ON NON.LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER: III. THE EQUATION ~ - k ( 1 -y~)j,+),=bftk
c o s@t+~)
FOR LARGE k, AND ITS GENERALIZATIONS
B Y
J. E. L I T T L E W O O D in Cambridge
I n t r o d u c t i o n ~
1. We are concerned with equations in real variables of the form
~] + l (y) ~j + g (y) = p (t),
w t ~ r e "I, g, P are s m o o t h functions of their a r g u m e n t s , a n d p has period ~ = 2.~///~ in t. A b o u t / we suppose t h a t l i m / > 0 as y - - > + o ~ ; t h a t is to say, we suppose t h e
" d a m p i n g " to be positive for large l yl. A b o u t g we suppose t h a t it has a " r e s t o r i n g "
effect, i.e. has t h e sign of y. T h e simplest case, a n d a specially i m p o r t a n t one, to be covered in a n y generalization, is g - a y for positive a. W e do in fact assume always t h a t g ( 0 ) = 0, a n d t h a t g' exists a n d has a positive lower bound.
There is some general t h e o r y of such equations. A t r a j e c t o r y (or " m o t i o n " ) w i t h initial conditions y ( t o ) - ~, ~)(to)=~ (~, ~ real) at some fixed t = t o is said to have t h e p o i n t P = (~, ~) as " r e p r e s e n t a t i v e p o i n t " . I f ~', 9' are t h e values of y, 9 at t - t 0 + 2 t h e t r a n s f o r m a t i o n T f r o m P to P ' = (~', ~]')= T P - T ( ~ , ~]) is 1 - 1 a n d continuous.
W i t h t h e condition l i m / > 0 a n d suitable conditions on g (fulfilled for g - y), e v e r y t r a j e c t o r y is b o u n d e d as t - ~ o o , a n d T t r a n s f o r m s a suitable large d o m a i n in t h e P space into a d o m a i n c o n t a i n e d in t h e original one. F u r t h e r , t h e v e c t o r V, or P - ~ T P , m a k e s e x a c t l y one r e v o l u t i o n as P m o v e s positively r o u n d t h e b o u n d a r y . T h e n a " f i x e d p o i n t " t h e o r e m holds, a n d t h e " i n d e x n u m b e r " proof of it is valid. 2
1 T h i s p a p e r is b a s e d on j o i n t w o r k w i t h M. L. CAgTWl~IGHT.
A p a p e r I , w i t h t h e s a m e general title, w a s p u b l i s h e d in the Journal London Math. Soc., 20 (1945), 180-189, j o i n t l y w i t h M. L. CARTW~IGHT. T h i s w a s w r i t t e n w i t h t h e s a m e a i m s as t h e p r e s e n t I n - t r o d u c t i o n , b u t i n d r a s t i c a l l y c o n d e n s e d f o r m . W e h a v e b o r r o w e d s o m e p a s s a g e s f r o m it.
:N. LEVINSOI% Journal o] Math. and Physics, 22 (1943), 41-48, a n d Annals o] Math., 45 (1945), 723-727.
268 J. E. LITTLEWOOD
T h e r e is a t l e a s t one " f i x e d p o i n t " (f.p. for s h o r t ) of t h e t r a n s f o r m a t i o n T, a n d t h e r e c o r r e s p o n d s a periodic motion {p.m. for s h o r t ) o/ period 2; t h i s n e e d n o t , h o w e v e r , be s t a b l e .
W e m u s t d e f i n e o u r v o c a b u l a r y in t h e m a t t e r of s t a b i l i t y . A f.p. P0 ( a n d its c o r r e s p o n d i n g p.m.) is c a l l e d s t a b l e if T r a P c o n v e r g e s t o P0 a s m - ~ for all P of a n e i g h b o u r h o o d of Po. I t is c a l l e d t o t a l l y u n s t a b l e if t h e r e v e r s e d t r a j e c t o r i e s a r e s t a b l e : t h i s is e q u i v a l e n t t o t h e s t a t e m e n t t h a t P ' s n e a r a n d n o t i d e n t i c a l w i t h P0 r e c e d e f r o m P0 u n d e r i t e r a t i o n of T. f.p. a n d p . m . t h a t a r e n e i t h e r s t a b l e n o r t o t a l l y u n s t a b l e we shall call n o n - s t a b l e . T h i s is n o t a t e r m in g e n e r a l use; b u t t h e s t r a n g e f a c t is t h a t o u r " n o n - s t a b l e " p . m . a r e for t h e m o s t p a r t i n f i n i t e l y s i n g u l a r (or
" m u l t i p l e " ) , a n d t h e r e is no p o i n t in classifying t h e m . (The o r d i n a r y " c o l " d o e s j u s t o c c u r a n d r e c e i v e a p a s s i n g m e n t i o n . )
A p e r i o d i c t r a j e c t o r y or m o t i o n (p.m.), of l e a s t p e r i o d n 2 , w i t h n > l , is called a subharmonic o/ order n. W e a r e i n t e r e s t e d in t h e class K of " l i m i t i n g t r a j e c t o r i e s " , 1 t h e class w h o s e r e p r e s e n t a t i v e p o i n t s a r e t h e set of l i m i t p o i n t s of T m ($, 7) as m ~ - > ~ ; t h i s class as a w h o l e is i n v a r i a n t u n d e r T, a n d i t s t r a j e c t o r i e s a r e all b o u n d e d in - ~ < t < ~ . T h e s i m p l e s t p o s s i b l e case is t h a t in w h i c h K is a single p o i n t ; t h e r e is t h e n a s t a b l e p . m . of o r d e r 1 a n d e v e r y t r a j e c t o r y c o n v e r g e s t o it.
T h e n e x t s i m p l e s t case is t h a t of a f i n i t e n u m b e r of p . m . , t o s o m e one of w h i c h e v e r y t r a j e c t o r y converges. I n t h e g e n e r a l t o p o l o g i c a l t h e o r y , h o w e v e r , o t h e r possi- bilities, i n d e e d v e r y " b a d " ones, h a v e t o b e c o n t e m p l a t e d , a n d i t can b e v e r y dif- f i c u l t in a g i v e n case t o r u l e t h e m out.
2. T h e t w o s i m p l e s t cases (in r e s p e c t of t h e f u n c t i o n s f a n d g) are:
(i) / > c > 0 for all y, t o g e t h e r w i t h s u i t a b l e c o n d i t i o n s on g, v a l i d w h e n g = y;
(if) small d e p a r t u r e f r o m l i n e a r i t y , / = ~ F , g = y + e G, w i t h f i x e d F , G, a n d s m a l l s. 2 W e m u s t n o t discuss h e r e w h a t is k n o w n a b o u t t h e s e cases, e x c e p t t o o b s e r v e t h a t t h e y d i f f e r f r o m e a c h o t h e r , a n d a r e b o t h c o m p l e t e l y u n l i k e a n y t h i n g in t h e p r e s e n t p a p e r . W e n o w ask: w h a t is t h e s i m p l e s t e q u a t i o n n o t i n c l u d e d i n e i t h e r of t h e s e cases?
Since ] is n o t t o b e of c o n s t a n t sign, a n d since i t is p o s i t i v e for l a r g e l Yl, i t m u s t h a v e t w o zeros a t l e a s t ; if we a d d s y m m e t r y a b o u t y = 0, a n d t a k e t h e s i m p l e s t p o s s i b l e g a n d p, o u r s e a r c h l e a d s t o a n e q u a t i o n t h a t c a n b e n o r m a l i z e d as
1 "Recurrent motions" of*BIRK~OFF, Dynamical Systems (New York, 1927), 198, and G. A.
HEDLUNI), Amer. Journal o] .Maths., 66 (1941), 605-620.
2 ] = a + e/~ for a > 0 is ruled out as a special case of (i).
T H E E Q U A T I O ~ y - - ]~ (1 - - y~) ~ + y = b/~/c cos (/~ t + ~.) 269 9 - / c ( 1 - y ~ ) ~ ) + y = b # k cos ( # t + ~ ) , / ~ = 2 ~ / 2 ; (E)
" v a n der P o l ' s e q u a t i o n " . /c m u s t n o t be small, or we are i n case (ii), t h e n e x t p o s s i b i l i t y of s i m p l i f i c a t i o n is t o suppose i t large, which gives us t h e e q u a t i o n of t h e title.
3. T h e e q u a t i o n has a c o n s i d e r a b l e l i t e r a t u r e , 1 a n d e x p e r i m e n t s h a v e s u g g e s t e d v e r y i n t e r e s t i n g b e h a v i o u r , especially i n t h e case of large /c. S t a b l e m o t i o n s occur which are s u b h a r m o n i c s of large o d d order ( c o m p a r a b l e w i t h /c), d e c r e a s i n g as b i n - creases. F u r t h e r , as b increases we h a v e a l t e r n a t e l y one set of periodic s t a b l e m o t i o n s , of order 2 n + 1, a n d t w o sets, of orders 2 n + 1 a n d 2 n - 1, t h e s h o r t e r g r o w i n g a t t h e ex- pense of t h e longer. I n t h e e x p e r i m e n t s n c a n be of t h e order of 100 or 200.
I t c a n be foreseen t h a t t h e t h e o r e t i c a l s t a t e of t h i n g s b e h i n d these f i n d i n g s of e x p e r i m e n t (two d i s t i n c t s t a b l e periods n e i t h e r a m u l t i p l e of t h e other) m u s t be of considerable c o m p l e x i t y . I f t h e r e is a s u b h a r m o n i c m o t i o n of order m t h e r e will be a " s e t " of m d i s t i n c t periodic t r a j e c t o r i e s , w i t h d i s t i n c t r e p r e s e n t a t i v e P , o b t a i n e d b y shifting t h e original o n e 0, 1 . . . ( m - 1 ) - p e r i o d s forward. N o w b e t w e e n t w o ad- j a c e n t " s t a b l e s " (p.m., or t h e i r r e p r e s e n t a t i v e P ) t h e r e m u s t exist some sort of
" w a t e r s h e d " . A s t a b l e f.p. P of order m has a " s p h e r e of i n f l u e n c e " , a n y p o i n t of which converges to P u n d e r i t e r a t i o n of Tin; t h e sphere of i n f l u e n c e is a d o m a i n b o u n d e d b y " l i n e s " of p o i n t s c o r r e s p o n d i n g to w a t e r s h e d b e h a v i o u r . U n d e r i t e r a t i o n of T (as opposed t o T ~) P a n d its sphere of i n f l u e n c e circulate t h r o u g h a set m i n n u m b e r , t h e p i c t u r e as a whole being i n v a r i a n t u n d e r T. I t is n o w a q u e s t i o n of t w o such sets, c i r c u l a t i n g a t d i f f e r e n t rates; w h a t is t h e n a t u r e of t h e w a t e r s h e d lines a n d t h e n o n - s t a b l e f.p. t h a t lie o n t h e m ? O u r m e t h o d s are n o n - t o p o l o g i c a l , a n d raise, n o t solve, topological questions. 2 W e find, however, t h a t t h e r e is a fine struc- t u r e of l i m i t i n g m o t i o n s , some periodic n o n - s t a b l e a n d some n o n - p e r i o d i c , of f a n t a s t i c c o m p l e x i t y , t h o u g h f o r t u n a t e l y d e s c r i b a b l e i n f a i r l y simple g e n e r a l t e r m s , a
1 See B. VAN DEI~ POL, PreC. Inst. Radio Engineers, 22 (1934), 1051-1086, where further re- ferences are given. See also D. L. HERR, Prec. Inst. Radio Engineers, 27 (1939), 396-402, for graphical solutions, and B. VAN DER POL and J. VAN DE~ MARK, Nature, 120 (1927), 363, for experimental re- sults. The experiments, however, are actually concerned with an extremely unsymmetrical system.
2 Cp. w 31 below, and M. L. CARTWnIGH~ and J. E. LITTLEWOOD, A n n a l s el Maths., 54 (195[), 1-37.
Since our preliminary announcement of our results, N. LEVINSON, A n n a l s of Mathematic~ ~, 50 (1949), 127-153, has published a study of the equation ~ + k ] ( y ) y + y = b k s i n t , and ](y)=-t 1 or
- 1 according as l Y I > 1 or l Y I < 1. The solutions are piecings together of expressions in finite terms, and he is able to show, in reasonable compass, that for some intervals of b there are two sets of stable periodic solutions. There is also a fine structure, and we may refer to his paper for further comments on its nature.
270 J . E . L I T T L E W O O D
T h e lesson w o u l d s e e m t o b e t h a t t h e worse p o s s i b i l i t i e s e n v i s a g e d in t h e g e n e r a l t h e o r y n o t m e r e l y can occur, b u t a r e n o r m a l o c c u r r e n c e s (once we d e p a r t f r o m cases (i) a n d (ii) of w 2). I t w o u l d be f a i r l y o b v i o u s in a n y case t h a t t h e g e n e r a l n a t u r e of t h e r e s u l t s is n o t b o u n d u p w i t h t h e p a r t i c u l a r i t i e s of v a n d e r P o l ' s e q u a t i o n , b u t in o u r d e t a i l e d a c c o u n t we d o in f a c t c o n s i d e r a n e q u a t i o n w i t h g e n e r a l i z e d ], g, p;
~ ] + k / ( y ) ~ / + g ( y ) = b l ~ p ( t ) , [, g, 10 i n d e p e n d e n t of /r /c large.
W e r e t a i n o n l y such f e a t u r e s of t h e o r i g i n a l as a r e n e c e s s a r y t o k e e p t h e b c h a v i o u r w i t h i n a single c a t e g o r y : t h e chief of t h e s e a r e (i) ] m u s t h a v e o n l y one p a i r of zeros (see w for t h e d r a s t i c changes if t h i s is d r o p p e d ) ; (ii)
pl(t)=fpdt
m u s t n o t a t t a i n its u p p e r or its l o w e r b o u n d m o r e t h a n once in a p e r i o d ; a n d (iii) we r e t a i n s y m m e t r y (] is even, g o d d , p ( 8 9 2 4 7 a n d p h a s m e a n v a l u e 0): if (ii) o r (iii) is d r o p p e d fresh c o m p l i c a t i o n s a r e i n v o l v e d . 1 I n t h e a c c o u n t t h a t follows i n t h i s I n t r o d u c t i o n , h o w e v e r , we k e e p t o t h e special v a n d e r P o l ' s e q u a t i o n t o a v o i d a m u l t i - p l i c i t y of c o n s t a n t s a n d o t h e r d i s t r a c t i o n s .F o r t h e r e s t we t r y , in t h e I n t r o d u c t i o n , t o g i v e a n a c c o u n t i n t e l l i g i b l e t o a r e a d e r w h o is p r e p a r e d t o t a k e t h e p r o o f s of m u c h of t h e c o m p l i c a t e d d e t a i l for g r a n t e d : g i v e n c e r t a i n k e y - l e m m a s ( r e a s o n a b l y p l a u s i b l e in t h e m s e l v e s ) , t h e m a i n lines of t h e a r g u m e n t , a n d e s p e c i a l l y t h o s e l e a d i n g t o t h e m o s t s t r i k i n g r e s u l t s , a r e f a i r l y clear a n d u n e n c u m b e r e d .
4. W e p r o c e e d t o d e s c r i b e t h e g e n e r a l n a t u r e of t h e l i m i t i n g m o t i o n s . E i t h e r one o r t w o sets of s t a b l e s u b h a r m o n i c s , as s u g g e s t e d b y t h e e x p e r i m e n t s , d o in f a c t e x i s t , b u t t h e r e is m u c h t o a d d . 2
I f b > ~,2 a n d k > k 0 (b, 2), ~ (E) shows t h e s i m p l e s t p o s s i b l e b e h a v i o u r : t h e r e is a s t a b l e p . m . of o r d e r 1, p e r i o d 2, t o w h i c h e v e r y t r a j e c t o r y converges.
I n t h e case b < ~ w e r e s t r i c t o u r s e l v e s t o t h e i n t e r v a l B d e f i n e d b y ~0~-< b _< ] 100" 1 W e h a v e n o w t o e x c l u d e c e r t a i n i n t e r v a l s of b, in all a s m a l l p r o p o r t i o n of ~ . T h e r e t h e n e x i s t e = e (2), s m a l l for k 0 =/c 0 (2) large, w i t h t h e following p r o p e r t i e s . I f /r _>/%
t h e r e is a set of e x c l u d e d i n t e r v a l s in B of t o t a l l e n g t h e a t m o s t . T h e r e m a i n d e r
x The new features involved when one of them (especially, perhaps, (ii)) is dropped might be very interesting.
Experiment could not be expected to do more than hint at the crudest of the non-stable p.m.
8 This is the simplest instance of a point that should be emphasized. We never assert that behaviour is more and more nearly such and such as k increases, always that it is exactly such and such so soon as k exceeds a certain k 0 [here k0 (b, ~)]. I n fact k is not "large", but only "large enough".
T H E E Q U A T I O N y - - k (1 -- y~) ~ + y = b/~ k cos (/~ t + ~) 271 of B is also a set of intervals, B, say; this varies w i t h k, b u t has l e n g t h at least
~-e. B divides i n t o t w o parts, B1 a n d B~, c o m p a r a b l e in size.
W h e n b belongs to an i n t e r v a l 11 of B1, (E) has a set of stable s u b h a r m o n i c s of order 2 n + 1 a n d m o s t 1 trajectories converge each to some one of these, n is con- s t a n t in I~ a n d is of order (~- b)k.
W h e n b belongs to a n interval 12 of ~2, (E) has a set of stable s u b h a r m o n i c s of order 2 n + l, a n d another of order 2 n - l ; m o s t trajectories converge each to s o m e m e m b e r of o n e of the t w o sets. It possesses a further set ~, infinite in n u m b e r , of non-stable p.m. of a great variety of "structures" described in m o r e detail later. It possesses further a set X , of the p o w e r of the c o n t i n u u m , of non-periodic limiting trajectories, of the type described as "discontinuous recurrent". If w e denote the sets of representative points P in the (~, 9) plane also b y ~] a n d X , then every point of ~ is a limit point of points of ~, a n d also a limit point of points of X . A point of Z is thus non-stable, a n d is clearly a highly singular, or multiple, f.p. T h e n u m b e r n (which is of the order of /c) is constant in Ie. M o r e o v e r the set K a n d its subsets
~, X r e m a i n topologically equivalent t h r o u g h o u t I~. T h u s a point of Y, remains "in- finitely multiple" for all b of the interval I2, contrary to the natural expectation that multiplicity w o u l d be confined to isolated values of b.
F o r b of an 11 (of ~1) there is a set of n o n - s t a b l e s u b h a r m o n i c s of order 2 n + 1.
F o r all b of @~, w ~0) there is a single f.p. of order 1, a n d it is t o t a l l y un- stable. W e shall call its representative p o i n t Pu, a n d d e n o t e b y K 0 t h e set K less t h e point P , .
As b increases in B (from I~0 to ~ - ~ ) , j u m p i n g t h e excluded intervals, t h e n u m b e r n decreases. W e h a v e n o t h i n g to s a y a b o u t t h e transitions f r o m one stable period (2 n + 1) to two stable periods (2 n __+ 1) a n d vice versa; these t a k e place in t h e excluded intervals, b u t we a d d finally t h a t for a n y b of B there are always sub- harmonics of some kind, of order c o m p a r a b l e with ]c.
5. W e shall be m a i n l y concerned in t h e rest of t h e I n t r o d u c t i o n with t h e range B, i . e . i~0_< b __<~3 - - 100'1 of b: t h e case of small b is a separate (and interesting) problem t h a t we do n o t discuss, a n d t h e case b > ~, in which b e h a v i o u r is simple, is t r e a t e d b y a d a p t a t i o n s of t h e a r g u m e n t s we shall be describing.
B y A (x, y) we d e n o t e a c o n s t a n t d e p e n d i n g o n l y on 2 a n d t h e p a r a m e t e r s shown explicitly; b y L a c o n s t a n t d e p e n d i n g o n l y on 2, b y D a c o n s t a n t A (d): all of these 1 The general sense of "most" is fairly obvious: to define it precisely would occupy too much space.
272 J . E . LITTLEWOOD
t o b e positive. 1 T h e c o n s t a n t of a n 0 is to be of t y p e L. W e use suffixes to i d e n t i f y c o n s t a n t s ; g e n e r a l l y o n l y for t h e m o m e n t , a n d s t a r t i n g afresh a t 1 o n t h e n e x t occasion.
D e p e n d e n c e o n b a l w a y s reduces to d e p e n d e n c e o n t h e lower b o u n d of b (i.e. ~ 0 ) o r t h e n e a r n e s s of b t o t h e critical ~. O u r p r e s e n t r e s t r i c t i o n to B m a k e s d e p e n d e n c e o n b " u n i f o r m " : O's a n d D ' s are " u n i f o r m i n b". T h e r e a d e r m a y therefore t r e a t it l i g h t l y . W e shall s o m e t i m e s allow ourselves t h e licence of u s i n g o ( I ) w i t h o u t precise d e f i n i t i o n . ~ T h e lower b o u n d /co, w h i c h is u l t i m a t e l y a D, is to be re-chosen a t a n y m o m e n t w h e n t h e r u n of t h e a r g u m e n t requires it.
W e use t h e s y m b o l Q for t h e c u r r e n t p o i n t (t,y) of a t r a j e c t o r y F, ~ for t h e p h a s e # t + ~ , ~1 for y - l , Yl for f y d t . L e t
y
F ( y ) = f / ( y ) d y = ~ y a - y = t t 2 + I t ] 3 - ~.
0
T h e n we h a v e a n i d e n t i t y a (which we shall call t h e " ~ - i d e n t i t y " ) w i t h t h e a l t e r n a - t i v e forms
- Yo - - k I F (y)]~. + b/C (sin ~ - sin q)o) - [Y~]~., t
F (y) = b sin q~ - (y~ + y)//c + C. ] (1)
6. I t is e a s y t o show t h a t for 0 < b < 2 e v e r y t r a j e c t o r y F e v e n t u a l l y satisfies [Yl < L ~ , ]y[ < L~k; a n d if a t r a j e c t o r y is s t a r t e d s u b j e c t to these i n e q u a l i t i e s a t t o, t h e n [Yl < L 2 , ]Yl < L 2 k for t_>t o , where we m a y f u r t h e r suppose (to cover small a c c i d e n t s ) t h a t L 2 > L ~ > 2 0 , say. W e will call a t r a j e c t o r y so s t a r t e d ( e x t e n d i n g t h e n a t u r a l m e a n i n g of t h e adjective) a n " e v e n t u a l " one. I f b > 0 i t crosses y = 0 i n f i n i t e l y often. T h e c o r r e s p o n d i n g r e s u l t s for t h e g e n e r a l e q u a t i o n are, for once, c o n s i d e r a b l y m o r e d i f f i c u l t ; b u t we c a n f o r t u n a t e l y a p p e a l to a proof we h a v e g i v e n elsewhere. 4 L e t us n o w t r y to look ahead. T h e regions y > l + 5 , y < - 1 - ~ h a v e large p o s i t i v e d a m p i n g . W e expect of a t r a j e c t o r y i n y > 1 + 5, once it has " s e t t l e d d o w n " , a n d so long as it r e m a i n s i n t h e region, first t h a t it is v e r y s t a b l e ; we r e t u r n t o t h i s p o i n t p r e s e n t l y . We expect, secondly, t h a t b o t h its v e l o c i t y a n d its a c c e l e r a t i o n will be b o u n d e d : ?), ~] = O(D). 5 G r a n t e d this, it follows f r o m (E) itself t h a t
i The constant of an inequality " < L " or " < D " is to be chosen "large enough", that of
" > L" or " > D ' , "small enough".
~- Actually all the o (1)'s are O (k A)'S.
a In which, the origin t--0 being at our disposal, we generally take t o = 0, 90-Y0 (0).
4 Annals el Math., 48 (1947), 472 494.
The D arises from the ~ just above. The fact is that we shall have to consider a number of different rcstrictions, each with a "5": in the interest of simplicity we make all ~'s the same.
The grounds for tile "expectation" itself are further considere(1 in w 7 below.
THE :EQUATION y -- k ( i - y2) # + y = b/t k cos (/~ t + :r Z
~g~~176
e~176176176176176176176
U y = l273
y = - l
Fig. 1. T h e d i p s U~ U2, Ua U4 a r e m u c h e x a g g e r a t e d .
J
(y~ - 1) 9 = b # cos ~ + O
(D/k),
(1)and from the #-identity t h a t
F (y) = C + b sin qJ - ~ [Yl]t0 § O 9 (2)
Of these (1) implies an a p p r o x i m a t e "linkage" between y and ~); it also implies t h a t the velocity adjusts itself to m a k e the d a m p i n g balance the forcing-term; a "settling"
of the trajectory: the regions of positive d a m p i n g have a strongly "settling" effect.
We shall find b o t h "linkage" a n d "settling" v e r y useful conceptions. (2)implies t h a t
~ 3 §
s i n ~ , (3)with error
O(D/k),
over a n y interval of time of length O(1). The curve ( 3 ) h a s period ~, and supposing C > b - ~ , it lies in y > 1 (it lies in y > 1 + 5 if C > b - ~ + =4 (5).I t has m a x i m a at q ~ z and minima at ~ - ~ J r . The effect of the t e r m in Yl, on the right-hand side of (2), is to m a k e a t r a j e c t o r y satisfying (2) consist, to error
O(D/k)
throughout, of successive ;t-waves of the form (3), the successive constants C diminishing b y[yl]/k,
the incrementYl =f Y d t
being t a k e n over the preceding wave:the trajectory (with C > b - ~ + A 5) gradually approaches y - 1 from above.
The neighbourhood of y - 1 (where the d a m p i n g changes sign) is naturally highly critica], and we have to spend m u c h time discussing it. B u t for clearness we state now w h a t in fact n e x t happens to the trajectory we have been following.
I t continues its downward trend until one of its ~-waves reaches y - 1 ; this happens, as we should expect from the foregoing account, at a point U~ where q~ is
18 -- 563804. Acta mathematica. 97. Imprlm6 le 8 aofit 1957.
274 J . E . LITTLEWOOD
a p p r o x i m a t e l y - - ~ ( m o d 2 ~ ) . I t t h e n (if b is not too large) makes a small " d i p "
(depth of order k -89 below y = l , emerges at U2, a n d executes a n o t h e r A-wave of t y p e (3), followed, perhaps, b y a n o t h e r " d i p " U a U4, and so on. After 0 ( 1 ) d i p s
U2n-1 U2n at most, the velocities remaining 0 (1) throughout, the trajectory arrives at U of the diagram, and, unless b y coincidence it belongs to a narrow exceptional class 1, it decisively enters l Yl < 1, is violently accelerated b y the negative damping, acquiring a velocity of order k which takes it right through l yl_< 1 to a m i n i m u m Z ' a t a d e p t h a p p r o x i m a t e l y y = - 2, still with ~ a p p r o x i m a t e l y - ~ z (rood 2 ~) (U Z ' occupies only a small time). The t r a j e c t o r y is slightly unsettled at E' (the region l Yl < 1 is "unsettling"), after a short time it is again settled, and it t h e n repeats, in inverted form, its earlier behaviour in y > 1 § ~.
We stop at this point in our description, observing t h a t so far we have to cover three stages; the "settled" long descent to
y=l,
the stage of dips, during which downward velocity at the successive dips is appreciably increasing (this is a subtlety, a n d not really intuitive~), and finally a " s h o o t - t h r o u g h " f r o m U to ~ ' .7. We have said t h a t the p a r t of the trajectory above y = 1 § is " v e r y stable". We do not here intend a v e r y precise sense of " s t a b l e " , a n d the m a t t e r is m o s t intelligible if we consider the reversed motion (r.m.). This is " v e r y unstable":
if it is slightly disturbed, t h e n (except for a coincidence) it "slices" or " p u l l s " a w a y from the original. (See w Fig. 7.)
I t is instructive (both here and in later contexts) to recall a n u m b e r of plati- tudes a b o u t the solution of the linear equations
~___ k~) § y = ]c# cos/~t, (I)
where k is large (and positive). The upper sign gives positive d a m p i n g a n d stability, the lower sign corresponds to the reversed motion of this (reckoned from t = 0 ) , with negative damping and instability. The complete direct motions (d.m.) with the posi- tive sign are
y = Y (t) + C 1 e -~kt + C 2 e -~t/k,
1 t 1
_#2 ]
where Y - 1 § (1 - #2)2/(]c2 #2) ~sin/~ t § - ~ - - cos jut I ,
~ = 1 ( 1 + ( 1 - 4 k - 2 ) ~ ) = 1 + O(k-2),
fl = ] / ~ = l + o (~-~),
1 There is, naturally, a borderland between trajectories that make one more dip and those that do not: we shall have much to do with it.
See w 15, f.n.
T H E E Q U A T I O N ~ - - ]r (1 -- y2) ~ + y = b # k cos (# t + :r 275 a n d C1, C 2 are a r b i t r a r y c o n s t a n t s which, for an " e v e n t u a l " t r a j e c t o r y , m a y be sup- posed 0 ( 1 ) . A f u r t h e r t i m e of t h e order of 1 n o w reduces t h e t e r m C l e - ~ to ex- ponential smallness; this is " s e t t l i n g " ; exact s e t t l e m e n t corresponds to C 1 = 0, b u t w h e n C 1 is n o t 0 there is a small " p l a y " (deviation f r o m exact settlement) d e p e n d i n g on t h e p a s t h i s t o r y of t h e t r a j e c t o r y . I t is o f t e n v e r y c o n v e n i e n t to speak of s e t t l e m e n t as if it were exact, ignoring t h e " p l a y " . 1 There is a similar " p l a y " in " l i n k a g e " of y, ?~ (for given t), a n d we s o m e t i m e s speak of linkage also as if it were exact.
I g n o r i n g t h e " p l a y " in settlement, i.e. m a k i n g C I = 0 , gives us t h e 1-parameter family
y = Y ~ C 2 e -~/~. (2)
I t s m e m b e r s converge t o t h e p.m. y = Y, b u t slowly, t h e time of half-decay being of t h e order ]~.
A small d i s t u r b a n c e of (2) can be of t w o v e r y different k i n d s ; t h e one involves a coincidence t h a t keeps C 1 = 0 a n d slightly alters C 2 o n l y ; here t h e new t r a j e c t o r y runs nearly parallel t o t h e old a n d slowly converges to it; t h e o t h e r k i n d creates in addition a small C1; here t h e n e w t r a j e c t o r y quickly arrives within an exponentially small distance of t h e t r a j e c t o r y (2) with t h e n e w C 2 (or of t h e original t r a j e c t o r y itself in t h e case where C 2 is n o t disturbed).
T h e r.m. (solutions of (1) with t h e negative sign) h a v e their s~parate lesson.
There is a 1-parameter family whose e q u a t i o n is
y = Y ( - t) + C 2 e ~t/k. (3)
T h e t r a j e c t o r y (3) is a p p r o x i m a t e l y periodic over a time of t h e order 1, a n d for C2> 0 t h e m e a n heights of t h e successive waves increase, v e r y slowly. A slight dis- t u r b a n c e can be t o a n e i g h b o u r i n g m e m b e r of t h e family, b u t a p a r t f r o m this coin- cidence will i n t r o d u c e a t e r m C~e ~k~ involving a violent " p u l l " or "slice". N o t e t h a t the d.m. (2) are " n o r m a l " , if we ignore play; their reversals (3) are abnormal.
8. T h e foregoing r e m a r k s a b o u t t h e linear e q u a t i o n illustrate w h a t we said a b o u t t h e " s e t t l e d " m o t i o n of trajectories of (E) in y > 1 + 8. W e n o w briefly con- sider m o t i o n in ] y l < 1 - 8 , which we do first in reversed form, w i t h its positive damping. So long as t h e (reversed) m o t i o n s t a y s in l y l < 1 - 8 , the conditions a p p l y t h a t led t o (1) a n d (2) of w T h e t r a j e c t o r y is a p p r o x i m a t e l y t h e periodic curve
l y 3 - y = F ( y ) = C + b sin ~ (1)
1 Exact settlement in the case of equation (E) could only be defined at the .very end of the story, not at the beginning.
276 J . E . LITTLEWOOD
o v e r a single h-wave a t a t i m e , t h e successive C ' s i n c r e a s i n g b y [yl]/k, w h e r e
[Yl]
is t h e i n c r e m e n t of f y d t o v e r t h e p r e c e d i n g w a v e . F o r t h e c u r v e ( 1 ) t o lie in [ y [ < l i t is n e c e s s a r y a n d sufficient t h a t C_+b b o t h lie in ( - a , ~) (the r e l e v a n t r a n g e of E ( y ) ) , or t h a t
- ( ~ - b ) < C < ~ - b ;
t h i s r e q u i r e s b < ~ ( a n d is one a s p e c t of t h e c r i t i c a l v a l u e ~ for b), a n d t h e n t h e in- e q u a l i t y for C c a n be satisfied. T h e r.m., a n d also t h e d.m., of p e r i o d ~ (the l a t t e r t o t a l l y u n s t a b l e w h e n b < ~), h a v e e a c h t h e s a m e e q u a t i o n y = y (~); t h i s is of t h e f o r m
ya _ y = b s i n ~ § O (/c-1), (2)
for all t, b u t ~ h a s t h e v a l u e s qg=~_+/~t for d.m. a n d r . m . r e s p e c t i v e l y .
I t is f a i r l y clear t h a t if C is i n i t i a l l y , say, p o s i t i v e , t h e n in t h e r . m . (where - ~ - , a t ) t h e [Yl] a r e n e g a t i v e , a n d C d e c r e a s e s (as l o n g as i t is p o s i t i v e ) . T h e r e is in f a c t c o n v e r g e n c e of t h e r.m. to t h e r . m . (2). To t h e c o n v e r g e n c e of a r . m . b y d e c r e a s e of C t o 0 c o r r e s p o n d s a d . m . whose successive ~ - w a v e s g r a d u a l l y rise u n t i l t h e i r m a x i m a a r e all n e a r y = 1, w h e n f r e s h c o m p l i c a t i o n s n a t u r a l l y arise i n t o w h i c h we n e e d n o t go. B u t it is t o be o b s e r v e d t h a t , in a c c o r d a n c e w i t h t h e e n d of w t h e s l o w l y rising d.m. is a b n o r m a l a n d u n s t a b l e , a n d if s l i g h t l y d i s t u r b e d will " p u l t " or
" s l i c e " a w a y . I n c i d e n t a l l y , all p r o l o n g e d d i r e c t m o t i o n in [y[_< 1 is a b n o r m a l a n d u n s t a b l e .
9. W e now p r o c e e d , a f t e r t h i s p a r t l y t e n t a t i v e s u r v e y , t o follow m o r e closely t h e s t r i c t a c c o u n t of t h e course of t h e t r a j e c t o r y of w a n d F i g . 1 u p t o i t s ar- r i v a l a t Z ' .
T h e discussion of t h e first t w o s t a g e s (the first t o U1, t h e s e c o n d f r o m U1 t o U) is b a s e d on two k e y l e m m a s whose p r o o f s we t a k e for g r a n t e d .
Tile first of t h e s e is
L ~ M . ~ A. (i) Let O < b < 2 (say), and let d be a non-negative and d' a positive constant. Then there is a k o (~, d, d') such that when k>_k o the/ollowing properties hold.
Suppose that an eventual trajectory F (/or which consequently [Y[ < L ) has a piece X Y Z ( t x < t y < t z ) lying entirely in y > _ l - d k - " ; suppose also that (a) X Y has a time
1 .
length at least d'; (b) Y Z contains a point at which q ; ~ - ~ , (c) Y Z has time length at least k -~ log k. 1 Then /or any Q o/ Y Z
1 (1) Conditions (a), (b), (e) are all fulfilled, with d'=L, if X Z has time length at least 32, :say (take Y so that X Y has time length 12).
I t would be sufficient (as the proof shows), so far as the inequality for [?)[ in (1) below is
I n the #-identity ~
THE EQUATIOS 9 -- k (1 -- y~) # + y = b # k cos (/z t + :r I~l[<A (d,d'), [~][<A (d,d')lc 89 [ y [ < A ( d , d ' ) k ;
?) (y2 _ 1) = b # cos ~0 + 0 (A (d, d')/c-~).
277
(1) (2)
i ya _ y = F ( y ) = C + b (1 + sin ~) - yl/lc - ~t/Ic ( 3 )
we may use any convenient point o/ X Y to determine the constant C, and in the stretch Y Z we m a y thus substitute ~ = 0 (A (d, d')).
(ii) Suppose that 0 < b < 2 and that d is a positive constant. A t a Q that has been preceded by a piece of an eventual trajectory I', lying above y = 1 + d and lasting a time /c -1 log2k at least, we have, provided that /c_>k0i;t, d),
I ~ I < A (d), [ ~ I < A ( d ) . . . (4) with various consequences, e.g. (2) is valid with error-term improved to O ( A (d, d')/c-1).
T h e d in each p a r t of L e m m a A is chosen in different w a y s on different occa- sions, generally as a l I D in p a r t (i), a n d generally either as an A or else as ~ in p a r t (ii). (d a n d d' are blank cheques.) I t will be observed t h a t L e m m a A (i) is specially designed to deal with t h e question of " d i p s " (it follows t h e facts v e r y closely).
T h e h y p o t h e s i s in (i) t h a t Y Z contains a point ~ - - - 8 9 should be n o t e d ; the result is false w i t h o u t it 2 (in fact all t h e a p p a r e n t complications follow t h e actual facts v e r y closely).
P a r t (ii) is t h e precise expression of t h e " s e t t l i n g " effect of a region y > l + d . 10. W e shall n o w follow t h e course of a F starting at a vertex a Z, with
yo=O, lyo-2[<Do ~-1, [Vo-~[<Do~-L
D o is u l t i m a t e l y going t o be a particular D. W e shall find t h a t (subject to an ex- ception) F presently arrives at a n i n v e r t e d v e r t e x Z ' satisfying
c o n c e r n e d , to r e p l a c e (a) b y t h e h y p o t h e s i s (in effect, m o r e g e n e r a l ) t h a t X Y c o n t a i n s a p o i n t R for w h i c h l Y/~ [ < d'. B u t t h e c o r r e s p o n d i n g r e s u l t s for I Y l, I
I
w o u l d b e less s i m p l e .(2) W e m a y fix i d e a s b y s u p p o s i n g t h a t Z is o n y = 1 - d k - 89 ( a n i m p o r t a n t e a s e in a n y e v e n t ) ; for if Z is a b o v e t h i s line we c a n p r o l o n g t h e t r a j e c t o r y u n t i l it m e e t s t h e line for t h e f i r s t t i m e (all c o n d i t i o n s r e m a i n i n g fulfilled a ]ortiori).
1 T h e r i g h t - h a n d side o f (3) c a n t a k e v a r i o u s f o r m s ; it m a y b e g i n , e.g. C + b s i n % C + b (1 + s i n ~0), or C - b ( 1 - s i n T) a c c o r d i n g to c o n v e n i e n c e .
T h i s is a s o p h i s t i c a t e d affair; t h e e x a m p l e s a r e c e r t a i n of t h e t r a j e c t o r i e s r i s i n g s l o w l y off t h e u n s t a b l e p . m . in [ y [ < 1. T h e s e c a n e n d w i t h a piece a b o v e y = 1 of t i m e - i n t e r v a l ~ + o (1), a n d a d o w n w a r d v e l o c i t y of a r r i v a l a t y = 1 g r e a t e r t h a n a n y a s s i g n e d L .
3 Z is t h e i n i t i a l of " z e n i t h " .
278 J. :E. LITTLEWOOD
y ~ = 0 , l y o + 2 1 < D ' k -1, 1 ~ 0 + ~ 7 ~ l < D ' k - i ,
where, provided k >_ k o (~, Do), D' is a particular D that does not depend on the choice o/ the original Do. 1 This independence on t h e p a r t of D', achieved b y t h e dependence of ko, is c o m p l e t e l y vital t o t h e a r g u m e n t . I t enables us, b r e a k i n g a vicious circle, t o choose D o = D ' (/c o t h e n b e c o m i n g a p a r t i c u l a r D); t h e n Z ' satisfies t h e same condi- t i o n (inverted) as Z, a n d we can s t a r t afresh f r o m E'. W e m e n t i o n all this as an a d v a n c e warning, because m u c h h a p p e n s (and t h e "circle" is a long one) before we reach Z' (in w 15).
I n t h e n o t a t i o n of L e m m a A we t a k e X (and t = 0 ) a t Z a n d we t a k e Y t o b e the t i m e - p o i n t log 2 k / k later, or t h e first arrival of F a t y = 2 - ~, whichever h a p p e n s
t first. Over X Y we have, setting ~ = k f / d t ,
0 d
d~t ( - ~/ e~) = u e ~, u = - b k tt cos q~ + y ,
t
- ? ) = f u(t') e x p { - v ( t ) + ' c ( t ' ) } d t ' .
0
Since l u [ < L k a n d v ( t ) - ~ ( t ' ) > _ L k ( t - t ' ) , this gives
t
- - ~ < L I c f e-Lk(t-t')dt' < L ,
0
a n d so y > y o - L t > 2 - D o k - l - L k -1 log ~ k > 2 - ~ .
H e n c e it is t = log 2 k / k t h a t h a p p e n s first, a n d X Y has t i m e - l e n g t h log 2 k / k . N e x t we h a v e (over t h e whole F)
F ( y ) - F ( 1 ) = C + b (I + sin q~)- k - l y ~ - ~ k -~, (1) in which, on s u b s t i t u t i n g t = 0 (and ?}o = 0), we h a v e
C = F (Yo) - F (1) - b (1 + s i n ~0)-
Here F (2) -- - $' (1) - ~ - ~ , F ( y o ) = F ( 2 ) + O ( y o - 2 ) = ~ + O ( D o k - ~ ) , s i n ~o o = 1 + 0 ( ( ~ o - 89 ~) 2) = 1 + 0
(D~k-1),
1 "D'" is a momentary notation, not to upset a chain D1, D2, ... we shall presently need.
a n d so
T h u s (1) becomes
T~E EQ~ATIO~ ~ -- k (1 -- y~) y + y = b # k cos ( # t § ~) C = ~ - 2 b + 0 (Do e k-~). 1
279
F (y) - F (1) = (~ - 2 b) + b (1 + sin ~) - k -1 Yl - 9 k -1 + O (D~ k - l ) . (2) Consider n o w t h e s t r e t c h from Y to t h e first a r r i v a l a t y = 1 + 1~. L e m m a A (ii) is valid /or this, with d = ~ , and consequently ] ? ~ l < L .
S u b s t i t u t i n g this i n (2) for t h e special t i m e , t say, of a r r i v a l a t y = 1 + ~ , we h a v e k - l y ~ > ( ~ - 2 b ) + O - L k - ~ - L D ~ k - ~ - ( F ( l + ~ ) - F(1))
2 212-1 11A_! (]1~3 § ) 29 LDOe/c-i I
> i s ~ - L Do - ( ~o ~ a ~J - ~ooo - > ,~o~,
p r o v i d e d t h e k 0 of k > k 0 ( 2 , D O ) is s u i t a b l y chosen. T h e n Yl > L k , a n d so t > L k . A /ortiori t h e s t r e t c h f r o m Y to t h e first arrival, a t U~, say, a t y = 1 has t i m e - l e n g t h a t least L k ( > 2 2 ) , a n d c o n t a i n s p o i n t s w i t h ~ v ~ - ~ 7 ~ . Lemma A (i) [with U 1 /or Z]
is accordingly valid, with d ' = l , a n d we h a v e ] # I < L for t h e s t r e t c h YU1, a n d i n p a r t i c u l a r ]#u,] < L (note t h a t t h e c o n s t a n t is L, n o t D); a n d f u r t h e r we m a y sub- s t i t u t e ]?~] < L i n (2).
Supposing, t h e n , t h a t F a r r i v e s a t y = I for t h e first t i m e a t U~, let
~ v ~ - 1 ~ - w l ( ] w ~ [ < z ) , v ~ = - # ~ , , (3)
a n d let V 1 = v I + b k (1 - cos wl). (4)
W e h a v e j u s t seen t h a t v l < L . N e x t , t h e r e is a n S of F , a t m o s t ~ before U1, where ~ - 8 9 t h e e q u a t i o n of S U 1 is, b y L e m m a A ( i ) (3), of t h e f o r m ~
~ y a - y + ~ = C ' § b (l + sin c f ) - y l k-~ + O (L k-~),
where we m a y r e c k o n y 1 = 0 a t S. S u b s t i t u t i n g ~ = ~ s ~ - ~ 7 ~ , y=ys>~ 1 i n this, we h a v e C ' > - L k -1. H e n c e
b ( 1 - cos w l ) = b ( l + sin q ~ , , ) < _ O - C ' + L k - l < L k - a ; a n d so also ] w l l < L b - 8 9 k-~-. a S u m m i n g u p we h a v e
O < v l < L , O < _ V I < L , [o)11 < L k-89 (5)
1 Supposing D o _> 1 [so D o = 0 (D~)].
2 This, like (2), is the y-identity with 0 (1) substituted for y, but this time with t = 0 at S.
3 We can get the result b[ sin eol[<Lk -~- at once from A(i) (2); but a special argument is then needed to show that ~o 1 is near 0 and not ~z.
2 8 0 J . E. LITTLEWOOD
11. We prove n e x t t h a t v a n d co, and so all three of v, V, w, are approxi- m a t e l y "linked". For this we need the second key-lemma, which covers the three questions of linkage at Ul, " d i p s " , and the final shoot-through from U. ( I t retains its form unaltered when we deal with generalized /, g, p.)
LEMMA B. 1 Consider the (Riccati) equation, /or x_>0, d ~ Z = z 2 - x 2 + l + ~ - 2 8 x , z ( 0 ) = 0 , d x
where :r >_ - 1, and 8/urther satis/ies 8 < 0 when ~ = - 1. z is positive/or small positive x.
There is a 8o (a) such that:
(i) i/ 8>8io [or 0 > 8 > 8 o when o ~ = - 1 ] , then z changes sign to negative at an x = xo (:r 8 ) > 0 ;
(ii) i/ 8 < 80, then z--> + ~ at an asymptote x = x o (~, 8) > 0;
(iii) i/ 8 = 8o, there is a solution in (0, oo) /or which z>_ 0 and z = x + 8 o + F ( x , ~),
where F is continuous in (x, ~), and F = O ( 1 / x ) as x--->~.
Further, 8o (:r and ~+8~(~) are continuous and monotonic increasing. 8o(:r large with large positive :r
Finally, 8e (cr has the sign o/ ~.
R e t u r n now to F and its " a r r i v a l a t UI". I f v* =ju (~- b) 89 ,
and 8o (r162 is the function of L e m m a B, we find a linkage expressed "implicitly" b y V l = v l + b k ( 1 - cos(ol), ~ + l = v l / v * , }
v1 = v* (1 + ~ + 8o ~ (~)) + o (1) (1)
(which connect v I (or V1) with (o 1 when the p a r a m e t e r ~ is "eliminated"). Equiva- lently we m a y t a k e the first equation of (1) together with three equations
f l = # - i (v* k)89 sin (ol, ~ + l = V l / V * , 8 = 8 o (cr + o (1), (1)' containing two parameters cr 8. I n the particular case r162 equations ( 1 ) g i v e for V1 when v 1= v*,
V1 (v*) = v* + o (1), (2)
1 The results are mostly fairly intuitive, but t h e y take a good deal of proving.
T H E E Q U A T I O N ~ - - k (1 -- y2) ~ + y = b,a/c cos (# t + :r 281 a n d , t o e r r o r o (1),
Vl~V*
a r e r e s p e c t i v e l y e q u i v a l e n t t o V l ~ v * . To e r r o r o (1), v I a n dV1 i n c r e a s e w i t h 0)1.1
T h e r e s u l t (1) d e p e n d s on L e m m a ]3 in a m a n n e r t h a t is r e a d i l y intelligible.
W e c o n s i d e r t h e r . m . f r o m U 1 o v e r a t i m e of s l i g h t l y l a r g e r o r d e r t h a n /c -89 I t s e q u a t i o n (in i n t e g r a l f o r m , a n d u s i n g r = - t for t h e n e w t i m e , r e c k o n e d f r o m U 1 as origin) is
d y
d r -- - - v I = lc (F(y) - F ( 1 ) ) - b k (sin ( - # r + ~u,) - sin ~u,) - ~ y d r .
0
W e w r i t e y = l + v * 8 9 r = v * 89189 e x p a n d in p o w e r s of z a n d x, a n d r e j e c t
T
t e r m s w i t h coefficients o(1), in p a r t i c u l a r t h e one a r i s i n g f r o m
f ydv.
T h e r e s u l t ofo
t h i s process, w h e n we w r i t e ~ = V l / V * - 1 , a n d w h e n /~ h a s t h e v a l u e in (1)', is t h e e q u a t i o n
d z
- - z 2 - x 2 + 1 + ~ - - 2 ~ x (3)
d x
of L e m m a B. I f we a d d a o (1) t o t h e r i g h t - h a n d side, t h i s is a c t u a l l y v a l i d for t h e y of I ' r e v e r s e d , o v e r a r a n g e of x of l e n g t h log k, say. lVIoreover t h e e r r o r is of t h e f o r m 0 (A (Do)/c-A), a n d b y s l i g h t l y d i m i n i s h i n g t h e i n d e x A, and reehoosing k o (L, Do), i t b e c o m e s 0 ( k - A ) , independent o/ Do.~ N o w a c c o r d i n g t o L e m m a B, if we h a v e /3 > /30 (:r t h e n z b e c o m e s n e g a t i v e a t a n x=xo(o~ , fl); if /3</30(cr t h e n z - - > ~ a t a n a s y m p t o t i c x = Xo (~,/3). B u t t h e z c o r r e s p o n d i n g t o F r e v e r s e d c e r t a i n l y does n e i t h e r of t h e s e things; we infer (restoring a o (1)) t h a t ~ a n d fl m u s t be c o n n e c t e d b y
fl = fl0 (~) + o (1). (4)
T h e r e s u l t (4) is in f a c t t r u e , w i t h a precise e r r o r - t e r m in p l a c e of o ( 1 ) t h a t we n e e d n o t p a r t i c u l a r i z e , a n d , like t h a t in (3), i n d e p e n d e n t of D o (since t h e b o u n d s for vl, co 1 are).
12. W e shall in future w o r k m o s t l y in terms, not of Vl'S, b u t of the V1's
"linked" with t h e m ; there is m o r e t h a n one reason for preferring VI, as will a p p e a r later. This being so, w e n e e d (to avoid m i x i n g s y m b o l s later) a s y m b o l for the V I
1 In the case of V 1 because ~ + fl02 (0r is increasing.
2 (1) We mean, of course, more precisely, that the upper bound implied in the error term is independent of D o (and similarly in future).
(2) A's cccurring as indices are always absolute constants.
282 J. :E. LITTLEWOOI)
l i n k e d w i t h v l = v * ; a n d (in t h e l i g h t of (2) of w 11) we de/ine V* t o be v*, so t h a t V 1 (v*) = v* + o (1) = V* + o (1). T h e r e a r e n o w t h r e e p o s s i b i l i t i e s a b o u t VI:
(i) V 1 > V * + 6 ; (ii) V 1 < V * - 6 ; (iii) I V x - V * ] - < 6 . T h e t h i r d we d e s c r i b e as a " g a p " case; V lies in a c r i t i c a l g a p r o u n d V*.
Consider n o w t h e d . m . f r o m u 1 o v e r a r a n g e of s l i g h t l y l a r g e r o r d e r t h a n k -89 a n d w r i t e y = 1 - v *89 k ~ ~, t = v *-89 k-89 x (~ w i t h a m i n u s sign since y is m o v i n g b e l o w y = 1); a g a i n we e x p a n d in ~ a n d x, a n d i g n o r e t e r m s w i t h coefficients o(1): t h e r e s u l t of t h e c h a n g e s of sign as a g a i n s t t h e (z, x) r . m . e q u a t i o n is
d ~ - ~ e - x ~ + 1 + ~ - 2 ( - / ~ ) x, (1) d x
t h a t is t o s a y t h e e q u a t i o n (3) of w w i t h $ for z a n d - f l for ft.
N o w w i t h error o ( 1 ) we h a v e fl=fl0(cr if f u r t h e r we a r e in case (i), t h e n ~ > 0 , 1 a n d fl h a v i n g ( b y L e m m a B) t h e sign of ~, is p o s i t i v e . T h e n - f l < 0 < f i 0 ( ~ ) . B y L e m m a B t h i s i m p l i e s 2 t h a t a ~ s a t i s f y i n g (1) t e n d s to + ~ a t a f i n i t e a s y m p t o t e x = x'0 (~, fl). C o r r e s p o n d i n g t o t h i s we e x p e c t t h e y - 1 of F t o b e c o m e a large n e g a - t i v e m u l t i p l e of k --~, a n d ~ to b e c o m e large a n d n e g a t i v e , a t a t i m e c o m p a r a b l e w i t h k -~. T h i s does h a p p e n (when we go p r o p e r l y i n t o t h e e r r o r - t e r m s ) ; t h e n e g a t i v e d a m p i n g t h e n t a k e s h o l d (as t h e r e a d e r will e a s i l y believe: t h e a r g u m e n t s b e c o m e cruder), a n d i t is t h e f a c t t h a t y a c q u i r e s a d o w n w a r d v e l o c i t y of o r d e r k a n d passes t h r o u g h ]y]_< 1 in a s h o r t t i m e .
Consider n o w case (ii). a H e r e cr a n d fi h a s t h e sign of r162 a n d is n e g a t i v e ; - f l > 0 > / ~ 0 ( ~ ). A ~ s a t i s f y i n g (1) t h e r e f o r e c h a n g e s sign a t x = x 0 ( ~ , /~). T h e corre- s p o n d i n g b e h a v i o u r of t h e y of I ~ is t o d e s c e n d b e l o w y = l t o a d e p t h D l k -89 a t m o s t , a n d e m e r g e u p w a r d s , a t U 2 say, a f t e r t i m e D k -~ a t m o s t . O b s e r v e t h a t , in a c c o r d a n c e w i t h w h a t was s a i d a t t h e e n d of w 11, D 1 and the D in D k -~- above do not depend on JD o.
S u p p o s e n o w t h e n e x t r e t u r n of P t o y = 1 is a t U S. B e t w e e n U 1 a n d U2, P con- /orms to the hypothesis o/ L e m m a A with d = D 1, provided k o is re-chosen to depend
1 More precisely cr > D (D arising from the ($ of the gap V*• (~).
2 Note the double use of Lemma B; first to establish an approximate equality fl=flo (a) be- cause "neither event happens"; then, because the equality is upset by changing the sign of fl, to infer that one or other event does happen in the new case.
a We shall here ignore the case v l = 0 , or r 1, which would need a special treatment in Lemma B. Its apparently critical character is spurious; since a small dip is found to turn upwards, it is obvious that a graze will do so also.
THE EQUATION ~ / - - k (1 - - y2) ~ + y = b/~ k cos (~t t + ~) 283
y = l
y = 1
y = - 2
C1
C2
F i g . 2.
on D 1. The e q u a t i o n of r between Ux Ua is a c c o r d i n g l y given b y L e m m a A (3), slightly rearranged;
8 9 sin ~ ) + o (1).
I n this, since y g = l a n d q s l - - - ~ s + o ( 1 ) , t h e c o n s t a n t C' is o(1). H e n c e the curve U 1 Ua satisfies
1 ~3 / ~ 2 l ~.3
~q T q = ~ y - y + ~ = b ( l + s i n ~ ) + o ( 1 ) . (2) 13. W e m u s t n o w enter on something of a digression. W h e n b <~ t h e locus
~ y a _ y + ~ = b ( l + sin F),
over a period 4, consists of three separate C1, C2, Ca, as shown in Fig. 2. I n accord- ance with w h a t was said earlier, C a a n d C1, which are in I Y I ~> 1, are a p p r o x i m a t i o n s to stable trajectories, a n d C2, which is in l yl -< 1, is an a p p r o x i m a t i o n to a stable r.m. or unstable d.m. W e shall h a v e more to s a y a b o u t these curves later. Meanwhile we expect F to follow a p p r o x i m a t e l y t h e curve C 1, a n d t o reach y = 1 at a Ua near t h e r i g h t - h a n d end of C~. B u t t w o points r e m a i n to be established.
(i) T h e a p p r o x i m a t i o n (2) of w 12 shews t h a t F is always near one or other of C1, C~, b u t n o t t h a t it c a n n o t shift f r o m C 1 to C 2 while still near U r To settle this we need t o go m o r e closely into t h e a c t u a l error-terms we h a v e presented as o (1)'s:
when we do this (and t h e extensions are n a t u r a l ones) we find t h a t F keeps a b o v e y = 1 for a time long e n o u g h after U2 for C 1 a n d C~ t o h a v e diverged m o r e t h a n t h e error- t e r m in y; a shift is impossible. So F continues along C 1, approximately, as far as
t h e r i g h t - h a n d end of C1.
2 8 ~ J . E . L I T T L E W O O D
(ii) W e h a v e t o s h o w t h a t I" does n o t t u r n u p w a r d s n e a r t h i s p o i n t b e f o r e a c t u a l l y r e a c h i n g y = 1. To d e a l w i t h t h i s 1 we n e e d t h e following g e n e r a l principle. 2
14. L EMMA C. Suppose Yl, Y2 are respectively solutions o[
~ = O ( y , t ) + R 1 , ~,
where ~ is continuous in (y, t), R1, 2 are continuous, and RI > R 2 /or t>_t o.
(i) I / now Yl (to)>-Y2(t), then Yl >y2 /or t > t o .
(ii) The conclusion is true if R I > R 2 /or t > t o only, provided we know inde- pendently that y~ > Y2 /or small positive t - t o.
W e s t a t e t h e c o m p l e t e t r u t h . T h e r e is a t o u c h of s u b t l e t y in t h a t (ii) is n o t t r u e w i t h o u t t h e f i n a l p r o v i s o .
W e give t h e e a s y p r o o f . F o r t > t o t h e Yl c u r v e is i n i t i a l l y a b o v e t h e Y2 one, b y h y p o t h e s i s in (ii), a n d for s l i g h t l y d i f f e r e n t r e a s o n s in (i) a c c o r d i n g as Yl (to)>Y2(to) or Yl (to)= Y2 (to). I f e v e r t h e r e is a n i n t e r s e c t i o n , l e t Q be t h e f i r s t one a f t e r t = t o.
A t Q we h a v e , as a m a t t e r of g e o m e t r y , Yl-< Y2, c o n t r a r y t o Y l - Y 2 = R 1 - R 2 > 0.
R e t u r n i n g t o F , l e t t o b e a t i m e l o n g e n o u g h b e f o r e t~, for y t o b e g r e a t e r t h a n 1 for s e v e r a l ~ - p e r i o d s a f t e r t o . F o r to<_t<_tu ~, l e t y(t) b e t h e y of F , a n d l e t y l = y ( t ) , y 2 - y ( t + ~ ) . T h e ~ - i d c n t i t i e s for Yl.2 a r e of t h e f o r m
w h e r e [see w 5, (1)] we h a v e
fl = - k F (y) + R1,2,
t+~
R 1 - R 2 = . [ y d t ,
t
a n d t h i s is c e r t a i n l y p o s i t i v e u n t i l y (t + 2) is 0 f o r t h e f i r s t t i m e , a t t = v, s~y. N e x t , i t is p o s s i b l e t o f i n d a t o as a b o v e s u c h t h a t y~ (to)> Y2 (to); for o v e r a su//iciently l o n g t i m e F c e r t a i n l y descends. I t follows f r o m L c m m a C t h a t Yl > Y~ for t o < t _< T.
Since Yl (tv~) = 1, w e c a n n o t h a v e y (t) >_ 1 for all t of tu, < t <_ tv, + ,~ [which w o u l d i m p l y b o t h T > t v , a n d Y2 (t~,)> 1 = Y l (tu,)]; a n d t h i s is t h e d e s i r e d r e s u l t .
15. W e f i n d , t h e n , t h a t F a r r i v e s d o w n w a r d s a t y = 1 for t h e f i r s t t i m e a f t e r U 1 a t a U2, n e a r U 1 + 2, a n d c o n f o r m i n g t o t h e h y p o t h e s i s of L e m m a A, w i t h d = D 1 (the n u m b e r a s s o c i a t e d w i t h t h e d i p a t U 1 U 2, w 12). There is a linkage o/v2, V2, 0)2 at U a; /urthera
1 The subtleties involved are curious, but we do not see how to avoid them.
2 Constantly used in the full account.
3 The identity in (1) is one reason for working with V rather than v.
T I I E E Q U A T I O N y - - k (1 -- y~) ~ + y = b # k cos (/~ t + g) 285
V 2 - V 1 = [yl]uu: = M -~ o ( 1 ) , (1)
where M = .f Y1 dt, and y = Yl(t) is the equation o/ C1; M is o/ the /orm A(,~).
o
The three alternatives concerning "V2" a n d t h e gap V* • arise again at Ua;
either V 2 is in the gap; or V~ > V* + ~ and the velocity !) becomes (negatively) large," or, /inally, V~ < V * - ~ and there is another dip, o/ depth D2]C-~ at most, P con/orms to Lemma A with d = M a x (D1, D2), and, i/ k o is rechosen to depend on De, there is a /resh arrival at U s = U a + ~ + o ( 1 ), with V a - V ~ = M + o ( 1 ) ; 1 and so on.
Now, b y (1), if we do n o t arrive at a V in t h e gap, we m u s t h a v e V~>V*+(5 after n = l + ( V * + ~ ) / ( M + o ( 1 ) ) dips a t most. A n d t h e n the /inal D~ and d are D's, independent o/ Do, and the /inal /Co is a ]co (,~, (5, Do).
L e t us s u m up, with p a r t l y c h a n g e d n o t a t i o n , 2 w h a t has been so far proved.
There is a ]c o (~, ~, Do) ; i/ lc >_/Co, F starting at a vertex of t y p e Y0 = 0, l Y - 2 1 < Do ]c ~, [ ~ 0 - ~ l < D0]c ~, descends in accordance with t h e description of w 6; after time at least L/c it arrives, after possible dips, at U, on y = l, w i t h
-~/u'=V, q ~ u = - l T e - ~ o , V = v + b ] c ( l - c o s a ) ) ,
v, V, co linked b y (1) of w 11, t h e errors h a v i n g b o u n d s i n d e p e n d e n t of Do, a n d finally either (i) V * - 5 < ~ V < V * + ( 5 , or else (ii) V * + 5 < V < V * + 5 + M + o ( 1 ) .
Suppose t h e gap case (i) does n o t occur, so t h a t case (ii) does. Then, as we said in w 12, t h e m o t i o n becomes c o m p a r a t i v e l y crude, a n d it is fairly easy to show t h a t F descends in t i m e O ( D k 89 to an i n v e r t e d v e r t e x E ' with y z . = 0 , at which 3
l y + 2 [ < A l ( 5 ) k -~, I ~-~- 1 ~ ] < A 2 ((5) ]c -1,
where A 1 and A~ are independent o/ D o. We are now able, without a vicious circle, to choose D o = Max (A1, A~). T h e n 1 ~, restricted b y t h e D o inequalities at t h e s t a r t i n g v e r t e x E,
1 W e c a n n o w , b u t o n l y n o w , i n f e r t h e i n c r e a s e of v, l i n k e d w i t h V, a t s u c c e s s i v e d i p s . 2 W e d o n o t c o n s i d e r d i p s a g a i n , a n d t h e U 1, U2, . . . a s s o c i a t e d w i t h t h e m d o n o t a p p e a r a g a i n , n o r do t h e a s s o c i a t e d V1, etc. I t will t h e r e f o r e n o t c a u s e c o n f u s i o n if, f r o m n o w on, we u s e v, V for t h e t h i n g s a s s o c i a t e d w i t h t h e " f i n a l " U, a n d d a s h e s ( U ' , v', V 1) f o r " i n v e r t e d " t h i n g s , a s s o c i a t e d w i t h y = - 1.
a W e f i r s t s h o w t h a t t h e t i m e is O(Db- 89 so t h a t I ~ z , + 8 9 a n d t h e n w e h a v e , b y t h ~ y - i d e n t i t y b e t w e e n U a n d Z t
0 = 9~, = b/c (1 + s i n ~z') - V - / c ( F (Yz') - ( F (1)) - .5 Y dt 0
= 0 ( k ( D b - 89 + 0 (1) - k ( F ( Y z ' ) - 2 ' ( - 2));
F ( y ~ , ) - F ( - 2 ) = O ( D k - 1 ) , a n d so ]y~,+21=O(Dk-1).
286 a. E. LITTLEWOOD
\ r
~=1
g = - I g
^A/
V ~ U '~
o . . O @ e o o a e o $ * 'Fig. 3.
has a r r i v e d a t a n i n v e r t e d v e r t e x Z ' o b e y i n g t h e same r e s t r i c t i o n (in a p p r o p r i a t e in- v e r t e d form). 1 It there[ore repeats in inverted [orm its [ormer behaviour, arriving, alter inverted dips, at U' on y = - 1 with
r 1 r V t = V I
y v , = v , ~ v , = ~ - w , + b k ( 1 - cos co'),
v', V', o~' are l i n k e d b y (1) of w i1 (with d a s h e d letters), a n d e i t h e r V' is i n t h e gap V* ~+(~, or else
V * + ~ < V ' < V * + ~ + M + o ( 1 ) .
16. W e d e n o t e b y Z1, Z~ . . . . t h e t i m e - p o i n t s a f t e r Z where ~ z ; N 1, N 2 .. . . those where F ~ - - ~ z , N 1 b e i n g b e t w e e n Z 1 a n d Z2, etc. These are a p p r o x i m a t e l y m a x i m a a n d m i n i m a in t h e long descent.~ T h e c o r r e s p o n d i n g Z ' , N ' s i m i l a r l y succeed Z ' , a n d h a v e ~ - ~ , 89 respectively. T h e s t r e t c h of a F f r o m Z 1 t o t h e e n s u i n g Z'I, or Z~ to t h e e n s u i n g Z1; or, a g a i n f r o m U t o t h e e n s u i n g U ' (or U' to U), we call a half-cycle.
W e call a F " g a p - f r e e " i n a r a n g e if it has, i n t h e r a n g e , n o V or V' i n t h e g a p V* + 3 .
T h e ?)-identity for F , b e t w e e n a U w i t h its V > V * + O t o t h e e n s u i n g U ' is
U"
V' + V = - ( ~ - 2 b ) - f y d t . (1)
U
Suppose n o w t h a t FL2 " s t a r t " i n t h e s t r e a m a t Z 0 s u b j e c t t o t h e "D0-inequali- ties" of w 15 (or t h a t t h e y j o i n t h i s s t r e a m after a n earlier s t a r t ) . Suppose f u r t h e r t h a t F~.~ are gap-free, a n d , f u r t h e r a g a i n , t h a t t h e y " m i s s t h e s a m e g a p s " , i.e. h a v e
1 Inverted behaviour may be inferred from direct by changing y into - y , and ~ into ~ + ~.
Our arguments implicitly prove, of course, that a trajectory, if started with not too violent initial velocity, and if remaining an unbroken time ~ ~ above y = 1, will behave similarly, i.e. will arrive, after possible dips, at y = 1 with a "gap" V, or else pass through an inverted vertex of the special kind.
2 Z is a real vertex, with exceptional behaviour; it is near Z0, where q~-:l~.
