Some examples of permutations modelling area preserving monotone twist maps

Physlca 28D (1987) 393-400 North-Holland, Amsterdam

SOME EXAMPLES OF PERMUTATIONS MODELLING AREA PRESERVING MONOTONE TWIST MAPS Glen R. HALL* Department of Mathemat, cs, Boston Unwerstty, Boston, MA 02215, USA Recexved 7 Apnl 1987

We present several sample examples of permutattons of a lattace whch model area preserving monotone twist maps of the annulus The behavaor of these examples ~s cons:derably chfferent from what the analogies with behavaor of their smooth counterparts would predict

1. Introduction

Some of the interesting recent work on area preserving monotone twist mappings of the annulus has been mouvated by "numerical experiments". A frequently used (and enlightening) techmque in such expenments is the computation and display of many iterates (e.g., tens of thousands) of a single point. The computer, of course, does not compute the exact value of the mapping, but rather it computes the orbits under a dynamical system on a finite set of points. Hence the monotone twist mapping computed by the maclune is more accurately considered a permutation on a fimte lattice of pomtst. In this note we present some (very) elementary examples of dynamical systems on a lattice which are the natural discrete counterparts of a family of area preservmg monotone twist maps of the annulus. The behavior of these permutations is seen to sometimes differ considerably from what analogtes with smooth maps would predict. The purpose of tlus note is not to criticize the numerical work-indeed, the pictures have been a remarkable aid m the study of area preserving twist maps and the examples we construct are somewhat pathological. Rather, we hope to point out that the dynamical systems on finite sets wluch computers compute can have intereslang and unexpected behavtor.

2. Notations and definitions

Following Lax [1] we will model area preserving mappings by permutat:ons of a lattice. We will consider maps of the lattice Z × Z to itself. We let

~2 : g x z - ' z : O ' J ) - ' *Parually supported by NSF and the Sloan Foundation tAdded m proof Thts techmque is exphtatly used m F Ratmou, Astron and Astrophys 31 (1974), see also I Perenval and F Vlvaldl, Physlca 25D (1987) 105

0167-2789/87/$03.50 © Elsevier Science Pubhshers B.V. (North-Holland Physics Publishing Division)

G R H a l l / Permutattons modelhng area preservmg monotone twtst maps

394

be the usual projections onto components. We wall be considering a specml farmly of maps of Z × Z to ~tself wtuch model the hfts of area preserving monotone twist maps of the annulus. Hence we say,

Definmon. A blject]ve map F: Z X Z ---,Z X Z will be said to have period N > 0 if (1) V ( I , . ] ) E Z X Z , FO+N,J)=F(t,j)+(N,O), (Ix) V(t, J ) ~ Z × Z,

~q(F(/, J + N ) ) - ~rl(FO, j ) ) m o d N and *r2(F(,, j + N ) ) = ,r2(F(, , j ) ) + N.

Remark. The x-coordinate corresponds to the angular coordinate in the annulus and condition (i) is the natural condition for hfts. Condxtion (n) is a penodlclty condition in the y or radical component. The assumption of such a periodicity m the y component allows the use of global compactness arguments and does not affect the type of phenomena we wish to study. Also tlus condmon is satisfied for "standard" famlhes of twist mappings which have been extenswely stud]ed (e.g., see Cturlkov [2], H e r m a n [3]) Define

T x. Z x Z - - - , Z x Z ,

~l: (,, J) ~ (, +S, S) and

T2: Z × Z - - * Z × Z , r2: (', J) -~ (t + =S, J) Then these are both examples of bljectxons with period N (for any N ) They are the discrete analogs of lntegrable monotone twist mappings

Definmon. F1x N > 0. We say a m a p ep: Z ~ Z sattsfies condtnons (a) and (b) if

(a)W z,

N-1

(b) Z

i=O

Notatton. Given 0: Z ~ Z satisfying (a) and (b) we define S¢,: Z X Z ~ Z X Z ,

(,, s) -" (,, s + Remark C o n d m o n (a) on ~ assures that S, has period N while condition (b) Is the analog of the " n o net flux" or "center of mass preservation" conditions for monotone twist maps (see H e r m a n [3], p. 47). We wall always consider bljecUve maps of Z --* Z as these seem m the natural model for area preserving maps.

3. Discrete standard maps In the first example we take advantage of an obvious resonance between the maps and the dlscrettzatlon. First, fix N = 2(2m + 1), 1.e, N is two times an odd number. Let T2 Z × Z ~ Z × Z be as m the last

G R Hall/Permutatwns modelhng area preservmg monotone tw,st maps

¢1(l) =

1

if t = 0 mod N,

--1

if

0

otherwise.

395

l=N/2modN,

Let So, be as in the last section and to simplify notation let S 1 = S~1. Then T2 o St is a bijecUon with peri(oh ~ .

Remark.

The maps T2 o S 1 is a reasonable model for the "standard" area preserving monotone twist map given on R 2 by

(x,y)~(x+

2(y+,cos(2-~)),y+,cos(2-~)

),

where we have divided the x and y axes into grids of equal size and c is just shghtly larger than half the grid size. Alternately, we would consider it a model for the map

(x,y)-*(x

+ y + , cos (2-x-), y + , cos (2-x-)),

where the grid length in the y direction is twice that in the x direction. Since the perturbation of the "integrable" map T2 is very s m a l l - a s small as a perturbation one may make an lifts f a m i ~ - one rmght expect the same stabalit2¢ p~opellaes given b2~ the K.A.M. theory (see Moser [4]) m the smooth case. However, it is easily seen that tins is not the case.

Prolposmon "k. ~ e r e

exksX~t, j'} ~ Z "f. X saxc~ t~taX

1 2((z2oal)n(,,j))l- oo Proof.

asn

oo

First we note that for any (k, l) ~ Z × Z

[k+21, ~rx((T2 * S1)(k, 1)) = ~ k + 2 1 - 2 ~k+21+2.

or

Hence, we may conclude by reduction that if k is even (respectively, odd) then ,q((T2 o s1)n(k, l)) is even (respectively, odd) for all n. Since N is even we see that if k l = ~rl((T2 * s1)n(k, I)) mod N, then k t Is even (respectively, odd) if and only if k is even (respectively, odd). Now consider any point of the form (0, j). Let (l 1, J1) = (T2 ° Sx)(0, J), so l 1 is even and di --J + 1. By the above, the value of the second component of ( T2 o S1)n(ll, J1) can only change when the first component is zero mod N (the first component cannot be congruent to the odd number N/2 mod N). The even numbers rood N is a finite set so (Po'mcar~ recurrences) there exists n > 0 such that •rl(ljT2 * S 1 ) ~ ' l b , ) ) ) - b mo/~ :R. "By pefioi~&~y ~n lMe hrst component we are back to t)~e ofigina) gJtuafion. H e n c e an ease m~tttcttott ar~ttmenX s'ctows t~tat ~or a.ay j , o Sl)

(o, j ) ) - , o ,

as n - -

396

G R Hall/ Permutattons modelhng area preservmg monotone twtst maps

SImalarly, for any j,

~rz((T2°S,)"(N/2, J))--*-oo s i n c e ,/?l((T2 o

as n ~ oo.

S1)n(N/2, j)) can only be odd mod N.

Clearly, the resonance between the choice of N and the map choice of N can yield even " m o r e " mstablhty.

Proposmon 2 If N = 2 p for p > 2

[]

T2 causes the mstabihty above. A " p o o r e r "

a prime number then for every

(t,j)~Z×Z

except

j ~ (0, p},

t ~ ( 0 , p ) rood N,

[~r2((r2°s~)n(t,J))l~oo

as n ---, oo

Proof. The proof follows exactly as in p r o p o s m o n (1) once we notice that for any (t, j ) ~ Z × Z if j is not in (0, p ) m o d N and t is even there extsts n > 0 with '/rl ( ( T 2 ° S1) n ( / , J ) ) - = 0 m o d

N,

while ff t is odd there exasts n > 0 with 'g/'l ( ( T 2 ° s 1 ) n ( l , j ) ) =- N / 2 mod N

[]

The choices made above ( N and T2) are already very special. However, we have considerable freedom in the choice of q~

Proposmon 3. Suppose q~ Z ~ Z satisfies conditions (a) and (b) of sectmn 2 wath N = 2(2m + 1) (1.e., two times odd). If X'(N-2)/2a, "-',=0 v ~tO, , - , / ~ 0 then there exists ( t , j ) ~ Z × Z such that I~ri((T2oSo)"O,j))l--, oo as /'/ "--'-~OO.

Proof. Let JV'= {0,1,2 . . . . N - l }

and 8 = ( 0 , 2 , 3 , ,N-2). The penodiclty of T2oS o and the fact that the points with even x-component form an mvarlant set implies we may define F o~× J V ' ~ g × .A/" by F(l, j ) = (tt, J1) where q ~ g, t 1 - ~rl(T2 o So(I ' j ) ) m o d N and A ~ .A/', J1 - IrE(T2 ° So(t, j ) ) m o d N. The set ~ × .A/" contains N2/2 points so we m a y define a measure # on ~ × .A/" by making each point an a t o m with measure 2/n 2 Since F is a permutation, # is an invarlant measure for F Define ~. o~× J V ' ~ Z by ~ ( l , j ) = q~(t) If we set ~ equal to the time average over orbits, i e n-1

~ ( l , j ) = hm 1 y , q~(Fk(t, j ) ) , n ~ o o /'/ k = 0

then the ergodlc theorem lmphes that the above hrmt exists and

Since f ~ d# = N E , =(N0 2)/2q'(20 4:0 we see that for some 0, j ) ~ o ~ × J V ", ~ ( t , j ) 4 : 0, so

n- 1 kO,J))l IEkfo~(F

G R Hall/Permutatwns modelhng areapreservmg monotone twtst maps

397

oo as n ~ oo. But since n-1

k=O []

the proof is complete.

The technique above could also have been used to prove proposltaon 1. We gaan some interesting adchtmnal information from the above proof•

Proposmon

4. For the maps T2 o S 1 and T2 o S, of propositions 1, 2 and 3, there exist pomts 0 , J) ~ Z X Z

such that

[~rR((T2 ° Sl)~ (,, j)) -JI

hm I* ---¢OO

> 2/N

and (N - 2)/2

n

lim

Proof.

(l,j))-j]

¢r2((T2oS*)

>

E

,b(2t)/N.

Using the notation from proposition 3, we know that (N- 2)/2

f,d

--

so that for

[

E ,(2,).1

t=O

some (t,

y) ~ oa× JV', 16(t, Y)l >

3"¢)

tim

NX~N-oZ~/zeP(2t)/(N2/2) so

(t,J))-cr2((T2os¢)k(t,J)

>2

for

E

some (t, j) E d'× .hr, ¢(2,)/N.

and s ~ e ~ e sx~m ~dez~.ope~ "~e ha'~e ( N - 2)/2

hm

l~r2((T2oS,c,)n(t,J))-Jl>_2

~

¢(2t)/N,

which gives the secottd tttequali~ of the propositiou. Tb.e &st meguality follows e~si~ hy appbdAg the • (N - 2)/2 second to the case ¢ = 0t since E,-0 ¢1(20 = 1. [] Finally, as a corollary to proposition (3) we note that a " r a n d o m " perturbaUon of T2 can frequently yield mstabihty. In particular, note that there are

N

E

I even

! )

,/2

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G R H a l l / Permutattons modellmg area preseromg monotone twtst maps

m a p s ~: .A/'---, { - 1 , 0 , 1 } with Ey__~ldp(/)=0. We m a y extend such a ¢ to a map Z---, { - 1 , 0 , 1 } periodicity. If we assign equal probability to each such ~ then

by

Proposmon

5. Again assume N = 2(2m + 1) ( = two times odd). Given a randomly chosen map ~ / ' ~ { - 1, 0,1 } with E,N_~lq~(t) = 0, the probablhty that there exists (l, j ) ~ Z × Z such that

,/72((~ o so) ° (l, J ))

~ OO as n ~ oo

~s greater than or equal to

_

'even N l/2 / ,"On~ (i)(l/21))

(N/2 ) ) 2 / ((,:' )~(0

and flus tends to one as N tends to lnfimty

Proof. The

number of maps ~. ,¢"--, { - 1, 0,1 } with

N-I

N-I

E ,(,)=0= E,(,)

triO i even

t=O zodd

is

=

( v ) ( 1/2'

and

t even

the first statement follows from p r o p o s m o n 3. The hmlt as N tends to infinity of the probabllltmS is one since (by elementary counting),

N

Z{-o i even

1

,/2

< ( 3 2 / N 2 ) -+ 0

as N ~

o0

[]

4. A n o t h e r e x a m p l e a n d invariant circles

The examples of the previous secUon rehed heavily on the resonance between the map T2 and the chome of N With one last example we show that this as not the only reason for mstablhty. Fix N > 0 and let [ .] denote the usual greatest integer function. Let TI: Z × Z ~ Z × Z (t, j ) ~ (t + j , j ) as in section 2

Proposmon 6

There exists ~" Z ---, ( - 1, 0,1 } satisfying conditions (a) and (b) of section 2 such that

.,((7"1o s, )~:'(o, o)) = [ N/21.

G R Hall/Permutauons modelhngareapreservmgmonotonetwist maps

399

Proof. D e f n e ~ ( l ) = + 1 for l---n(n + 1)/2 mod N, n = 0,1, 2, . . , [ N / 2 ] and then extend ~ to a map from Z to { - 1, 0 , 1 } satisfying con&tions (a) and (b) of section 2. Then, an easy induction shows ~h((T1 o S,)"(0,0)) = ~ k = n(n+ 1) 2 ' kffil

,:((T~ o S , ) " + x ( 0 , 0 ) ) - ¢5((T, o S , ) ' ( 0 , 0 ) ) = 1 wl~'cfi compleles the proof. Hence a "small" perturbation S~, in the sense that [(~(l)[ < 1 for all t, yields a map with orbits whose y coordinate varies a great deal. Fiua[[,~. we state a DtqDositiou whick ~ s ~at tke cxmceot Qf "i~,~a~-i~.~t~tclc.~ an~)iv~.~ _to_d~id rotatlon" occurs only for trivialmaps in the class wc have bccn consldcnng.

Proposmon 7. Fix N > 0 arbitrary. Suppose ~) satlsficscondition (a) and (b) of scctlon 2 and suppose there cxlsts ~: Z --*Z satisfying

(1) w

z,

¢O+N)=¢(t),

(11)

TlOScb({,,¢(I)): I E Z } ) - - - - { ( l , t ~ ( t ) ) : , ~ Z } ,

Off)

' f ' < '1 then ¢q((rl o S,)(,, ~b(,))) < ¢q((rlo S , ) ( q , ~ ( q ) ) ) .

Then # ~ 9 so T l o $~ = T~.

¢r2(TlO S÷)(t, ~b(,)) ~ ~ ( , ) so ~ cannot be constant. Fix , so that ~(t) > ~(~ + 1). Then ¢rl((T~ o S~)-I(,, ~0))) -< ¢rl((T1 ° S~)-10 + 1, ~(, + 1))) - 2 and we see condition (in) must be violated for the point ( q , ~ ( q ) ) with ¢rl((T~ o S,)-~(t, * ( , ) ) ) < tl < ¢q((T~ o S÷)-1(, + 1 , , ( , + 1))). Hence we have a contradiction, so ~ - O.

Remark. T~e. ~mc~f ~

sta~s

~',a~ ff ~

[] = ~a~t

ha~ a ~ a p h " ~ c ~ is m v a r ~ a m unde~ T 1 o S~ ti~en

twist maps and the functional form of invariant circles for "standard" famihes of monotone twist maps

(sc~ ~ - n . w , I~9.

400

G R Hall/Permutations modelhng area preservmg monotone twist maps

5. Concluding remarks We again emphasize that the simple examples above are not meant to cast doubt on the numerical "experiments" on area preserving monotone twist maps. The escape rates are very slow and resonances are clearly needed for these examples. Also it is not clear when the perturbaUons of "integrable" maps described above are models of smooth perturbations of the assooated continuous dynamical systems. We only wish to point out that blanket generahzat~ons between the theorems for smooth maps and the maps on finite sets which computers compute are not jusufied and that these maps on finite sets can display comphcated and interesting dynamics worthy of study.

Acknowledgements These examples arose from conversations with R McGehee and R Moeckel and a lecture of P Lax

References [1] P Lax, Approxtmatlon of measure preservangtransformations, Comm Pure and Appl Math 24 (1971) 133-135 [2] B V Chnkov, A umversal mstabdlty of many-dimensional oscdlator systems, Phys Rep 52 No 5 (1979) 263-379 [3] M R Herman, Sur les courbes mvanantes par les &ff~omorplusmesde l'anneau, Vol Asterlsque 1 (1983) 103-104 [4] J Moser, Stable and random motions m dynamacalsystems with specml emphasis on celestaal mechamcs, Annals of MathemaUcs Studies # 77 (Pnnceton Umv Press, 1973)