numerical study of billiard motion in an annulus bounded by non-concentric circles

Physica 5D (1982) 273-286 ~ North-Holland Publishing Company

NUMEIUCAL STUDY OF BILLIARD MOTION IN AN ANNULUS BOUNDED BY NON-CONCENTRIC CIRCLES

N. SAIT(~

Department of Applied Physics, Waseda University, Tokyo 160, Japan

H. HIROOKA

Faculty of General Education, Hosei University, Tokyo 102, Japan

J. FORD and F. VIVALDI

School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

and

G. H. WALKER Department of Physics, University of Tennessee.Chattanooga, Chattanooga, Tennessee 37402, USA

Received 6 October 1981

This paper exposes a simple, focusing--dispersing billiard system whose phase space simultaneously exhibits the full range of possible Hamiltonian system orbit behavior over a continuous range of system parameter values. Specifically, we find that billiard motion between two non-concentric circles is characterized by three distinct phase space regions: (I) a rigorously integrable region in which billiard orbits u,dergo collisions only with the outer circle boundary; (2) a KAM near-integrable region in which billiard collisions strictly alternate between inner and outer circles; and (3) a chaotic region produced by orbital sequences of billiard collisions which randomly alternate between the integrable and near-integrable patterns. The properties of the integrable, near-integrable, and chaotic regions, the presence of island chains, and the homoclinic points associated with certain hyperbolic fixed points are discussed. Perhaps the most interesting feature of this billiard system, however, is the fresh view of the source of chaos it provides; specifically, the chaotic orbits of region (3) exhibit "random" jumping between the extended invariant curves of regions (I) and (2).

1. Introduction

In the present paper, we consider a billiard moving in the annular region bounded by two non-concentric circles as shown in fig. 1. By a billiard, of course, we mean a dynamical system consisting of a mass poipt freely moving inside a plane, bat ,lded domain and undergoing elastic collisions w~.h the boundaries which latter are ~ I U LU U ~ U t ~ t p l g ; t ~ l t i i ~ , a . v ~ , u o a a a ~ , v a s a w w ~ a ~ a ~ -

they are col ~.ave, convex, or straight relative to the boundm: interior. The billiard potential energy is ever)where constant except at the boundaries where it becomes infinite; thus, the billiard is a discontinuous, conservative Hamil- tonian system having two degrees of freedom.

For over a century it has been recognized [3] that the simple billiard can be used to study and illustrate tthe full range of possible Hamiltonian system behavior [1] without the need to intro- duce more complicated Hamiltonians.

Strictly integrable Hamiltonian behavior is illustrated [3] by the billiard moving within a circle or an ellipse, both of which have everywhere focusing boundaries. In the early 1960s, Ya,G, Sin~ [2] proved that a billiard moving in a domain having everywhere dispersing boundaries is a K-system, implying not only ergodic and mixing behavior but also everywhere exponentially separating orbits as in geodesic flow on surfaces of everywhere negatb e Gaus- sian curvature. More recently, Bunimovich [4]

016%2789/82/0000-0000/$02.75 © 1982 North-Holland

2".4 N. Sail6 el aLl Numerical study of billiard motion in an annulus

i P ( l . ~ )

, 1 0 R ,/ ]

/

Fig. i. This figure shows the annular region of bi!liard motion :,ounded by "he inner circle of radius r' and the outer circle of radius R. The centers of the two circles are ,eparated by :,:1¢ distance ,5'. Meaning is given to the other symbols in thi:: drawing in the accompanying textual nar- ratix e.

has rigorously established that K-system behavior is retained even when the billiard moves within a class of boundaries which are a com- bination of focusing and neutral only. The most famu, us Bunimovich boundary has the shape of a raceti ack or a stadium, i.e, two half circles joined by tx.vo straight lines. By letting the straight line segments of the stadium shape tend to zero length, one can observe a sharp, discontinuous transition in billiard system motion from chaotic to integrable (the circle). In a pioneering work, Benettin and Strelcyn [5] used the billiard to study the more usual continuous transition to chaes of Hamiltonian dynamics. Specifically, they continuously deformed a circle into the stadium via a sequence of generalized stadium boundaries composed of four circular arcs, except at the circle and stadium endpoints, of course. They found ergodic components in the phase space of generalized stadia intermediate between circle (integrable) and stadium (ergodic). In this paper, we devote our attention to establishing the properties of billiard motion in a particular disjoint boundary consisting of

one focusing and one dispersing element, a boundary type not previously studied by others to our knowledge.

In our system, when the distance 8' between centers of the two non-concentric circles tends to zero, the billiard reduces to the integrable case of concentric circles [3], since angular momentum is then a constant of the motion in addition to the energy. On the other hand, when 8' > 0 , our numerical results show that some chaotic orbit behavior appears. Nonetheless, our system is never fully ergodic because of a non-zero set of billiard orbits which always completely miss the inner circle. These orbits conserve angu:ar momentum and are therefore identical to the integrable orbits associated with a billiard moving interior to a single circle. Since the measure of this set of integrable orbits is always positive, full ergodicity is precluded. Finally, over a certain range of 8' values, the periodic orbit along the line OP in fig. I is numerically observed to be surrounded by KAM invariant surfaces. Thus, our system can simultaneously exhibit strictly integrable, KAM near-integrable, and chaotic behavior.

Since our non-concentric circle billiard is a Hamiltonian system having two degrees of freedom, we could reduce the motion to a plane, area-preserving r:apping through use of the Poincar~ surface of section technique [1] com- monly applied to continuous Hamiltonian systems. However, each sraight segment of a billiard orbit terminates or, ly at a billiard collision with the boundary; thus we choose to follow Birklloff [3] and Bcnetlin and Strelcyn [5] and use the more conveni,.nt area-preserving mapping variables (l ,s) , where I is normalized length around the or ter boundary and s is the sine of the reflectior angle at collision with the outer circle. For ncn-concentric circles, it is a matter of simple I;eometry to determine the mapping equation~, linking (I,s) values at sequential billiard, allisions with the boundary; a furthel simplific:.tion results when one notes that the t all I: ill:.:,, d orbit is completely specified

N. 5ait6 et al. l Numerical study of Mlliard motion in an annulus 275

by the sequence of ( I , s ) values for collisions with the outer circle only. In the following, w e numerically iterate the (I, s) mapping equations for selected values of the non-concentric circle parameters r = r ' lR and 8 =LS'/R, where R and r' are the radii of outer and inner circles respectively and where 8' is the distance between centers of the two circles. For the case r > 8, regions of integrability, near-integrability, and chaos coexist as mentioned previously. When r < 8, the periodic orbit OP in fig. 1 becomes unstable and the region of near-integrability surrounding it vanishes; in th;s parameter range, the whole plane is chaotic except for the always present finite region of integrable orbits.

Section 2 presents and disucsses the mapping equations; section 3 investigates the size of the near-integrable region as a function of the limiting rotation number of its central fixed point and also studies the homoclinic properties near an vnstable fixed point of the mapping. Finally, era:eluding remarks appear in section 4.

2. Mapping for the non-concentr ic billiard problem

We consider a billiard in the domain Q enclosed between a circular outer boundary O and a circular inner boundary I which latter is shifted to the left by an eccentricity 8' as shown in fig. I. The velocity of the billiard is normalized to yield unit sr~ed. The billiard moves in a straight line between collisions and, upon a collision with a boundary, it is reflected elastic- ally according to the rule: angle of incidence equals angle of reflection.

One may identify the billiard in Q with a flow on a three-dimensional phase space M = Q × S', where S t denotes the unit circle of possible two-dimensional unit velocity vectors. A point in this phase space is described by the coordinates (x, y, 0), where 0 is the angle between the unit velocity vector of S t and the x-axis. In this space, the measure (dx dy dO) is preserved.

But the dynamics of a billiard can be more conveniently described by the coordinate system (I, s); here l = ool2,rR, where o0 is arc length measured around the outer circle O from point P in fig. 1, where 2,rR is the circumference of the outer circle yielding III ~ 0.5, and where s = sin a with a being the angle of reflection at the point of collision with the outer circle yielding lal < ,rl2. The angle a is measured from the inward drawn normal to the outer boundary. In fig. 1, "he billiard leaving the point Po(lo, So) next collides with the outer circle at Pt(It, st) either directly without striking the inner boundary (hereafter called a-motion) or immediately following a collision with the inner circle I (called b-motion). The mapping from Po(Io, sD to Pt(It, sl) in the (l ,s) plane we denote by the symbol T. Successive mapping iterates are, as usual, denoted P0, TPo. T2Po, etc., and we determine a full system orbit by computing all prior and subsequent iterates of the initial Po.

For our system, the transformation T can be easily written down using geometrk, at , ~ u ments. For a-motion we find

T,t: A~ I = O~0,

oo~ = COo+ (It - 2c~o);

while for b-motion we have

(la)

I'.: sin a0 + 8 s in(no- oo) = r sin ~,.

sin at + 8 sin(a~ + or) = ," sm p.

2/3 = cx0 - COo + at + o~t. (Ib)

where/3 is the angle of reflection of the orbit at the inner circle. We may specify the geomet:y of the non-concentric circles using the two normalized parameters r = f i r and ~ = 8'/R. Billiard motion between two non-concentric circles is thus reduced to the mappings T,, and I".. In particular, the geometrical tangenc y con- dition dictating the use of T~ rather than T. is given by

[sin a + 8 sin(cx - oo)[ <~ r, (2)

2"~ N..Seilb el al. I Numerical sludy of b i l l i a r d motion in an annulus

where equality in inequality (2) means that the billiard orbit is tangen' to the inner circle. Here the choice between T~ or %, is made depending on whether the given (!, s) coordinates lie between or outside the dotted lines (see fig. 2 for example) in the (i, s) plane computed using the equalily in inequality (2). The mapping T can be shown to be area-preserving with the measure (dl ds) which is easily proved directly using eqs. (la) and (Ib), but a general proof is given by Arnerd and Avez [6]. A typical picture of the mapping generated by eqs. (la) and (lb) using

the parameter values r=0 .5 and 8 =0,25 is shown in fig. 2. Here we may distinguish four regions in the mappi'lg plane of fig. 2.

First, there is region D, defined by the con- dition

Isl = Isin al ~> r + a, (3)

which consists cf points requiring only the mapping 7",. For an orbit in this reeion, the straight line s eq~mls a constant is an invariant curve. In other ,cords, in regior. D the billiard

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Fig 2. Here we set a typical mapping plane describing billiard motion for the paramet<'rs r = 1/2 and a = 114. Region D, which ~ ,:o~ered by horizontal straight line invariant curves, is ~'igorously integrable. Region A, which here appears completely , overed by oval invariant curves, is actually oaly near-integr:tble. An orbit initiated with (I, s) values lying on the dotted curves ~ill deparl the outer circle on a straight line path tangent to the inner circle. Orbits initiated at points in region C proceed to comp!e'ely miss the inner circle; while orbits initiated at points in region B proceed to immediately hit the inner circle. The chaotic orbits ,..n regions B + C are those which randomly mires or strike the inner circle. Note, however, that there is a periodic orbit at the center of the chain of six islands which periodically misses and hits the inner circle.

N. ,~ i t6 el al.l NamericdstUdy o.f,.biiliard motiOn in an annulus 277

stays outside the dashed circle shown in fig. I of radius r '+8' and it repeats the a-motion indefinitely. Here, a given billiard orbit is always range:it to a circle concentric with 0 which has

~ad cons( regio Final to region D have finite measure, the non-concentric circle billiard system is not fully ergodic.

Let us now examine orbit behavior in the region /5 complert~entary to D for which Isl < r + 8. Examing/5 in fig. 2, we observe two types of billiard motion: (i) an uninterrupted sequence of b-motions and (ii) regularly and irregularly alternating sequences of a- az~d b-motions. For the trivial, completely integrable case 8 ffi 0, I5 is also integrable. Thus, we now only consider the case 8 > 0. In fig. 2, neglecting the ergodic look- ing splatter of points, the most prominent feature in D is the set of invariant curves, generated by type (i) pure b-motion, which appear in subregion A. Subregion A is in turn dominated by the fixed point ~t its center. Thus, let us examine the Tb of eq. (lb) for fixed points. We find two such fixed points, one at (0, 0) and one at (-+112,0) in the (l,s) plane. Since the character of the mapping in subregion A of region 15 is determined by the stability properties of these two, period one orbits, let us now briefly turn to a study of their stability character.

Linearizing the Th mapping of eq. (lb) about the origin of the (to, a) plane, we find

( : , ) 1 ( , + 2 r 8 - 2 8 - 2 8 : 2(1+1~)(!-,+8)~(a~o) (4) , = r 2~(g-r) r + 2 r S - 2 8 - 2 ~ 2} ao "

D - . ^ , - - , , . ^ [ A ~ -'~ ¢ ^ I I ^ . . , . ~ ¢ k ^ ¢ ~ k ~ , - - , ^ , 4 , , ^ ~ : ^ C . . , , . , ~ , , , l ' l U i l i ~ g l " 1~"9], ILL I U I i l T V Y 3 t i l g g t LIgc; t.JU, QUICgLtq~ J L U t ~ l i ~

(281r)(r- 8)to 2 + (2(1 + 8) / r ) ( l - r * 8)a 2 = constant (5)

are invariant curves under the linearized mapping. In regard to stability, we note that the

conditions r > 8, r < 8, and r ffi 8 correspond to the fixed point at the origin being elliptic, reflection hyperbolic, or parabolic respectively. Similarly, linearizing the Tb mapping about the fixed point (-+~, 0) gives

(~,~ ~) , /r-2rs+ 28-28: 2 , , -8 , ( , - , -8 ,~ = r \ 28(r + 8) r - 2r8 + 21~ - 28:]

\ ao /

yielding the hyperbolic invariant curves

(-281r)(r + 8)(~ - ~r) 2 + (2(1 - 8)/r)(l - r - 8)a: = constant. (7)

The fixed point is ordinary hyperbolic for all allowed valuues of r and 8.

When (as in fig. 2) r > 8, the central fixed point at the origin of the (l,s) plane is then rigorously known to be elliptic. As a consequence, KAM theory [1] guarantees that this elliptic fixed poin|, will be surrounded by a non- dense set of invariant curves. In fig. 2, the computer indicates that these KAM curves extend all the way out to the dotted curve given by using equality i~ eq. (2). Although elliptic near the origin, these curves become increasingly distorted as they lie increasingly far from the origin. It is perhaps worth emphasizing again that the orbits generating these invariant curves in subregion A involve type b-motion only. When r ffi 8 or when r < 8, shown in fig. 3 and fig. 4 respectively, the elliptic fixed point at the origin of the (I, s) plane turns first parabolic and then hyperbolic causing the stable subregion A to disappear into the chaos intermingling subregions B and C. In figs. 3 and 4, region D appears to be almost completely filled by a single ergodic component, although increased compul~er accuracy could possibly reveal many extremely small island chains of KAM st~L~,ility. Indeed,, fig. 3 shows chains ~f four and six islands which have become unstable for the parameter values used in fig. 4. These periodic orbits are composed of the transformations

~'~ N. Sails e! al. / N~merical study of billiard motion in an almulws

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Fig. ~. This ~gure shows the mapping plane for the parameter values r - 318 and 6 = 3/8. Here the stable region A of fig. 2 has ,,~ni~hed since the central fixed point at the origin has become parabolically unstable. As a consequence, the whole plane • ~rre:~r, to be chaotic except for region D and the two very small island chains of T6(C)) and T<(O). Note that the chain of six islands here is quite different from that appearing in fig. 2.

(T]T~): and (T~Tb):; the shapes of these billiard orbits in configuration space are shown in fig. 5. Further discussion of subregion A in /5 appears in the following section.

Finally. let us consider subregions B and C of /). If an o:bit could remain in the subregion B lying on either side of subregion A and between .~.. ~ . . . . ~ ,_^_ fig -,, ~,. uu,.tcu cuFves ~ . ~j computed from eq. (2). then only type b-motion would occur and the invariant curves of subregion A would extend on out to cover the whole of subregion B. Similarly, if an orbit could remain in subregion C lying between region D and the dotted curve, then only type a-motion would occur and

horizontal l ine invariant curves would cover all of subregion C. However, pure type t,-motion can at most occur only within a region bounded by the outermost KAM invariant curve surrounding the origin which, in fig. 2, touches the dotted curve. Orbits in the B subregion therefore must eventually miss the inner circle and cross into subregion C. o:_:,~.,. . • • .k.. o,,,,,ai,y, orbits In . ,~ subregion C conserve s = sin a = so until eventually they strike the inner circle moving into subregion B. Fig. 2 shows that the sequences of combined type a-motion and type b-motion which can occur in subregions B + C are quite random indeed. I~: fact, the ergodic splatter of

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Fig. 4. At r = I/4 and 8 = 1/2, the central fixed point at the or! in has gone hyperbolic creating what appears to be a completely chaotic sea between the two stable subregions comprising teL., ~ .'~. Specifically, the fixed points of T6(Q ) and T4(4~) are now hyperbohc also•

points in fig. 2 represents a single orbit. The chaotic subregions shown in fig. 2 axe much like those observed in the well-known Poincax6 map for the H6non-Heiles system [7].

However, as opposed to other chaotic sys-

Fig. 5. On the left, we see the position space appearance of an orbit yielding the fixed points of TS: while on the right, we see an orbit generating the fixed points of T 4.

terns previously studied, the non-concentric circle billiard system exposes an especially clear and simple view of the source of its chaos• It is this fact which makes it worthy of study and which offers hope for a deeper understanding of chaos. Specifically, let us recall that an orbit, during its sojourn in subregion C, has mapping iterates of the a-motion type only and that these iterates lie along the horizontal line s = sin a = So. Two especially clear examples of such orbit segments appear near the s = 0 axis in fig. 4. On the other hand, when this same orbit crosses the dotted curve of eq, (2)and 0asses into subregion B, then the mapping iterates are cf the b-motion type only, d u ~ the stay i~ subregion B, which conserve the near-constant of the motion asso-

280 N. Saitd et ai. i Numerical study of billiard motion in an annulus

elated w=th the KAM subregion A. Finally, when this orbit again crosses the dotted curve and returns back to region C, it now conserves s = s l rather than the original s = so. In short, chaos in this system is due to random hopping between constant of the motion curves in subregions B ÷ C. Alternatively stated, if we write the symbolic dynamic representation for an orbit a s . ~. aabbbabbaa . . . . then we note that chaotic orbits are described by (a, b) sequences as random as those of a coin toss. Not all such (a, b)- sequences need be random, for, as fig. 2 reveals, there is a chain of six islands corresponding to the periodic (a,b)-sequence . . . aabaabaabaab . . . It. remains to be seen whether or not. the full allowed set of (a,b)-sequences actually occur in this mapping.

Finally, it must be emphasized that this orbital jumping between constant of the motion curves is not merely a curiosity cf this simple billiard system .All Hamiltonian systems have a full set of local constants of the motion everywhere in their phase space; thus the chaos in thege systems may also be regarded as a random jumping belween local constant of the motion cur~es. The virtue of our non-cc~ncentric circle billiard system is that it illustrates this view of chaos in its simplest form where the jumps occur between only two con.~tants of the motion. Strictiy speaking of course, the KAM near-integrable constant of the motion is only an approximate o~ae, but, as figs. 2 and 7 reveal, the approximation is quite good. Moreover, once the basic notion is clear, one may easily con- struct a billiard s~stem having ~,,,vo rigorously integrable mapping regions as we demonstrate in section 4. Regardless of the particular model, however, these simple systems provide a new and especially clear view of the source ofchaos in all Hamiltonina systems.

3. Near-integrability of the region A

Since we are now done with the study of region /), we no longer have need to call A a

subregion; thus, in the following, we speak of region A. The dominant, governing features of region A axe the fixed point at the origin of the (!, s) plane and, to a slightly lesser extent, the other fixed point at (-+½, 0). In order to assay the integrability character of region A, we now turn to a study of these fixed points.

The nature of a fixed point is determined by its residue R, [8]. The eigenvalues A of the linearized m a p p i n g M about a specified point are given in terms of the residue R, by the equation

A = I - 2R°+-2VR,(R,- i ) (8)

For the fixed points at (0, 0) and at (---~, 0), the residues are

6 Re = r ( 8 - r + 1) (9)

and

8 = + r - l ) , ( 10 )

respectively. The fixed point (0,0) is elliptic (0 < Re < 1) for r > 8, and reflection hyperbolic

4 . I ( R e > l ) for r < 8 . The fixed point (_~,,0) is ordinary hyperbolic since Re < 0 always. As 8 or r vary, the behavior of the elliptic point at (0, 0) is best understood by studying its limiting rotation number 3, given by

2 cos 27r3' = l - R ~ = - r ( 8 2 + 8 - r S - ( r / 2 ) ) . (11)

Note that T becomes complex when the elliptic fixed point goes reflection hyperbolic.

I~. fig. 6, the curves of equal 3, for the elliptic case are shown for variqvs values of r and 8. As revealed by fig. 6, the case r = 1/2 and 8 = I/4 which is shown in fig. 2 has a limiting rotation number at the origin which is greater than 1/5. Fig. 7 shows an ~nlargement of the region A in

!

N, Soit6 et aLl Numerical study of billiard motion inan ann; lus 281

tO with the invariant curves of region A in fig. 2 vary quite slowly (see fig. 7); thus, if islanc! chains actually exist in the region A of fig. 7,

0.3

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0 0 0 .5 r t.O

Fig. 6. Residue and limiting rotation number of the central fixed point at the origin of the (I, s) plane. The cross marks show the cases treated in this paper.

fig. 2; even with enlargement, notice that no island chains indicating near-integrability appear. Since the island chain with rotation number 1/6 is far out in the ergodic sea, the rotation numbers (lying between 1/5 and 1/6) associated

the sense that if

Tb(It, SO = (12, sa), (12)

then

T b ( - - I I , - - S t ) " - ( - - 1 2 , - - Sz ) . (13)

Finally, the chaotic sea shown in fig. 2 lies just beyond the outermost invariant curve of fig. 7.

On the other hand a quite differen~t situatio~ is pictured in the enlargement of fig. 8 which uses r = 1/2 and 8 = 1/8. Here the limiting rotation number ,/ is slightly larger than 1/8 and the rotation numbers of the invariant c ~:rves slcwly decrease from that value as one moves away from the origin. Were this an integrable sy~,tem, the invariant curve with rotat;on number 1/8 would consist purely of period-eight fixed

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"8" N. Sail6 et al. i Numerical study of billiard molion in an annulus

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' . 9 . , ? • ,

' 7

x .o ~ e x T . , '

' . l

T 1 .

0 8 , I

T . .

• .

~

i " . 6' 6~ 6. .

-0 3 0 2 -0.1 0 O 1

2

Fig 8. Mapping of the upper half-plane of region A for r = !12 and fi = !18. As opposed to fig. 7. island chains as well as an ~uTer fr inge of c h a o s are now quite visible, indicating but not proving near-integrability.

points. In fig. 8. however, we note that the pos fition of this period-eight, invariant curve is occupied by a chain of eight islands, indicating but not proving near-integrability. Moreover, further away from the origin where one might expect to find an invariant curve with rotation nu,-nber 1/9, one finds two, interlaced chains of nine islands, again indicating but not proving near-integrability. The interlaced property of thii,~ island chain is due to the symmetry described in eqs. (12) and (13). Beyond the outermost inv:riant curve in fig. 8 lies the ergodic sea containing a complex network of invariant cur- yes associated with hyperbolic fixed points, an example of which we discuss in the following.

In most cases having limiting rotation number ,~ = l / n (n an il~teger), we obtain mappings similar to those previously shown, including interlaced island chains when the number of

islands is odd. When the lin,iting rotation number of the linearized maptfing about the fixed point at the origin is near one gf the values 112, 1/3, or 1/4, then a crises may ~ccur in which neglected, higher order, nonlinear terms determine mapping behavior near the origin [9]. Thus we find a strikingly different behavior when ~ is slightly greater than 114. The mapping for this case using r = 0.5 and 8 = 0.31 is shown in fig. 9. Here the outermost "elliptic" invarian: curve is diamond shap,.a ~,~a ;~ h,,,,,,a,.a h,, ¢,,,,, h,,,,°,. v v ~ a a ~ v 3 A ~ . ~ a a a j V ~ a

bolic fixed points of T~,. The hyperbolic invariant curves lying outside the hyperbolic fixed points all extend to near region C where they disappear. Indeed, if oae starts a point on one of these hyperbolic invariant curves, iterates move along the curve until they enter region C; eventually mapping iterates emerge from region C and begin to trace out another hyperbolic in-

N. Sail6 el al.I Numertcal study of billiard motion in an annulus 283

l ' t ' ' ~ t . ~ i _

01- .:,

• \ , / •

o i .........

. .

-0! i . . • ' ~ '~ 1

T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t . . . . . . . . . . . . . . . . . . . . . . . . . T . . . . . . . . . . . . . . .

0

Fig. 9. In the mapping picture shown here the limiting rotation number is slightly above I/4 and r - 0.5,/5--0.31. Indeed here we see that a crisis has occurred, making this figure dramatically different in appear~ince from that of fig. 2. The outermost elliptic curve is here diamond shaped, and hyperbolic invarinat curves exist outside the hyperbolic fixed points sitting just outsioe the vertices of the diamond. As the value of rotation number decreases toward I/4, the size of the stable A region tends to zero because the four hyperbolic fixed points then march toward and finally merge into the origin. The behavior shown here is typical of the richness contained in this seemingly elementary billiard system.

variant curve. This provides a striking example of jumping from one constant of the motion curve to another. It perhaps should be not~-d here that these four hyperbolic fixed points of T~ are not associated with a corresponding set of elliptic fixed points; thus, there exists no closed invariant curve outside the hyperbolic fixed points. Finally, the most dramatic feature . .t it,a 5 t.At-....:~.. ut ,-at, t, t,~.awut near v = I I',IA :,~ . . . . . . . .~vva,~u"A k..uy starting at the mapping of fig. 9 and then obtain- ing a sequence of mapping pictures for which 7 approaches 1/4. As ~/decreases toward 1/4, the f, ur hyperbolic fixed points of T~ march in along their "espective axes toward the origin, thereby decreasing the area of elliptic invariant curves about the origin and increasing the area of hyperbolic invariant curves outside the

hyperbofic fixed points themselves. Throughout this whole sequence, the hyperbolic invariant

immediately reemerge as the 1,-value is further lowered below 1/4 but now they form the vertices of a square rather than a diamond. The behavior being illustrated here is quite typical of that which occurs in nonlinear oscillator systems whose frequencies are moved into low order resonance.

Fig. 10 shows that the billiard mapping plane at r -- 0.5 and , / - 0.03 appears to be complete!y integrable. Nonetheless, an experi¢,ced numerical investigator is convinc.-.d tha~ such is not the case by the mere presence of the island chains shown in fig. 8. To actually Ioca*e and verify the exis , ~,'e of island chains in fig. I0, however, would be numericr.lly tedious. Thus, to definitively establish the KAM near-integrable character of the fig. 10 mapping, we

0 5

o .

o .

y " .

j ,

/ ' /

l •

\ ,~\\

• .

i

p

/

\ ' ' i 1 /

, , I

-o.s 0 l 0.s Fig. 10. At r - 0.5 and 8 - 0.03, the .,lapping equations of eq. (Ix, b) appear integrable, although extremely high computer accuracy can be used to reveal island chains and a small band of chaos near the dotted curve. For an American, the eye shape centered on :he origin reminds one of the trademark of an America:, television network or of the tuning eye on old radios.

2S4 N, Soil~; et al. I Numerical study of billiard molion ~n an annulus

. 9 " ~

4,

-'0 5 it - Q,,95

Fig. l i. Accurate in~tegratie'~ of the incomip.g and oulgeing invariant curves associated with the hyperbolic fixed point al (+~,OJ re,~'eais the homoclinic points of intersection be- t~veen the invariant curves. As a consequence, region A in fig. I0 is only near4ntegrable.

computed a h enlargement of fig. 10 near the hyperbolic fixed poim (-+~,0). If the invariant curves associated with this hyperbolic fixed point exhibit homoc!inic points [6] of intersection, then even mathematicians become co.r~vinced of KAM near-integrabitity in fig. 10. Fig. II shows that indeed the hyperbolic in- v,~r~ant curves do intersect, assuring near-in- l egrab i l i t y o f r e g i o n A. T o o b t a i n fig. i l , w e

forward integrated the upper, outgoing invariant curve from ( _ t 0) in fig. l0 until it approached (~.~,0): we then backward integrated the upper, ;..~cohling in~ariant curve to (~,0). In the magl~itied view provided by fig. I l, the homoclinic intersection points of these two distinct curves are quite obvious. Finally, the previously unnoticed small regions of chaos near (+[, 0) in fig. 10 are exposed in the enlargement of fig. 12.

4, C o n c l u d i n g r e m a r k s

In this paper, we have exposed a billiard system which can simultaneously exhibit all possible Hamiltonian system orbit behavior-

oo,!

i "..

% . . . .

0 , ,

L ~i~ < "

- 0 . 4 9

Fig. 12. The homoclinic structure shown in fig. II implies the existence of narrow bands of chaos in fig. 10 about the invariant curves connecting the fixed points (±!,0). Under magnification, these small regions of chaos can be rendered visible as shown here.

( i ,")

H

/ .

A I '

, ,'/I/ \ /

\ . °

(A)

• H '

• /

' t i t , ^ i

t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 13. In (a) we see two, concentric, mutually per- pendicular, intersecting ellipses• By selectively choosing part of this system, we can obtain a billiard boundary having t w c ~ ~ t r l e t l v i n t o o r ~ h l , r ,~e, l n n e I n f l ~ t h - . e .~l~,4 o . , . ~ , . ~ ~ h . ~ . , e . . . . . . . . ~ . . j . . . ~ l ~ . . . . . ~ . ~ . l ~ . v , m o • i , , ~ , . . ; L u g . . . w • ~ i . o l | v o :

the part of {a) which must be retained to obtain a billiard having two strictly integrable regions. This is achieved by :

i [etaining the two stable minor axes and eliminating the two ~

unstable major axes. Finally in (c), the solid curve shows the part of (a) retained in order to yield a billiard having great instability. Here only the unstable major axes are retained; preliminary evidence indicates that this system may be a K-system.

. '-° ' " ' , ; ' : ," . . , . ' ,~. ~ ' ~ ; " '" " , . " .~ . " " , ." . : ; : ~ . : . . . ' " ' : , ~ . " ", " , . ~ ' ~ : " _'_* ~,. ' ; '.7; % , , -~ ' , , " * ' , .~,*..,., .:.:~y ,: ",." ~. " . } % Z , ' , . ~ , . - ..¢~, ~' . .~ , .~ . .~ " . , , , ' . . - , ' . . " ~ ~.,.,~.. . . . . . - : .~ ; . , | ,,' . , / . - , , , . , , " ' , ~ , e ' . . r ' , . . r . , " . ' . , '" ' ~ " " " , , V ; . . . . . . " . . 1 , ' , i _ v - - . • . . . . - f . •

, , . - . " , ~ , ~ , . f . : . • , - . , . , , ~ ' . . . . . , , - , , ; . ' . : . . , ; . ; . . , " . z . , ' , . . • , , z . . . . .~ - , ~ , , , , l ~ , . , , . • . . . . _.,.. , . . . , . . . , . . . . . , . . . : : . . . , . . . . . " ' . ~ . . ~ : . . , . : : . . . . . . . • : ' , . . & ~ ; : • .

• . . ' % , , . , ' , , - ' " . . " " . . " ' ~ ' l : , ~ f " . . ' • , ~ - . ~ . . • . . . - , . -

• . • o . ! • ~ • . .

• . *o

4 * * a

• . .. o? ,,..". .... ..,.. oo .".. ) ~ " - . " . . . . " "" 0 " ' . " " " . " " c ~ . . . . . "" " ": ' , • . . . . . • , . . . . , _ - . . . . , . . . . . . . . , . . . . . , . . . . ~ . . . - . . . , . . . . . . .

• ".'.. ". .... " • ".....-"~,'~"'._ ";'..,..~ :'...."" ;",<t : ""..: " " ." "'~"- ' ' • '. ." "" . . . . . . :... : . . . . - • ~. , . . . . . ~,. ..,.: . . . . . . . . .

/ "~... ,', ,_. .. .." \ _ ,,..:.'" ~ .... , ..., . : .. ... ,._ ..,.,.....

• .,~'i i~ / .~ ~, "

I ":" ! ".':" "<'" ~ "" "' "'"- [ . t : ; . . " . ! s . _ ~ " " . ' :

• ,~ .,~' "~:: : i'"' ' "#\ :~. ; . .

- / ~ . " . / ' - -

. 4',:

72" ": . . . . ; ' : ' . . . . . . . . . . . ' " " ' . \ • """ . . .~. . .

~" ' . ' " "'~ . ~ " " .: '" '" ":i .... ": "~

o ; ' ..... o " . . ,

• ' . • • °

." °

, * ; , " . ° , •

• ' . . . f j . . . • , • , • . * , " i " q . "

, . . . - " a: . . .~ . " " " • "'~ " ' " . "z'l " '," • ' " ' " ~' , ' o . . . . .~ ~.~ . . ' ~ s . . . ~ " . " : ' ~ . " . . . . . ; . . . . . . • • . . . . .

• . ' . . , , , , i t , , ' , , "~ , • • . . . . . . " , . : - . . . o , t

- 0 . $

! . -

...:'; 0 ~ "" .... " C , ,~. ::'.. , . . : . : ' ~ " ~ , . • . . , . . , . . . . . • • , . . ~ ~ - . , " . , . . . . . .

i " , . " ' " ~ " . " " " . " : ' ¢ : ) " . . " " " . " " " , " ' ~ " ~ - . : - . - . . , .

• o.

• . . . ' , D o " . . . . . . . . . ' " o 0 .. , , . . ' ° ,

• ..

• . " . l ~ e , . " . . . '

. . , . ' . o .~ : . . . . • , . " ~ . - . .

• . " , . : . . . : " . " : ~ , . . : : , , . . , , : . . . , . . . . ' ; , . - ' : . . . " , . " . ~ . ~ . . . e . , . : . . . ' , ' ~ ' : . . , . , . ' "" 4 " : . . , : ,~. ' , , " ; . - ,

t. e - o , ~

I - t

O . 5

Fig. 14. This is the mapping plane for the billiard of fig. 13b. The four stable regions along th,~ s = 0 axis are the rigorously integrable regions surrounding the two straight line, stable periodic orbits AA' and BB' of fig. 13b. The four large stable regions above (and below) the s - 0 axis are centered on the diamond shaped periodic orbit ABA'B' (and its time reverse) of fig. 13b. Preliminary computer evidence indicates that all the stable island chains shown in this figure are rigorously integrable, a quite startling and remarkable result, if true.

: i e = o . S

. i

, h ' '

. i ,.,o " . , ' . "

' ' . ' q ' " t ' *,"

' ' : ' '" " " . . . : " . .~, ' • • : c • . : , . • . . . . . . . ~

• • . • . • . . .

• . o • . . . ~ ' . . , o . , . , • ! ~ ' , , ~ ' ' , .

, , . o . . • . . . e . ' " . . • . o . . o • " : ~ " ' ' • ' i . . . . . "~. ,

- 0 . S o o , S

Fig. 15. This ~' he mapping plane for the billiard confined to the boundary shown in fly. 13c. To the computer accuracy used, no regions of s bility could be found. Because of derivattvc d,scontinuities in this o~tterwise strictly convex boundary, the KAM theorem need not apply; thus, there is a possibility that this is a K-system billiard.

285

2S¢, N. Salt6 et al. INum, ,'ical study of billiard motion in an annulus

integrable, near-integrable, and chaotic. Specifically, ,a billiard bouncing between two non-concentric circles can display three distinct iypes of orbits: l) coilision with outside circle only (integrable): 2) strictly al ternating collisions between inner and outer circles (KAM near- integrable): and 3) random xeqo-~,nces of inner- outer circle collisions (chaotic). In case 1) above, an orbit conserves the constant of the motion s = sin c~ = so; in case 2), an orbit conserves an approximate, computable , near-integrable constant of the motion [ 10]: and in case 3), an orbit sequentially and randomly "conserves" the constants of the motion for cases 1) and 2). !t is this obvious random hopping between only two constants of the motion which provides the hope that this system, or ones like it, can become the paradigm for chan~. Fur- thermore, when the elliptic point at the origin of the ( l , s ) plane goes reflection hyperbolic, we might expect a period doubling (of the Feigen- baum type) resulting in chaos in the neighbour- hood of the origin. This will be discussed else- where.

Another simple system exhibiting constant of the motion jumping is that of two coupled, harmonic oscillators which can undergo hard roint collisions. Between collisions, the phase space orbit lies on some two-dimensional torus [i]: upon collision, the orbit jumps to another torus. Chaos has already been observed in this system [I 1], and we intend to discuss this chaos in terms of torus jumping in another place.

One defect of the no"~-concentric circle billiard is that o , of its constants of the motion is not exact. This defect can be remedied by con- sidering the intersecting, concentric ellipse bil- l ~ , ~ , - , 4 I . . ..... .,I . . . . . t . . . . .

,,,~,u vuuuu, t~y ..,Jit, wn in fig. 13b. Reca l l tha t the

billiard in a single ellipse is r igorously integrable and that the stable periodic orbit along the minor axis is surrounded by a large region of stability. The intersecting ellipse boundary shown in fig. 13b is now seen to contain both minor axes and both regions of strict integrability %r which the ,'ons*.ants of the motion

are known. Fig. 14 shows the intersect ing ellipse billiard mapping for the boundary of Fig. 13b, where the two ellipses have equal eccentricity. The striking feature of fig. 14, at least to the numerical accmacy we used, is not merely that two rigorously integrable regions exist but that all the stable island chain regions also appear to be integrable. Should this feature survive more accurate investigation, it would be a quite remarkable result indeed. Finally, the billiard boundary shown in fig. 13c retains only the unstable, major axis orbits; fig. 15 thus hints that the four delta-function negative curvature points embedded in the otherwise strict!y con-. vex focusing aoundary may yield K-system behavior.

Reterences

111

[21

[3]

[4]

[5]

[61

[71 [8) [9]

[10] [Ill

The following papers provide general reviews of integrable, near-integrable, and chaotic behavior in dynamical systems: J. Ford, in Fundamental Problems in Statistical Mechanics 1II, E.G.D. Cohen, ed. (North Holland. Amsterdam, 1975), p. 215; M.V. Berry. ;n Topics in Nonlinear Dynamics, A.I P. Conference Proceedings No. 46 (American Institute of Physics, New York, 1978), p. 16: B.V. Chirikov, Phys. Repts. 52 (1979) 263; and R.H.G. Helleman, in Fundamental Problems in Statistical Mechanics V, E.G.D. Cohen, ed. (North-Holland, Amsterdam, 1980). Ya.G. Sinai, Soviet Math. Dokl. 4 (1963) 1818: Russ. Math. Su,'~. 25 (1970) 137. See, for example, either G.D. Birkhoff, Dynamical Systems (American Mathematical Society, Providence 1927), or Ya.G. Sinai, Introduction to Ergodic Theory (Princeton University Press, Princeton, 1976), Lecture 10. L.A. Bunimovich, Funct. Anal. Appl. 8 (1974) 254; Commun. Math Phys. 65 (1979) 295. G. Benettin and J,M. Strelcyn, Phys. Rev. AI" (1978) 773. V.I. Arnol'd and A. Avex, Ergodic Problems of Classi- cal Mechanics (Benjamin, New York, 1968), App. 31. M. H~non and C. Heiles, Astron. J. 69 (1964) 73. J.M. Greene, J. Math. Phys. 9 (1968) 760. C.L. Siegel and J.K. Moser, Lectures on Celestial Mechanics (Springer, New York, 19711, §31 and §32. J. Moser, Memoirs Am. Math. Soc. 81 (1968) 1. R.S. Northcote and R.B. Potts, J. Math. Phys. 5 (1964) 383.

numerical study of billiard motion in an annulus bounded by non-concentric circles

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