shall e n d e a v o r t o elaborate this line of argument.
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In what follows
2. PHASE TRAJECTORY
W e will c o n s i d e r first t h e differential equation (1.9) of t h e p h a s e trajectory assuming e << 1. Expanding t h e r i g h t h a n d of (1.9) into a series o n e gets do
dO
-
2pl-e(1 - p Cos2 0) Sin"- 0 + d(1 - p Cos"- 0)2 X X S i n 3 0 C o s 0 + d(1 - p Cos"- 0)3Sin4 0 Cos'-' 0 + . . . ] ,
(2.1)
w h e r e p = p(O) is a n u n k n o w n periodic function of 0 w h i c h we will e n d e a v o r t o determine b y t h e perturbation method. If one s e t s o = p0 + ep~ + do2 + ".- in t h e preceding e q u a t i o n , o n e o b t a i n s a series of successive approximations arranged according t o t h e o r d e r of e.
(a) Approximation of order zero dpo -0; dO
oo = K0 = C o n s t .
(b) First approximation From (2.1) one has
dpl dO
2(1 - po Cos"- O)po Sin s 0,
whence p~ = K1 - po(1 - -~po)O + ,~po Sin 20 - -~p0"- Sin 40,
(2.2)
K1 being a n integration c o n s t a n t . The s e c o n d term o n t h e r i g h t of (2.2) is clearly a secular term a n d c a n n o t exist in a stationary s t a t e . Determining t h e c o n s t a n t p0 so as t o eliminate this term o n e o b t a i n s a g a i n p0 = 4 so t h a t (2.2) becomes m = K I + 2 Sin 20 - Sin 40.
(2.3)
The c o n s t a n t K1 is determined in tile next approximation.
(c) Second approximation Equating t h e t e r m s with ~2 in (2.1) one o b t a i n s dp.z
dO
-
-
2pt Sin2 0 + 4pip0 Sin2 0 Cos" 0 - 2p0 Sin a 0 Cos 0 + + 4p02 Sin~ 0 Cos~ 0 - 2pd Sin3 0 Cos~ 0.
Replacing pt a n d p0 b y t h e i r v a l u e s , p a s s i n g t o t h e m u l t i p l e arguments, integrating, a n d eliminating t h e s e c u l a r term (which yields K1 = 0)
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one gets 3 1 .~ 1 p2 = K 2 - ~ C o s 2 0 + ~ C o s 4 0 + l ~ C ° s 6 0 - ~ C ° s 8 0 '
(2.4)
where K~ is a constant of integration.
(d) Third approximation One obtains K2 = ~ and
1 5 Sin 40 + 23 37 Sin 80 p3 = - - - S i n 2 0 - 192 ~ Sin 60 - 192 12 -485Sin100+~4Sin120+K3"
(2.5)
If one limits the approximation up to the terms of the order ,4, the periodic function p(0) is then given by the following expression arranged by the orders of ~: p( O) = 4 +
+ ,(2 Sin 20 - Sin 40)+ +~
~-~Cos20+~Cos40+
+,3 K 3 -
'
)
Cos 60 - ~ Cos 80 +
1 5 23 37 ~ S i n 2 0 - 192 Sin 40 + ~-~ Sin 60 - --192 X
(2.6)
X Sin 80 - 5 Sin 100 + 5Sin 120) + ~
-
•
•
..
If the terms of this array are arranged according to the order of harmonics one gets p(O) = (4+-~2+ .. . ) + + [ ( 2 ~ - - 1 e3+ ...
) Sin 2 0 + ( - ~3 ~2+. • . ) Cos 2O]+
+[(-~-1-~3+'")Sin40+(1~+""
) Cos 40]+
(2.7)
) Sin 6 0 + ( 5 , 2 + ... ) Cos 60]+
+ Jr-
[ ( ,_~_@~3+... 7 •
•
) S i n S 0 + ( - ~1, ~ + . . . ) Cos 80]+
..
The non-written terms in brackets are ascending powers of e. Unfortunately, the calculations beyond the third approximation are so
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209
long t h a t i t b e c o m e s i m p r a c t i c a b l e t o c o n t i n u e t h e m u n l e s s , p e r h a p s , with the aid of s o m e m e c h a n i c a l c o m p u t e r s . I t is p o s s i b l e , h o w e v e r , t o form c e r t a i n g e n e r a l c o n c l u s i o n s r e g a r d i n g the s e r i e s (2.7). 1. T h e coefficients ao, al, bl, as, b2 . . • of the t r i g o n o m e t r i c t e r m s in (2.7) a r e p o w e r s e r i e s in ~ a p p r o a c h i n g zero w h e n E --* 0, e x c e p t a0 w h i c h a p p r o a c h e s the v a l u e 4. T h e a m p l i t u d e s ~/al" + b~2 . . . of the v a r i o u s h a r m o n i c s also a p p r o a c h zero w h e n ~--+ 0. A s t o t h e i r p h a s e s , they a p p r o a c h a t the l i m i t ~ --~ 0 the p h a s e of the t e r m w h o s e coefficient cont a i n s a l o w e r p o w e r of ~. 2. I t s e e m s l i k e l y ( a l t h o u g h the a p p r o x i m a t i o n s in o u r case d i d n o t p r o g r e s s far e n o u g h t o be able t o a s s e r t t h i s ) t h a t for a sufficiently s m a l l ~, the coefficients a , , b, of the t r i g o n o m e t r i c s e r i e s (2.7) a p p r o a c h zero w h e n n --+ oo. If s u c h is the case the t r i g o n o m e t r i c s e r i e s (2.7) is actually a Fourier series. 3. T h e c o n s t a n t t e r m (4 + }~" -i- " " ) m a y be r e g a r d e d a s a " c o m p o n e n t " of a h a r m o n i c oscillator w h i l e the t r i g o n o m e t r i c t e r m s r e p r e s e n t m e r e l y the energy fluctuations b e t w e e n the V D P oscillator a n d the e x t e r n a l s o u r c e . I t s h o u l d be n o t e d t h a t the p r e c e d i n g d i s c u s s i o n r e l a t e s only t o the i n t e g r a l c u r v e p(O) of the V D P o s c i l l a t o r i n a s m u c h as time does n o t e n t e r h e r e . I n the f o l l o w i n g s e c t i o n we s h a l l i n v e s t i g a t e the d e p e n d e n c e on t i m e , t h a t is, the a c t u a l f l u c t u a t i o n s of e n e r g y . It m a y be u s e f u l t o s a y a few w o r d s r e g a r d i n g the c o n v e r g e n c e of the exp a n s i o n (2.1). I n a s m u c h a s this e x p a n s i o n was o b t a i n e d from the f u n c t i o n a p p e a r i n g on the r i g h t side of (1.9) it is c l e a r t h a t the only case w h e n i t is d i v e r g e n t is w h e n dO/dt = 0. This case is o b v i o u s l y r u l e d out for ~ << 1 in w h i c h we are i n t e r e s t e d h e r e . It can be s h o w n , howe v e r , (section 5 b e l o w ) t h a t , even in the case w h e n ~ is l a r g e , dO/dt ~ 0 a l t h o u g h it m a y b e c o m e of the o r d e r of 1/~2 for some s p e c i a l i n t e r v a l s of 0. In t h e s e i n t e r v a l s the c o n v e r g e n c e b e c o m e s so slow t h a t t h e pert u r b a t i o n m e t h o d b e c o m e s i m p r a c t i c a b l e in v i e w of the i m p o s s i b i l i t y of c a r r y i n g the s u b s e q u e n t a p p r o x i m a t i o n s indefinitely on a c c o u n t of a r a p i d l y i n c r e a s i n g c o m p l e x i t y of c a l c u l a t i o n s . F o r this r e a s o n the p r e c e d i n g p r o c e d u r e is a p p l i c a b l e only for ~ < 1 a n d , p r e f e r a b l y , for E << 1. In m a n y a p p l i c a t i o n s , h o w e v e r , one is i n t e r e s t e d p r e c i s e l y in the r a n g e of s m a l l E so t h a t the p e r t u r b a t i o n p r o c e d u r e c a n be u s e d t h e n . A s r e g a r d s the so-called " r e l a x a t i o n r a n g e " (e >> 1), a l t h o u g h the p e r t u r b a t i o n m e t h o d c e a s e s t o be a p p l i c a b l e , the e n e r g y f l u c t u a t i o n s in a V D P o s c i l l a t o r can be s t i l l e x p l o r e d in c o n n e c t i o n with the d a t a o b t a i n e d by the i s o c l i n e m e t h o d a s will be s h o w n in s e c t i o n 5. 3. I~..NERGY FLUCTUATIONS
T h e p r e c e d i n g s t u d y c o n c e r n s the f u n c t i o n p(0) in w h i c h the time does n o t a p p e a r . In o r d e r t o i n v e s t i g a t e the e n e r g y f l u c t u a t i o n s o(t) in
2IO
N. MINORSKY
[J, F. I.
a V D P oscillator it is necessary t o d e a l directly with E q s . 1.7 a n d 1.8 containing t i m e . In o t h e r w o r d s , it is necessary t o determine t h e m o t i o n p(t) o n t h e trajectory o(0). The perturbation procedure remains t h e same as before with t h e exception t h a t it is t o b e carried o u t simultaneously in b o t h variables p a n d 0 considered as functions of t i m e . Equations 1.7 a n d 1.8 c a n b e w r i t t e n
do
d-)- = el0(1 - Cos 20) - 102(1 - Cos 4 0 ) ] ,
(3.1)
dO d-t
=
e[(½ -- ¼o) Sin 20 - -~o Sin 403 - 1.
(3.2)
Since t h e time does n o t a p p e a r explicitly in t h e s e e q u a t i o n s , t h e r e is n o difference between t a n d t + to, to being arbitrary. This m e a n s t h a t t o a given trajectory (or integral curve) o(0) c o r r e s p o n d s a n infinity of motions o(t), O(t) differing b y a n a r b i t r a r y p h a s e to d e p e n d i n g o n t h e initial c o n d i t i o n s . The perturbation procedure is carried o u t b y s e t t i n g = po + col + e202 -t- " " , 0 = 00 -4- e01 -I- d02 -k- " " .
p
Since 0 appears only u n d e r t h e sign of t h e trigonometric functions we h a v e t o e x p a n d Sin 20, Cos 20, • • • into a Taylor's series a r o u n d 0 = 0o, w h i c h gives Cos 20 = Cos 200 - 2(~01(t) + ~202(t) + . . . ) Sin 200 + . . . , Sin 20 = Sin 200 + 2(e01(t) + ¢~02(t) + . . . ) Cos 200 + . . . . Rearranging t h e results according t o t h e powers of e one finds d-2 = el-p0(1 - Cos 200) - 1oo2(1
dt
Cos 400)] -4-
+ ~2~2po01 Sin 20o + o1(1 - Cos 200) - 0o201 Sin 400 - ½ p o m [ 1 - Cos 400)3 + < . . . 3 + . . .
(3.3)
dO = _ 1 + d-(½ - Ip0) Sin 200 - kp0 Sin 40o] +
dt
+ #[-(1 - ½oo)01 Cos 200 - lol Sin 200 -
10001 Cos 400 - -~plSin 40o-] + Of-...-1 + " " .
The approximation of t h e zero o r d e r yields a g a i n
do--2 = O; dt
__dO° = _ 1, dt
t h a t is
00 =
- t,
(3.4)
Sept., ~949.]
FLUCTUATIONS
I N A VAN DER P O L
OSCILLATOR
211
w h i c h c o i n c i d e s with Eqs. 1.5 for the h a r m o n i c oscillators. The e l i m i n a t i o n of the s e c u l a r t e r m in the f i r s t a p p r o x i m a t i o n g i v e s a g a i n p0 = 4. T h e r e a p p e a r s , h o w e v e r , a n o t h e r c o n s t a n t of i n t e g r a t i o n in the s e c o n d e q u a t i o n w h i c h m u s t be d e t e r m i n e d by the initial conditions, the i n d e p e n d e n t v a r i a b l e b e i n g now t. T h e s i m p l e s t w a y of i n t r o d u c i n g the i n i t i a l c o n d i t i o n is t o a s s u m e t h a t for t -- 0, 0(0) = 01(0) = 02(0) =
" ° •
~-
0,
T h e f i r s t a p p r o x i m a t i o n b e c o m e s then
pl(t) = KI - 2 S i n 2t + Sin 4 t ; 01(t) = ~- - } Cos 2t - { - C o s 4t,
(3.5)
w h e r e K1 is a n i n t e g r a t i o n c o n s t a n t d e t e r m i n e d in the s e c o n d a p p r o x i m a t i o n . T h e p r o c e d u r e from now on b e c o m e s o b v i o u s , viz: for t h e s e c o n d a p p r o x i m a t i o n we c o l l e c t the t e r m s with e2 in (3.3) a n d (3.4) a n d it is o b s e r v e d t h a t do2/dt a n d dO2/dt a r e e x p r e s s i b l e now in t e r m s of po, 0o, t)~ a n d 01. I n t e g r a t i n g t h e s e e q u a t i o n s , e l i m i n a t i n g the s e c u l a r t e r m in 02 ( w h i c h l e a d s t o the d e t e r m i n a t i o n of K1) a n d d e t e r m i n i n g the c o n s t a n t of i n t e g r a t i o n in 02 by the a s s u m e d i n i t i a l c o n d i t i o n s , one o b t a i n s the s e c o n d a p p r o x i m a t i o n p2 and 02; one f i n d s 7 2 02 = K 2 + C o s 2 t - ~ C o s 4 t + ~ C o s 6 t , w h e r e K1 = 0. I t is t o be n o t e d t h a t the e x p r e s s i o n for o2(t) is n o t the s a m e as t h a t for p~(0) a n d l i k e w i s e for o t h e r a p p r o x i m a t i o n s so t h a t , in g e n e r a l , the t r i g o n o m e t r i c s e r i e s for p(t) is n o t the s a m e a s t h a t for p(O). T h e r e a s o n for t h a t is due t o the n o n - u n i f o r m i t y of r o t a t i o n of the r a d i u s v e c t o r as is seen from the e x p r e s s i o n for O~(t) w h i c h g i v e s in this a p p r o x i m a t i o n 0 =
- t +
~(~- ¼ C o s 2 t - ~Cos4t).
T h i s s h o w s t h a t t h e r e e x i s t s a kind of a " p h a s e m o d u l a t i o n " a f f e c t i n g all t r i g o n o m e t r i c t e r m s . T h e f l u c t u a t i o n of e n e r g y in time up t o the t e r m s of the o r d e r e'~ is then
p(t) = 4 + ~ ( - 2 S i n 2 t + S i n 4 t ) + + ~2
(
Cos2t-~Cos4t+~Cos6t
)
+K2,
w h e r e the c o n s t a n t of i n t e g r a t i o n K2 is d e t e r m i n e d in the next a p p r o x i m a t i o n a n d so on. 4. E F F E C T O F T H E
FORM O F T H E V A N DER POL E Q U A T I O N O N T H E
FREQUENCY SPECTRUM
So far we h a v e b e e n c o n c e r n e d with Eq. 1.2 o r i g i n a l l y f o r m u l a t e d by v a n d e r Pol (1). L a t e r on, this e q u a t i o n was g e n e r a l i z e d by A.
N. MINORSKY
212
[J. F . I.
L i 6 n a r d (2), E. & H. C a r t a n (3), N. L e v i n s o n a n d O. K. S m i t h (4) a n d o t h e r s . T h e g e n e r a l form of the v a n d e r Pol e q u a t i o n is 2 -- ef(x, 2)2 + g(x) = O,
(4.1)
w h e r e the f u n c t i o n s f a n d g m u s t s a t i s f y c e r t a i n c o n d i t i o n s (4). W e s h a l l give a few e x a m p l e s r e l a t i v e t o the e l e c t r o n t u b e c i r c u i t s a s s u m i n g t h a t they do n o t i n v o l v e a n y n o n - l i n e a r c a p a c i t i e s in w h i c h case g(x) = x, a n d c o n s i d e r f ( x , 2) = f ( x ) in the L i 6 n a r d s e n s e , viz: f ( x ) is a n even f u n c t i o n of x so t h a t F(x) = f f ( x ) d x is o d d a n d p o s i t i v e for s m a l l v a l u e s of x a n d b e c o m e s n e g a t i v e b e g i n n i n g with x = x0, d e c r e a s i n g m o n o t o n i c a l l y for x > x0. T h e p o l y n o m i a l s of the form F(x) = a x - b x a ; F(x) = ax + bxa - cx5 e t c . , c l e a r l y , s a t i s f y this c o n d i t i o n a n d are f r e q u e n t l y u s e d in a p p l i c a t i o n s . T h u s , for i n s t a n c e , w h e n F(x) is a p o l y n o m i a l of the fifth d e g r e e , the v a n d e r Pol e q u a t i o n h a s the form 2 - ~(1 + ~ x2 - 5x4)2 + x = 0.
(4.2)
U s i n g t h e p r e c e d i n g p r o c e d u r e one gets dp -dO
2p[-¢( ) Sin 2 0 + ~2( )= S i na 0 Cos 0 + ca( )a Sin 4 0 C o s~ 0 +
. . . ],
w h e r e ( ) = (1 + ap Cos 2 0 - /~p2 Cos 4 0). T h e p e r t u r b a t i o n m e t h o d l i m i t e d t o the z e r o , a n d the f i r s t a p p r o x i m a t i o n g i v e s p0 = K0 = C o n s t ; pl = -- 2 [ ' p o f S i n s OdO 4- o~po2 f Cos 2 0 Sin 2 OdO -- 3po3 f Cos 40 S i ns OdO] + K1 a n d the e l i m i n a t i o n of the s e c u l a r t e r m g i v e s oo =
+
+
so t h a t , up t o the t e r m s of the o r d e r e~ one gets
'
o =p0+~pl =p0- ~
+
p0+~-~/~00'
)
Sin 2 0 +
1
l po~ ( a - -~ [3 ) Sin 40 -
l ~po' Sin 60 ] + K2.
T h e r e e x i s t s o b v i o u s l y n o p0 for j3 = 0 s i n c e the d a m p i n g t e r m is t h e n n e g a t i v e for all v a l u e s of x. F o r a = 0, B = 1, (4.2) b e c o m e s -
~(1 - x')2 + x = 0,
(4.5)
in w h i c h case o0 = 2V2. I n t e r e s t i n g c a s e s a r i s e w h e n the form of the v a n d e r Pol e q u a t i o n for a n e l e c t r o n t u b e oscillator is m o d i f i e d by the p r e s e n c e of c e r t a i n
Sept., 1949.]
FLUCTUATIONS
I N A VAN DER POL
OSCILLATOR
2I 3
variable conductors in the oscillating circuit. We shall consider one such case in connection with the circuit shown in Fig. 1. If all p a r a m eters (L, C and R) are constant, it is well known (5) that the van der Pol equation in this case is of the form d2v dr2
dv (fl + 2~v - 3~v2) dr + v = O,
(4.6)
where v = vu/v~ is a dimensionless variable (v~ is the grid voltage ; v~ the
t_ R
FIG. 1.
constant "saturation voltage"), r = ~0t ; c00 = ~ / - f / L C ; ~ = ( ~ 1 -- RC)o~o '*/ ----- ~k'Y1600;
~ ----- ~k~l~O0;
81 = S 1 ;
"~'1 = ,~2Ps;
~1 ~" - - S 3 p s 2,
and the non-linear function, the a n o d e current, is assumed to be represented by the polynomial /~ = ¢ ( ~ ) = I0 + $ 1 ~ + $2~o~ + $ 3 ~ / . Let us modify the problem by replacing the constant resistor R by a variable one R ( I i l ), depending on the absolute v a l u e of the current and
214
N. MINORSKY
[J. 1~. I.
a s s u m e , m o r e o v e r , t h a t the non-linearity of R ( ] i ] ) a p p e a r s before the non-linearity of the e l e c t r o n t u b e is felt. In o t h e r w o r d s , we wish t o c o n s i d e r the case w h e n the e l e c t r o n t u b e o p e r a t e s s t i l l on the r e c t i l i n e a r p a r t of its c h a r a c t e r i s t i c ( t h a t is, "r = ~ --~ 0). E q u a t i o n 4.6 in s u c h a case b e c o m e s d2v
d~ 2
Ex~l -
R ( [ i l ) C ] d~ dr + v = 0.
(4.7)
It is necessary to make a certain assumption regarding the function R([i]). Assume, for example, that the resistance of the non-linear resistor is of the form R(Ii[)
= m +niil,
where m and n are positive constants, that is, the resistance increases as a linear function of the absolute v a l u e of the current which is generally the case for conductors having a positive temperature coefficient. On the other hand vo =
whence
{f
idt,
dp (t v
so that
Ill = Cv,cOo d~ and (4.7) becomes dr ~
~
dr ] dr + v = O,
where = (X~I - . m C ) ;
b = nC2v,~Oo.
The preceding equation becomes d2v
-tiT2 ---
~
(
1 - K ddr~ ] d~ ~r
+ v =0,
K = -b"
The equivalent system is dv
--Y;
dy ~(1 -- K l y ] ) y - v. d-; =
Assuming that e and r are small, Eq. 1.9 in this case is do 2e( )o Sin2 0 dO ~( ) S i n 0 C o s 0 - 1 -
-
=
=
- - 2 o [ - e ( ) S i n 2 0 + e2( )2 S i n a 0 C o s 0 + ~3( )3 S i n ~ 0 C o s2 0 + . . - ~ ,
where ( ) = (1 -- K r l S i n 0[) = (1 -- Ko~[Sin 01).
Sept., I949.]
ITLUCTUATIONS I N A VAN DER I'OL
OSCILLATOR
215
T h e perturbation method gives dP-2° = 0 ; dO
p0 = K0 = C o n s t .
d m = _ 2(1 - K p 0 ~ ] S i n 0])p0 S i n2 0 = dO = - p0 + o0 Cos 20 + K p 0 ~ ( I S i n 01 - Cos 2 0 1 S i n 01). Replacing ISin 0[
. . . .7
C o s 2 0 ] S i n 0] =
7r
~ C o s 2 0 +
Cos 4 0 +
Cos 6 0 + . . .
76 Cos 40 . . . . - 3~r -2- + 28 ~ Cos 20 - 105---~
a n d e l i m i n a t i n g the s e c u l a r term one o b t a i n s p0
=
r02
=
8 K /
'
w h i c h s t a r t s the p e r t u r b a t i o n p r o c e d u r e w h i c h we will n o t c o n t i n u e h e r e . I t is w o r t h m e n t i o n i n g m e r e l y t h a t in this case we e n c o u n t e r a s i t u a t i o n in w h i c h a l r e a d y in the f i r s t a p p r o x i m a t i o n t h e r e a p p e a r s the w h o l e t r i g o n o m e t r i c s e r i e s i n s t e a d of t r i g o n o m e t r i c p o l y n o m i a l s cont a i n i n g b u t a few t e r m s a s this was the case for Eqs. 1.2 a n d 4.5. T h e f r e q u e n c y s p e c t r u m of e n e r g y f l u c t u a t i o n s in a n oscillator of this kind is, therefore, far r i c h e r in h a r m o n i c s t h a n in the c a s e s j u s t m e n t i o n e d . T h e s e e x a m p l e s show t h a t , f o l l o w i n g this p r o c e d u r e , it is a l w a y s p o s s i b l e t o form a n idea a s t o t h e f r e q u e n c y s p e c t r u m of e n e r g y f l u c t u a t i o n s in a V D P o s c i l l a t o r for a g i v e n n o n - l i n e a r f u n c t i o n F(x) p r o v i d e d it satisfies the L i ~ n a r d c o n d i t i o n a n d p r o v i d e d ~ is sufficiently s m a l l . 5.
cAs~
~
~ is
L~GE
T h e p e r t u r b a t i o n m e t h o d c e a s e s t o be a p p l i c a b l e w h e n ~ is n o t s m a l l . C e r t a i n a d d i t i o n a l c o n c l u s i o n s c a n be s t i l l o b t a i n e d from (1.7), (1.8) a n d (1.9) a l s o , in this c a s e , w i t h the aid of the d a t a y i e l d e d by the m e t h o d of isoclines. I t is t o be n o t e d t h a t the m a x i m a a n d m i n i m a of both p o l a r c u r v e s 0(0) a n d r(O) are s i t u a t e d on t h e s a m e r a d i i - v e c t o r s s i n c e the 0(0) c u r v e is o b t a i n e d m e r e l y by s q u a r i n g the r a d i i v e c t o r s of the r(O) c u r v e for each 0 a n d , c o n v e r s e l y if p(0) c u r v e is g i v e n , the r(O) c u r v e can be o b t a i n e d b y a n i n v e r s e o p e r a t i o n . \,Ve s h a l l s p e a k p r e f e r a b l y of the e n e r g y c u r v e p(0) i n a s m u c h a s in this m a n n e r it will be p o s s i b l e t o d e r i v e certain p h y s i c a l c o n c l u s i o n s r e g a r d i n g the V D P oscillator b u t o c c a s i o n a l l y we s h a l l r e f e r also t o the i n t e g r a l c u r v e r(O) in view of the a b o v e r e l a t i o n b e t w e e n t h e s e two c l o s e d c u r v e s . F r o m (1.7) it f o l l o w s t h a t the m a x i m a a n d m i n i m a of e n e r g y o c c u r w h e n p(0) = l / C o s20 w h i c h in c a r t e s i a n c o o r d i n a t e s is o b v i o u s l y x = =t= 1.
216
N. 1V[INORSKY
[J. F. I.
For these particular values of 0 the term with i in (1.2) vanishes and the VDP oscillator coincides instantaneously with the harmonic oscillator of this particular (Om~x or pmi,,) energy content. One can also verify this circumstance by replacing p = l/Cos" 0 in (1.8) w h i c h gives 0 = - 1 w h i c h in conjunction with the condition ~ = 0 gives conditions (1.5) for a harmonic oscillator. In o r d e r to distinguish m a x i m a from m i n i m a it is sufficient to calculate )5 which gives = 2d-~(Sin2 0 - 20 Sin2 0 Cos2 0) + 200(Sin 0 Cos 0 + p Sin3 0 Cos 0 - 0 Sin 0 Cos3 0)7. Replacing in this, expressions ~ = 0, 0 = - 1, p = 1 / C o s~ o, one finds )5 = - 4 ~ T a n 30, w h i c h shows that the m a x i m a of energy are located in the first and the t h i r d quadrants of the p o l a r curve 0(0) and m i n i m a in the second and in the fourth quadrants w h i c h also agrees with the data yielded by the m e t h o d of isoclines. Similarly, when ~ increases the isocline m e t h o d shows that the m a x i m a increase and the m i n i m a decrease but since the condition p Cos2 0 = 1 is independent of ~, it is clear that the values of 0 corresponding to Pm~ approach the y axis and those for pmi, recede to the x axis while still being in t h e i r respective quadrants. This also agrees with the data furnished by the isocline method. T h e r e exists thus a definite relation between the magnitude of the m a x i m a and m i n i m a of energy and t h e i r phase in the p o l a r curve. For a sufficiently large , and in the neighborhood of m a x i m a of the energy content, the rate of change of energy ~ varies very r a p i d l y even for a small variation 50 around the value 0 = 0 . . . . Thus, for instance, from the isocline curve for ~ = 30, pm~ ~ 1300 and 0 .... ~ 88.4°. For A0 = 4- 0.1°, # changes from about - 1 3 0 0 0 (for A0 = -- 0.1 °) to a b o u t + 1 1 0 0 0 (for A0 = + 0.1 °) in the dimensionless units used here, whereas in the neighborhood of m i n i m a the same variation A0 accounts for a small part of one per cent of the above variation. If one wishes to give a physical analogue of the behavior of the VDP oscillator in the neighborhood of its m a x i m a energy points, at least as far as energy fluctuations are concerned, perhaps a pneumatic hammer or a similar device may give an adequate analogy. In fact in a device of this kind a r a p i d acceleration of the hammer with the incident increase in its energy content (which is the kinetic energy here) is followed by a not less r a p i d deceleration when the hammer strikes an energy absorbing m e d i u m and thus loses its energy. In addition to one maximum and one minimum of energy per one half cycle (the o t h e r half cycle being identical) the VDP oscillator has also one stationary point for 0 = 0 (for the other half-cycle this stationa r t point occurs for 0 = 7r) a t w h i c h the energy passes through a stat i o n a r y v a l u e without being either maximum or minimum. One sees
~el)t., I949.]
FLUCTUATIONS
I N A VAN DER
PoL O,gCILLATOI{
2I 7
this from (1.7) w h i c h s h o w s t h a t # p a s s e s t h r o u g h zero a t t h e s e p o i n t s w i t h o u t c h a n g i n g its s i g n . O n e n o t e s also t h a t ? = 0 a t this p o i n t . W h e n e is l a r g e t h e s e s t a t i o n a r y p o i n t s are c l o s e t o the m i n i m a p o i n t s of e n e r g y . T h e i n v e s t i g a t i o n of the b e h a v i o r of a n i n t e g r a l c u r v e in the n e i g h b o r h o o d of t h e s e s t a t i o n a r y p o i n t s is far from b e i n g s i m p l e , if E
5q ~2
50 28 2a 22
20 18
-i6
J - i s -x.'o -o:5
-
lO a?
zo
z3
2.o
FIG. 2.
is l a r g e e n o u g h a n d this is p r o b a b l y one of the p r i n c i p a l difficulties of the v a n d e r Pol e q u a t i o n . T h e m e t h o d of i s o c l i n e s s h o w s , in f a c t , t h a t t h e r e is a s h a r p b e n d of i n t e g r a l c u r v e s in this " r e l a x a t i o n r e g i o n " a s c a n be seen from Fig. 2, r e p r e s e n t i n g the i n t e g r a l c u r v e r(O) for E = 30 cons t r u c t e d b y the m e t h o d of isoclines. ~ If one f o l l o w s this i n t e g r a l c u r v e The author is indebted to Miss E. Yost for this construction.
218
N. MINORSKY
[J. F. I.
in the positive direction ( t h a t is, clockwise) one finds that it cuts the x-axis at r i g h t angles (point A) but immediately a f t e r that the integral curve has an a l m o s t 90° bend and follows practically the x-axis t o w a r d the origin. The accuracy of the graphical construction is obviously insufficient to be able to specify the details of what happens in this critical region and it is necessary to make a closer analysis. We shall indicate first an approximation based on the results of the isocline data and shall analyze the situation l a t e r independently of the m e t h o d of isoclines. Let us assume, as an approximation, that in this "relaxation region" (say between x ~ 1.9 and x ----- 1.2 in Fig. 2) the integral curve is parallel to the x-axis, that is, coincides with the zero isocline whose equation is y
= x/
(1
-
If one replaces this value of y into the expression for 0 in the x, y coordinates, viz: 0 = - 1 + ~(1 - x~)xy/o, a f t e r a few transformations, one finds the following approximate expression for 0 in this region 0=
1
d(1
-
x ~ ) 2'
which shows that 0 is small of the order 1 / d and negative as everywhere else on the integral curve. In view of this, from (1.8) we get ,(1 - 0 Cos~ o) ~ 1/Sin 0 Cos O, so that (1.7) becomes dpp ~ 20 Tan 0 --~ 200.
do--
(5.1)
Since 0 < 0 in this region is small and changes very slowly, if one t a k e s its average v a l u e 0 in this interval, (5.1) yields an approximate formula p = poe~°~
(5.2)
w h e r e p0 corresponds to the beginning of this "relaxation" region. This approximate relation shows that the energy is slowly drained away .from the system in a m a n n e r in which the electrostatic energy of a charged condenser disappears gradually owing to an imperfect dielectric. This somewhat crude analysis is merely a physical interpretation of the results yielded by the m e t h o d of isoclines. For that reason it is preferable to investigate the behavior of the functions b(0) and 0(0) in a small interval (01, -01); 01 > 0 a r o u n d 0 = 0 (the a r g u m e n t applies also to 0 = ~r). This will permit obtaining certain conclusions directly without relying on the m e t h o d of isoclines.
Sept., 1949.]
F L U C T U A T I O N S IN A VAN D E R P O L
OSCILLATOR
2I 9
Differentiating (1.8) with respect to 0 and setting 0 = 0 a f t e r the differentiation one gets
(
(dO)d_O o = - ~ ( p o - 1)" =
d20)o = O;
=
dO~ / o so that 02
0(0) -- 00 + 0
0 ÷
( " ) 0
03
= -- 1 -- ~(p0-- 1)0 + 4 5 p 0 - - 2)-~ + . . . .
(5.3)
Since 0 is small and p0 is of the o r d e r 4 one can limit this expansion to the l i n e a r term in 0. U n d e r this assumption, 0 becomes very small for 0<0 such that ~(p0-1)] 01 is of the o r d e r of unity, that is [al ~0(1/3~). In such a case the term with ~03 can be neglected as we did. If, therefore, the interval 01, -01 in which we propose to investigate the behavior of ~(0) and 0(0) is fixed on this basis, 0 < 0 is a monotonically increasing function. In fact 0 -- - 1 - K 0 has the values 10101 = = 1 + K 0 1 ; ]0Io = l a n d [ 0 [ - o , = l - K 0 1 . In a similar m a n n e r differentiating (1.7) with respect to 0 and setting 0 = 0 a f t e r the differentiation, one gets =
=
-
-
As, moreover, ~0 = ~(0) = 0, one gets ~(0) - ~(0) + 0
~
0 + ~
~-~/0 +
"
~ - 2~p0(p0 - 1)¢-.
(5.4)
Since ~I 0] is assumed to be of the o r d e r one, the preceding expression shows that ~(0) undergoes a small variation of the o r d e r 0 in the interval (01, -01). As first approximation we can assume, therefore that p ~ p0 in this interval. We shall indicate l a t e r how a second approxim a t i o n can be obtained. It is useful now to specify the interval (01, - 0 1 ) , 01 > 0 in which the preceding assumptions apply. In the construction of the curve of Fig. 2, ~ = 30 was assumed which is a purely a r b i t r a r y value selected with a view to go a little beyond the v a l u e (~ = 10) w h i c h van der Pol used in his construction. This permits obtaining a more pronounced "relaxation region" between the points A and B. On the o t h e r hand, van der Pol, in his analysis of a relaxation oscillation circuit with constants commonly encountered in applications, finds ~ ~ 3.10L that is K ~ 1 0 6. This gives the value of 0 < 0, for w h i c h 0 becomes very
:220
N. MINORSKY
[J. F. 1,
s m a l l , of the o r d e r of 1 sec. of arc. T h i s m e a n s t h a t the m o t i o n of the r e p r e s e n t a t i v e p o i n t w h i c h is p r a c t i c a l l y t a n g e n t i a l ( w i t h r e s p e c t t o the r a d i u s v e c t o r ) for 0 = 0, b e c o m e s r a d i a l for 0 ~ - 1 s e c . , w h i c h g i v e s a n i d e a a s t o the o r d e r of m a g n i t u d e of the i n t e r v a l (01, - 0~) in q u e s t i o n for the v a r i o u s v a l u e s of ~. A s r e g a r d s the f u n c t i o n ~(0), (5.4) s h o w s t h a t , u n d e r the a s s u m e d a p p r o x i m a t i o n , it is a n even f u n c t i o n of 0; m o r e o v e r ~(0) is n e g a t i v e in this i n t e r v a l , t h a t is, the e n e r g y d e c r e a s e s b e t w e e n 01, a n d - 0 1 . W e p r o p o s e now t o c a l c u l a t e the loss of e n e r g y - A p l , a n d - A p 2 in t h e h a l f - i n t e r v a l s (0~, 0) a n d (0, - 0 1 ) , respectively, w h e n the r a d i u s v e c t o r t r a v e r s e s the s m a l l arc (01, - 0 1 ) . O n e h a s -
-
Apl ----
f0 i
i~dt;
-- 5p2 =
ixtt.
F r o m (5.3) a n d (5.4) we h a v e o =
w h e r e K = e(p0 -- 1).
= _ 2poK92,
- (1 + Ko);
Thus 02
[Jdt = 2poK 1 nt- K~O dO a n d , therefore, -- Apl = 2poK fo° O2KodO; ~1+
-- Ap2 = 2poK fo -°~
dO. 0~ I +KO
U n d e r this a s s u m p t i o n --Am=
2p0 Ks[½(I+K01) 2 - 2 ( 1 + K 0 , ) + l n ( 1 + K 0 1 ) +~],
(5.5)
--Ap2 =
~ [ ½ ( 1 - - K 0 1 )2 - 2 ( 1 - K 0 , ) + l n ( 1 - K O 1 ) + ~ ] .
(5.6)
S i n c e in t h e half-interval (0, - 0 1 ) the q u a n t i t y 1 - KOt > 0 b e c o m e s v e r y s m a l l , In (1 ~ KO~) is a v e r y l a r g e n e g a t i v e n u m b e r so t h a t Ap2 >> Apl. On the o t h e r hand, since --Apl and --Ap2 represent the decrease of the r a d i u s vector in (01, 0) and (0, - 0 1 ) , respectively, one obtains a situation depicted in Fig. 3 which explains the origin of a s h a r p bend in the curve of energy p(0) (and also the integral curve r(O)) in the negative neighborhood (0 < 0) of the stationary point A. Once the dissymmetry of 0(0) has been ascertained starting from the assumption that p ~ p0 in this angular interval (0t, - 0 1 ) , it is possible to c a r r y out the second approximation using, for example, the values p0 + Am/2, po - A p 2 / 2 instead of p0 (and the corresponding values Kt and K2) in both half-intervals w h i c h gives a more correct estimate for Apl and Ap2 and thus calculate the shape of the p(0) curves in this neighborhood.
Sept., 1949.]
I~LUCTUATIONS I N A VAN DER POL
OSCILLATOR
22I
T h e s h a r p b e n d of the c u r v e s 0(0) a n d r(O) is t h u s due t o a v e r y m a r k e d a s s y m m e t r y of the f u n c t i o n 0(0) in this r e g i o n w h i l e the f u n c tion #(0) r e m a i n s s y m m e t r i c a l a b o u t t h e s e s t a t i o n a r y p o i n t s . F r o m a p h y s i c a l s t a n d p o i n t this c i r c u m s t a n c e m e a n s t h a t , on a c c o u n t of a s m a l l e r v e l o c i t y 0 in the (0, - 01) half i n t e r v a l the s a m e d r a i n of e n e r g y (for the s a m e 0) p e r s i s t s for a l o n g e r time in the n e g a t i v e half i n t e r v a l than in the p o s i t i v e one w h i c h is t r a v e r s e d r a p i d l y , so t h a t the a d t u a l d e c r e a s e of e n e r g y is g r e a t e r in (0, - 0 1 ) t h a n in (01, 0). It m u s t be n o t e d t h a t for v e r y l a r g e v a l u e s of E, like t h o s e m e n t i o n e d by v a n d e r Pol, the b e n d of i n t e g r a l c u r v e s b e c o m e s so s h a r p t h a t the a n a l y t i c i t y of the c u r v e in this r e g i o n is bad. F o r t h a t r e a s o n the T a y l o r e x p a n s i o n in this r e g i o n b e c o m e s r a t h e r c o m p l i c a t e d a s c o m p a r e d t o the r e m a i n i n g p o r t i o n s of the c u r v e . I t is l i k e l y t h a t t h e s e local c o m p l i c a t i o n s in the
I
Fro. 3.
a n a l y t i c a l s t r u c t u r e of t h e i n t e g r a l c u r v e a t the b e g i n n i n g of this " r e l a x a t i o n " r e g i o n , is the m a i n r e a s o n w h y n o e x p l i c i t s o l u t i o n ~of] the v a n d e r Pol e q u a t i o n (for e x a m p l e , in the form of a s e r i e s ) c o u l d be f o u n d a s y e t , in s p i t e of n e a r l y t w e n t y y e a r s of effort, a l t h o u g h c o n s i d e r a b l e p r o g r e s s h a s r e c e n t l y b e e n a c c o m p l i s h e d in c o n n e c t i o n with a q u a l i t a tive a n a l y s i s of this d i f f i c u l t p r o b l e m (6). CONCLUDING REMARKS
T h e m o s t i n t e r e s t i n g f e a t u r e of the V D P oscillator is the fact t h a t its b e h a v i o r is w i d e l y d i f f e r e n t for d i f f e r e n t v a l u e s of the p a r a m e t e r ~. A t one end of the r a n g e (~ << 1) the V D P oscillator d i f f e r s v e r y little from the h a r m o n i c oscillator a n d t h u s offers p r a c t i c a l l y a p e r f e c t p h y s i c a l i m a g e of the l a t t e r . P r o b a b l y it is n o t a n e x a g g e r a t i o n t o s a y t h a t , s i n c e the time of G a l i l e o w h o i n t r o d u c e d the c o n c e p t of the
222
N. MINORSKY
[J. F. I.
h a r m o n i c oscillator, we w e r e a b l e t o a c t u a l l y o b s e r v e a p r a c t i c a l l y p e r f e c t p h y s i c a l i m a g e of the l a t t e r only w h e n the V D P o s c i l l a t o r b e c a m e a v a i l a b l e r e l a t i v e l y a s h o r t time ago. In f a c t , v e r y likely, t h e r e e x i s t s no b e t t e r p h y s i c a l i m a g e of a s i m p l e h a r m o n i c m o t i o n t h a n t h a t •w h i c h is p r o d u c e d by m o d e r n h i g h q u a l i t y e l e c t r o n t u b e oscillators, p a r t i c u l a r l y t h o s e w h o s e f r e q u e n c y is s t a b i l i z e d b y q u a r t z u n i t s . B u t this a l m o s t p e r f e c t p h y s i c a l i m a g e c o n t a i n s a g e r m of a n i m m e n s e comp l e x i t y due t o the p r e s e n c e of a n infinite s p e c t r u m of f r e q u e n c i e s with w h i c h the e n e r g y f l u c t u a t e s b e t w e e n the oscillator a n d t h e s o u r c e of e n e r g y . T h e s e f l u c t u a t i o n s e s c a p e o u r o b s e r v a t i o n , h o w e v e r , if ~ is very s m a l l a n d , as the r e s u l t of t h i s , w h a t is a c t u a l l y o b s e r v e d is only the c o n s t a n t t e r m in the infinite a r r a y of t e r m s (2.7). H o w e v e r , i t is p r e c i s e l y this h i d d e n c o m p l e x i t y of v a n i s h i n g l y s m a l l e n e r g y f l u c t u a t i o n s w h i c h p e r m i t s o b t a i n i n g a s i m p l e h a r m o n i c oscillation in its apparently pure form. On the o t h e r end of the r a n g e (~ >> 1) the b e h a v i o r of t h e V D P o s c i l l a t o r is r a d i c a l l y d i f f e r e n t from t h a t of a h a r m o n i c oscillator. If one h a s t o look for a n a n a l o g y , a s far as e n e r g y f l u c t u a t i o n s are conc e r n e d , in this r a n g e the V D P o s c i l l a t o r r e s e m b l e s a p n e u m a t i c h a m m e r or a s i m i l a r d e v i c e in w h i c h the e n e r g y f l u c t u a t i o n s ( f r o m a b s o r p t i o n t o d i s s i p a t i o n a n d vice v e r s a ) o c c u r in a q u a s i - d i s c o n t i n u o u s m a n n e r . On the o t h e r h a n d , a c l o s e r i n v e s t i g a t i o n of the b e h a v i o r of the o s c i l l a t o r r e v e a l s t h a t in s p i t e of this a p p a r e n t difference of its b e h a v i o r for d i f f e r e n t v a l u e s of ~, t h e r e e x i s t s a f e a t u r e c o m m o n t o all c a s e s , n a m e l y the e n e r g y r e s u m e s exactly the s a m e v a l u e a f t e r one p e r i o d a l t h o u g h d u r i n g the p e r i o d it f l u c t u a t e s b e t w e e n the o s c i l l a t o r a n d the s o u r c e with infinitely m a n y f r e q u e n c i e s of e v e n h a r m o n i c s . A t t i m e s the e n e r g y is a b s o r b e d , a t t i m e s it is d i s s i p a t e d b u t the time i n t e g r a l s of all t h e s e f l u c t u a t i o n s w i t h d i f f e r e n t f r e q u e n c i e s over the p e r i o d are zero a n d t h e r e r e m a i n s only the a v e r a g e c o n s t a n t t e r m a r o u n d w h i c h t h e s e f l u c t u a t i o n s o c c u r . If t h e s e f l u c t u a t i o n s are v a n i s h i n g l y s m a l l in comp a r i s o n with the c o n s t a n t e n e r g y c o n t e n t of the V D P oscillator, it b e h a v e s a p p r o x i m a t e l y a s a c o n s e r v a t i v e s y s t e m . If ~>> 1, o n the c o n t r a r y , the f l u c t u a t i o n s of e n e r g y d o m i n a t e e v e r y t h i n g else t o such a n e x t e n t t h a t it b e c o m e s even i m p o s s i b l e t o a n s w e r the q u e s t i o n : a r o u n d w h i c h a v e r a g e c o n s t a n t v a l u e of e n e r g y do t h e s e f l u c t u a t i o n s o c c u r ? In a d d i t i o n t o t h e s e f l u c t u a t i o n s of e n e r g y , t h e r e e x i s t also f l u c t u a t i o n s in the a n g u l a r v e l o c i t y of the r a d i u s v e c t o r of t h e r e p r e s e n t a t i v e p o i n t m o v i n g on the i n t e g r a l c u r v e . If ~ is s m a l l e n o u g h t h e s e f l u c t u a t i o n s are also s m a l l so t h a t , from this s t a n d p o i n t , the m o t i o n on i n t e g r a l c u r v e s d o e s n o t d i f f e r m u c h from t h e c o r r e s p o n d i n g m o t i o n for t h e h a r m o n i c o s c i l l a t o r in w h i c h case t h e s e f l u c t u a t i o n s a r e r i g o r o u s l y z e r o . F o r a n i n c r e a s i n g ~ t h e s e f l u c t u a t i o n s in a n g u l a r v e l o c i t y b e c o m e m o r e a n d m o r e p r o n o u n c e d a n d , for v e r y l a r g e v a l u e s of ~, the a n g u l a r v e l o c i t y b e c o m e s of the o r d e r 1 / , 2, t h a t is v e r y s m a l l , in the n e i g h b o r h o o d of the
Sept., 1949.]
FLUCTUATIONS
I N A VAN DER POL
OSCILLATOR
22 3
p o i n t s a t w h i c h the e n e r g y p a s s e s t h r o u g h a s t a t i o n a r y v a l u e . In this n e i g h b o r h o o d one o b s e r v e s the i n t e r e s t i n g p h e n o m e n o n of a " r e l a x a t i o n a l " d r a i n of e n e r g y from the s y s t e m a n d r e s u l t s in a s h a r p b e n d in the i n t e g r a l c u r v e . In s p i t e of a n u m b e r of c o m m o n f e a t u r e s b e t w e e n a h a r m o n i c , a n d a V D P o s c i l l a t o r for ~ << 1, t h e r e e x i s t s a f u n d a m e n t a l difference bet w e e n the two oscillators, n a m e l y : a h a r m o n i c o s c i l l a t o r can oscillate with any e n e r g y c o n t e n t p r e s c r i b e d b y the initial c o n d i t i o n s , w h e r e a s the e n e r g y c o n t e n t of a V D P oscillator does n o t d e p e n d a t all on t h e s e c o n d i t i o n s . In fact, the o s c i l l a t o r " s e l e c t s " so t o s p e a k its own e n e r g y c o n t e n t c o n s i s t e n t with the form of the n o n - l i n e a r f u n c t i o n F(x) as w e l l as with the p a r a m e t e r s of the s y s t e m . This r e m a r k a b l e p e c u l i a r i t y of the V D P oscillator, a s we saw, is due t o the fact t h a t a s t a t i o n a r y p e r i o d i c m o t i o n in this case is p o s s i b l e only in the case w h e n all s e c u l a r t e r m s are r e d u c e d t o z e r o , if we wish t o t h i n k in t e r m s of the p e r t u r b a tion t h e o r y . T h e r e q u i r e m e n t for the e l i m i n a t i o n of s e c u l a r t e r m s in the infinite f r e q u e n c y s p e c t r u m of e n e r g y f l u c t u a t i o n s t h u s i m p o s e s a r e q u i s i t e n u m b e r of a d d i t i o n a l c o n d i t i o n s o w i n g t o w h i c h only one p e r i o d i c m o t i o n is p o s s i b l e , i n s t e a d of infinity of s u c h m o t i o n s a s in the case of a h a r m o n i c oscillator. Acknowledgments T h e a u t h o r is i n d e b t e d t o P r o f . M . S c h i f f e r for v a l u a b l e d i s c u s s i o n s of this m a t t e r . This w o r k was c a r r i e d out u n d e r a r e s e a r c h p r o g r a m of the Office of N a v a l R e s e a r c h , N a v y D e p a r t m e n t . REFERENCES
(1) BALTH, VAN DER POL, "On Relaxation Oscillations," Phil. ,1fag., Vol. 27 (1926). - - - - - "Oscillations Sinusoidales et de Relaxation," l'Onde Eleclrique, Vol. 9 (1930). - "The Non-Linear Theory of Electrical Oscillations," Proc. I R E , Vol. 22 (1934). (2) A. LIft;NARD, "Etude des Oscillations Entretenues," Revue G~ndrale de l'Electric#d, Vol. 23 (1928). (3) E . ANDH. CARTAN, "Note sur la G6n~ratlon des Oscillations Entretenues," A n n des P. T. T., Vol. 12 (1925). (4) N . LEVINSON AND O. K. SMITH, " A General Equation for Relaxation Oscillations," Duke Math..Tour., Vol. 9 (1942). (.5) A. ANDRONOW AND S. CHAIKIN, "Theory of Oscillations," Chapter VII, Moscow (1937); English language edition of this book will a p p e a r shortly in Princeton Univ. Press. (6) D. I'LANDERS AND J. STOKER, "The L i m i t Case of Relaxation Oscillations," New York University (1946). J. LA SALLE, "Relaxation Oscillations" (to appear shortly).