Almost periodic Philip F r ~

e By

rent motions1).

Camb~e, M~. (U.S.A.)~. w Introduction.

In his study of dynamical systemsS), Bir]chofl was led t o the consideration of' certain ]imlting motions w]fich he called recurrent. While every recurrent motion which he explicitly constructed for am analytic syste~ has an a/most per/od/o character'), he constructed a non-analytic system with a recurrent motion not of this type, and conjectured t h a t analogous analytic systems existe~ The concepts of recurrent motion and atmcet t ~ o d i c motion thus appear to be related, but not identical. In the present note we show that every stable almost pe~odir motion is "a reourr~t motion ~z~ssessi~ a certain ~ability property, and conversdy, every reoz~en~ motion possessing this property is a n ~ periodio motion. w Definitions and k n o w n theorems. We first recall some of Birkhofl's definitions and resultsS). The motions he disouss~ are defined by a set of differential equations dZm = . .

(1)

d~

where each of the n functions ,Yj is s real and single-valued function of the n variables z~. These functions are usually assumed to be analytic, 1) a) s) 4)

Communicated to the American Mathematical Society, January, 1928. Pro tern. st Zfirich, Switzerland. G. D. Birkhoff, Bulletin iSoc. math. France 40 (1912), p. 805--823. H. Bohr, Act~ Mathematica 4~ (1924), p. 29-127.

326

Ph. Franklin.

though most of the discussion applies if they are merely subjec~.d toga Lipschitz condition n S=I

Here x k is an abbreviation for zl, x~. . . . , x~ and there are n equations given by i~I,2,.., or n. The x~ of (I) may be thought of as the Hamiltonian co6rdinates and momenta for the originalmechanical problem, t being the time. W e shall refer to them a~ co6rdinates in the sequel. A motion is said to be poai~ivdy and negatively atab/e if all the points xk(t), -- ~ t <~ ~ , for this motion may be included in a closed, anal hence finite domain D, composed entirely of regular points, L e. where the Xj are analytic in the case which principally interests us or, in the more general case, subjected to such conditions that the theorems of existence, uniqueness and conti~uous dependence of the solution on the initial conditions and parameters appearing in the X# apply to the system (1). The poin~ L of the x~ space such that every neighborhood of a point L contains points Of a particular motion with t arbitrarily great positively (or negatively) are called omega (or alpha) limit points of that motion, The motions determined by these points are called omega (or alpha) derived motions., For stable motions ~) these alpha and omega derived motions are also stable. A closed set of stable motions such that the set of omega derived motions, as well as the set of alpha derived motions, of any one motion of the set Coincides with the original set is called a minimal set, and every motion of such a set is said to be recurrenL S u c h a motion is characterized by e) Theor~.m A. A necessary and sufficient condition for a positively and negatively s-table motion to be recurrent is that to every given positive quantity ~, there should exist a time interval T, ~uch :that the portion of the motion corresponding to every time interval of length T has some points at distance less than ~/from a n y point of 'the entire

r~0tio~ ~n general thel solutions of (1) are continuous in the initial conditions in the sense that if a me,ion defined by x~(to)~ a~ is capable of extexision throughout the time intervals I ~ - to[ ~ ~, then for any given Positive quantity e, there exists a ~(e,E,a~) such that if l ~ - - a k [ ~ ~, the motion defined by ~ ( t e ) ~ ~ is capable of similar extension, and throughout the time intervals for which It--tel___8, we 'shall have l~k (~) ~ z~ (t~ 1 < ~. Consequently, for a stable motion, this relation will 6) lee. eft. 8), p. 309. e) leo. eft. a), p. 812.

Almost periodio re~m~nt motions.

827

hold for any 8, if the proper ~ (e, B, ak) is taken. But, as all the points of a stable motion may be included in a closed region D, in which the X~ are analytic, the partial derivatives of the X~ will be bounded in. this region, or in particular throughout the motion. As the G (%) of (2) are related ~o these derivatives by the law of the mean we may derive a uniform Lipschitz condition. We have, namely, s

!

(8)

ZrLa,.~.;a'.l<~ Xl %(-,)I =.);=: .=,

I%(,,§

where z~ is a point intermediate between x~ and % § a x) on some path 9joining them, and hence may be taken in D if both these points are in D or in particular if x k is a point of the motion and Az~ is small enough,9 j A x ~ I < r G is a bound for the absolute value o f T ~ ~ in D. But, from (1), we have

(4)

~~i x j..... ,~(~J+axJ) dt

d(~J)---dt %(x'+

~x~) -- X j ( ~ ) ,

which, combined with (3) gives

I dAzj d*

(5)

~--GZI~='I" 8--1

As the quantities Jxk(t ) restricted by this relation and also by

(6)

l ax~ (to) l _ b,

will clearly be greatest when both equalities hold or for t > t 0, (7)

d~x~dt ----riGA%,

we have f o r [ t - -

to] ~

(8)

8,

J %(t)~ be"~(Ha,

~1 < e

l~(,)-~.~(t)l=IAxk(t)[K_be"~-*or ~'be"aS ~_~,

provided that

(9')

l a, - a, I --I ~ ~(~o) I < b ~

~e

-')as

.

Thus for a stable motion, we may take (10)

a(e, 8, a~) = e ~ e - ' ~

e~ = rain (e, c),

which is independen~ of ak, and will be denoted by a(e, 8). The quantity just found for a stable motion does, however, depend on 8. The spedal stability condition which we need in our discussion is the existence of such a quantity which is independen~ of both 8 and a~. 9As we shall only need. it for a motion compared with itself, we formulate the following definition.

828

Ph. Fr~.~i.~

A stable motion is said to possess the s~bility property 8 i~ foz evelT positive e, there e~lsts a co~esponding h(e), such that if I z~ (~1) -- zk (t,) I < h (e), the motions defined by taking z~ (t1) and z~ (t g) as initial points will have their coordinates diOer by less than e for all times, i.e. ]xi,(t~+t)--x~(~+~)[
If( 9 + ") -- f ( z ) I ~ ~"

If, for every positive e, there is an interval L (e) such that every interval of length L(e) contains at least one displacement number z(e), f(x) is said to be almost periodic. By an almost periodic mo[io~, we shall, mean a motion each M whose coordinates ~s an ~lmost veriodic function of the time. w Characterization of stable almost periodic motions. We shall now prove T h e o r e m I. A ~.~e~aary a~ut au//ie~e~ condition/or a aab~ motion

to be cdraost periodic is that it be a re~urrem motion poss~si~] t ~ pro. perry ~. We begin with the necessity. As our motion is almost periodic, each of the n functions x~(t) is. Hence s) these are simultaneously almost periodic, i.e. for each given ,, there is a single L (8) such that in each interval o f this length there is a ~ingle 9 (e) which may be used for all the n fimctions. If then t1 < t < t1 + L is any interval of this length, and ts is any other time, there exists a displacement number ~(e) in the interval t 1 -- t~ < 9 < ~1 -- tg + L. Consequently

(12)

Jz~Ct, + , )

"=~(t,)[

and as tg + 9 is in the arbitrav/ time interval we started with, we see that on taking T = L(e), the conditions of theorem A ate met. and the motion is recur~nt. We must next show that our motion possesses the property 8. As the motion is stable, by ( 1 0 ) w e may find a ~(-~,-~q) for any finite S. ~) loc. cir. '), p. 80. s) foe. cir. '), p, 87--40.

Almost I~iodie ~ t

329

motiona

In particular, take for 8 a COmmOn L(~- / .

W e assert that

this

~ ( ~ , L ( ~ ) ) will serve a s t h e h(e) demanded bythe property 8. For, if

we have from the definition of 0(r 8),

Ix~(tl'+u)-x~(t~+u)l~-,

(14)

if lul
Moreover, for any t, there exists a common ~(~-) in the interval <,

(15)

+

for which

I ~ ( t l + t ) - ~ ( t l + t - 7)1 _~ ~7

and

(16)

s

Ix~ (t, + t) - ~ (t, + t - ,)1 < T"

But, as ~ t - , t < L ( ~ - ) ,

we may set u = t - ,

in (14), and on com-

bining the result with ( 1 5 ) a n d (16), we obtain

(17)

I~ (t, + t) -

~

(t, + t) l ~ ~,

which holds for all t and shows that the propertT ~ holds. For the sufficiency, we assume that our motion is recurrent and satisfies the condition ~q. By this condition, there exists an /t (e) for our motion. We assert that the T of theorem A cox;responding to ~ / = h ( 8 ) is the L (e) required by the definition of ~lmost periodioitY. For, by the fundamental ,property of T, in any interval of the motion with time duration of this length there is a point at distance less than h (e) from ~every point of the motion, in particular, let x ~ ( O ) b e the particular point of the motion selected, and t1 the time for the point in the interval at distance less than h (~) from it. We have:

(is)

I~ ( t l ) - ~(0)1 < h(~).

Consequently, by the fundamental property of h(e),

(Io)

Ix, (t~ +t).- xj(t)I <

for all t, which shows that we may take ~ ( e ) = tx. w Further properties. From the presence of a single stable almost periodic motion in a system, we may draw certain conclusions about other motions of the system. We begin with

330

Ph. Franklin.

T h e o r e m II. I / a stable motion is almost periodic, each o/its omega (o r alpha) deri~ed motions is al~o almost periodic. Let ~k(O) be a n omega limit point of the stable almos~ periodic motion z~(t), and ~ (t) the omega derived motion defined by it. If ~(e) is a common displacement number for the xk(t), and t1 is any fixed v a l u e o f t~ by ( 1 0 ) w e may find a ~(~/,[~]-}-]ta[-~-l).for the first motion. As ~ ( 0 ) w a s an omega limit point, for some t~ we shall have

(20)

] x~(t,)- ~ ( 0 ) l < a(~, I~l +]t=l ~ 1 ) . A,

As a consequence of this and the definition of ~ (e, ~),

(21) and (22)

I x~ (t, A- t,) - ~(t~)l:< I xk(ts + t~ - ~ , ) -- ~k(t~ + z)] < ~.

Recalling that v is a displacement number corresponding to e ~Ior the r~(t), we get f~om (21) and ( 2 2 ) :

(23)

i ~ (t, + ~) - :~ (t~)l <

2 ~ -4- e.

Since ~ is arbitrary, this impfies

"(24)

I ~ ( t ~ + , ) -- ~(tl)[ ~ ~.

This holds for all t l , and shows that ~is also a displacement number correspoading to 9 for the ~ ( t ) , so that the motion defined by them is almost periodic. C o r o l l a r y 1 . I ] one motion o/ a m i n i m a l set is almost periodic, all o/ the motians in the set are. For, the motions of the set are the derived motions of any one of them. C o r o l l a r y 2. l / a system o/motions contains a single stable almost periodic motion which is not purd# periodic it contains a non-enumerable number el such motions. For, by our theorem I the" motion is recurrent. Hence it is a member of a minimal set, which by corollary 1 consists entirely of Almost periodic motions. But 6 every minimal set which does not-consist of a single purely periodic motion has the power of the continuum. C o r o l l a r y 3. E a c h o/ the motions o/ a minimal set o/ almost periodic motions luts the same characteristic frequencies as an!~ "other. The corresponding (complex) coe/Jicients have the same absolute values. "For, let ~ ( t ) be any derived motion of zt~(t), as in the theorem. A~ g~(t) is stable and almost periodic, by theorem T it possetmes the propert~ E. From its propertT as a limit m o t i o n we have for some tx

(2t~)

Ix~ (t,) - ~ (o) 1 < h (~),

Almost periodic recurrent mot/one.

331

and hence for all t,

(26) I ~ (t~ + z) - .e~ (t)I < so that, race l e-~c,l+,~l-- 1,

~,

T On taking the, limlt for T = ~ ,

we find from this for

eachx k

and hence equals zero since e is arbitrary. The characteristic frequencies agree, since they are the 2'$ with nonvanishing coefficients a(2). We note t h a t ~[br realquantities, if ,4~. are cosine and B~ are sine coefficients, (29)

l a(~) t' = A: + BZ.

In conclusion, we note that a simple ex~aple of a system which illustrates o u r theorems is furnished by the motions of a particle on a smooth circular b~lliard table. For each motion, the angle which the path makes with the tangent to the circle at a point of reflection from the circumference is constant. Each set of motions obtained by shifthlg the p o i n t of reflection, but keepi~ag this angle and the velocity in the path fixedyields either a set of periodic motions or a minimal set. It is easy to see directly that these motions are stable, possess the property ~ and are almost periodic. Massachusetts Institute of Technology. (Eingegangen am 2. Mal 1928.)