Chaotic dynamics of a bouncing ball

Physica 19D (1986) 355-383 North-Holland, Amsterdam

CHAOTIC DYNAMICS OF A BOUNCING BALL R.M. EVERSON Department of Applied Mathematical Studies, The School of Mathematics, The University of Leeds, Leeds LS2 9JT, UK Received 18 July 1985 Revised 1 November 1985

A detailed study of a mapping on a two-dimensional manifold is made. The mapping describes a system subject to periodic forcing, in particular an imperfectly elastic ball bouncing on a vibrating platform. Quasiperiodic motion on a one-dimensional manifold is proven, and observed numerically, at low forcing, while at higher forcing Smale horseshoes are present. We examine the evolution of the attracting set with changing parameter. Spatial structure is organised by fixed points of the mapping and sudden changes occur by crises. A new type of chaos, in which a trajectory alternates between two distinct chaotic regions, is described and explained in terms of manifold collisions. Throughout we are concerned to examine the behaviour of Lyaptmov exponents. Typical behaviour of Lyapunov exponents in the quasiperiodic regime under the influence of external noise is discussed. At higher forcing a certain symmetry of the attractor allows an analytic expression for the exponents to be given.

1. Introduction

In recent years it has become strikingly evident that dissipative nonlinear systems subject to periodic forcing do not display the relatively simple phenomena associated with linear systems. The Duffing equation [1] and van der Pol's equation [2] are primary examples from the field of differential equations exhibiting chaotic motion. The Poincar6 map provides a convenient means of studying such systems as it yields a mapping on a manifold with dimension one smaller than the original phase space. Chaotic motion, which in dissipative systems requires local expansion and contraction normal to the trajectory, occurs in differential equations with at least three phase space dimensions or a mapping on a manifold of dimension at least two. In this paper we examine a system which directly gives a mapping of a half cylinder, R ~ X S t, onto itself. The map, f, is framed in terms of the following pair of difference equations: v.+ I = ev. + (1 + e)(1 + sin #.),

#.+1 =0,, +By.+1

mod2cr.

(1)

As we describe below, these equations were originally derived to model an imperfectly elastic ball bouncing on a vibrating platform, though they are similar in form to many other common nonlinear mappings and we believe the behaviours we describe are typical. The state variables #n and v, represent a phase and a velocity, e ~ (0,1) and B > 0 are parameters which we are at liberty to vary. Each application of the map contracts areas by a factor e and B is in some senses a nonlinearity parameter. As the parameters are varied a wide variety of trajectories, including periodic, quasiperiodic and chaotic motion, are followed by iterates of f. These phenomena are displayed by many other mappings on a two-dimensional manifold and have been described in isolation. A principal objective of this paper is to examine the way in which the dynamics change from one behaviour to another as a parameter is varied. For certain parameter ranges we are able to prove facts about the dynamics; in between we make recourse to numerical studies. Lyapunov exponents are powerful indicators of a trajectory's nature since they distinguish

0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

356

R.M. Eoerson / Chaotic dynamics of a bouncing ball

regular and chaotic motion and quantify the rate of loss of information about initial conditions. Another objective of this paper is to show how Lyapunov exponents vary with parameters. In several regions we are able to make analytic estimates. For the most part we shall fix e at 0.4 and concentrate on varying B. To give some idea of the possible motions we present in fig. 1 a numerically generated phase bifurcation diagram. It was composed by plotting the phases, 0,, visited by 3500 iterates of f for each of 1000 values of B, with e fixed at 0.4 throughout. Each trajectory began at the same phase and velocity, and the first 500 iterates at each B were not plotted so as to show only persistent motion. Also plotted in fig. lb is the largest Lyapunov exponent calculated at each B; a positive exponent signals the divergence of neighbouring trajectories, whilst a negative exponent indicates a motion stable against small perturbations. Of course, fig. la is only half the story because the corresponding velocities, o, at each B cannot be shown. Figs. 2-7 show typical trajectories plotted at fixed B and e for a few representative points in fig. la. The velocity and phase are plotted as (r, 0) polar coordinates. The two major regions for which one can prove facts about the dynamics are for small B or e and for B > 5.6. As fig. 3 suggests the attractor when B is small consists of an invariant curve surrounding the origin. Both entrainment (or phase locking) to the forcing frequency and ergodic trajectories are common in this region. Quite a lot can be said about the periodic and quasiperiodic trajectories. When B is greater than 5.6 it is possible to show the existence of Smale horseshoes, which is usually taken to imply chaotic motion. This region is open to a statistical analysis. For B between about 0.83 and 3.5 the motion is very complicated as the invariant curve breaks up and a strange attractor created (fig. 4). We are able to follow its development from the form in fig. 4 through several periodic orbits to the form shown in fig. 5 and then to the spiral structures at higher B (figs. 6 and 7).

The organisation of the remainder of the paper is as follows. In section 2 we give the physical basis for the mapping and collect some elementary facts about it. Periodic orbits, which are crucial in determining the map's behaviour, bifurcations from them and the corresponding Lyapunov exponents are examined in section 3. Section 4 deals with Smale horseshoes when B is greater than 5.6. We show the existence of an invariant manifold surrounding the origin in section 5, and consider the dynamics upon the manifold and the manner in which the manifold breaks up to form the strange attractor of fig. 4. The effect of noise on quasiperiodic trajectories is also considered here. The link between periodic orbits, the attractor at B = 1.55 (fig. 4) and the chaotic attractors when horseshoes are present is made in section 6. We will primarily be concerned with collisions between •invariant manifolds and the resulting changes in trajectories. We find that intermittency and chaotic transients are characteristic of qualitative changes in behaviour. This work is closely related to the work of Grebogi et al. on crises [3]. Finally, in section 7, we show how Lyapunov exponents may be calculated when B is not small (B > 3.5) and phase correlations are low.

2. The mapping In this section we give a brief description of the derivation of the mapping (1) and collect a few facts about it. We model a small ball bouncing on a massive platform whose velocity Vp(t) varies periodically with frequency to. On collision with the table the hall's velocity relative to the table is reversed and reduced to ~ times its impact velocity, where e < 1 is the coefficient of restitution between the ball and the platform. Between bounces the ball moves vertically under gravity. The ball's motion may be characterised by its downward velocity V, as it makes the n th impact at time 0 J t o , where 0~ is the phase of the platform. Successive bounces are given in terms of the

R.M. Everson/ Chaotic dynamics of a bouncing ball

357

/

0.4

0.8

1-2

1.6

210

214

2:8

I 0.4

3:2

,f

i

-O4

b 0.4

0.8

1.2

1.6

2'.0

2.4

2.8

3.2

Fig. 1. (a) Numerically generated phase bifurcation diagram for eqs. (1). The 0, visited by a typical trajectory are plotted for 1000 different B between 0 and 3.6. • - 0.4. The top and bottom edges should be identified as O, runs between 0 and 2¢r. (b) Variation of the largest Lyapunov exponent with B, corresponding to the trajectories of fig. la.

R.M. Everson / Chaotic dynamics of a bouncingball

358

"'•..,.

\

< .=

,..•"•" f

........................~'l..,•....•"

\

) \

/ //

Fig. 2.

Fig. 3.

Figs• 2-7• Paths of typical trajectories at e = 0.4 and different B. In each figure the velocity and phase (v,, ~,) are plotted as (r, t~) polar coordinates. The centre of each figure corresponds to v, ffi 0 and the side of each figure has length 10. [] mark simple fixed points. Fig. 2. B = 0.3. The insert, in rectangular coordinates 4.5 _< O < 5.5 and 0 _< v _< 0.6, shows behaviour close to the fixed point (0, 3~r/2). Fig. 3. B ffi 0.5. Fig. 4. B = 1.55. Fig. 5. B ffi 3.0. Fig. 6. B = 5.6. Fig. 7. B = 10.0.

II

ql

II

!

[]

Pl

%\ \

g

i."-),, #

I

o

l,t\! '~

.;'/:i

I/".:; ../t~j>

\ ~'\ • "x--,.

....::'-~"}k

,/~

F'~

25£~ :.~°"

Fig. 4.

,,:

: : .::%.

Fig. 5.

R.M. Everson / Chaotic dynamics o f a bouncing ball

359

3

...-.

O

• ..:.~;,:;:;"i!'r'1' :: .?'.3:.

;~: :~i ,! •. 2..

,i?[

;::

" . ,

i,

'

":;'?2~

=%

: .2;i ........

F i g . 6.

Fig.

last by the formulae

V.+~ = eV. + (I + e)Vp(##~), (2)

0.+1

=

2to {"ev"+ (1 + e ) + --i-

( o . / ,o ) ) rood 2 ~'.

Here g is the gravitational acceleration and we identify bounces separated by an integral number of platform cycles by reducing the phase modulo 2¢r. Also, we have made the approximation that the platform's position at each bounce is unchanged. This approximation simplifies the form of the difference equations and yields great computational savings. However, if a V~ is sufficiently small and Vp(O,/,0) negative the approximation leads to negative times between bounces and the ball being projected downwards on leaving the platform. Holmes, who models a similar problem [4], is content to allow trajectories with negative velocities, but we insist that all velocities are positive by choosing Vp to be non-negative. One thus thinks of a ball bouncing on a platform which

7.

delivers an impul~ to the ball, the magnitude of which varies periodically with time. We choose Vp(t) =A(1 + sin~0t) and write (2) in terms of v~, the bali's velocity normalised to the platform velocity, A. This yields the mapping (1) with B = 2oJA/g. The coefficient of restitution e and B are two dimensionless parameters controlling the problem. B measures the violence of the platform's oscillations, ~A has the dimensions of acceleration and so we call B the reduced acceleration. Eqs. (3) describe a smooth invertible mapping ] of a half cylinder into itself, f: R~ × S 1 -~ R~- x S 1 and the trajectory followed by a ball that initially hits the platform at phase 0 o with velocity vo is completely described by successive compositions of f. We define x n = (v,, # ~ ) = f t " ) ( x 0 ) = f o ft"-t)(Xo) and f(°)(x) = x. The determinant of the Jacobian of the mapping is just the coefficient of restitution: e

det ( D r ) = det Be

(1 + e)cos#. 1 +B(1 +e)cos#.

] =e. (3)

R.M. Everson / Chaotic dynamics of a bouncing ball

360

Thus for a perfectly elastic ball the map preserves area, whilst if imperfectly elastic some dissipation is introduced and the map shrinks areas by a factor e, implying that all motion, after transients have decayed, must lie on some sort of attractor. Energy is removed from the ball solely through imperfect elasticity; the vibrating platform only supplies energy to the ball. Indeed, in the area preserving case, when no energy is removed, it is easy to show that all orbits (except for a set of measure zero) are unbounded. When dissipation is present all long term behaviour is confined to a compact trapping region, D:

D= ((V,O)IV<--Vm~,=2(1 + e ) / ( 1 - - e)}. If the bali's initial velocity is greater than Vmax its velocity on succeeding bounces decreases monotonically until it enters D. Once in D, a trajectory never leaves. Even confined to this bounded region the range of qualitatively different trajectories available to the ball is vast. Since the map contracts areas there is always an attracting set

A =

N ft"(D), n>_O

which is eventually approached by all iterates. The exact nature of A may, however, be extremely complex.

3. P e r i o d i c o r b i t s

We call a trajectory in which the n + 1st collision velocity and phase are the same as the n th, but m platform oscillations later, a simple periodic orbit of order m. Such orbits are fixed points of f(x). They are to be contrasted with more complicated periodic orbits in which f~P)(x)= x, but ftk}(x) 4: x for all k < p . These complicated orbits

are not open to easy analysis, but fortunately are of much smaller importance than the simple orbits, except when e or B is small. This regime is examined in section 5. Simple periodic orbits are given by x* = (v*, ~*),

v* = ---if,2~rrn

0 " = arcsin I~--ff~2~rrn f -

},

where -/= (1 + e)/(1 - e). The two branches of arcsin yield two fixed points if B > ~rm/T; one, qm, with ~r/2 < t~* < 3qr/2 and a second Pro, lying in the first or fourth quadrant. A linear stability analysis shows that Pm is always a saddle point, but qm is a sink in the range ~rm

--

T

< B <

rrm T

T

+ --.

~rm

(4)

While eq. (4) shows that the periodic orbits exist over quite large ranges of parameter space, they often coexist with other attractors. The rn = 1 and m = 2 orbits both exist in the parameter range spanned in fig. 1, but trajectories only find the m = 1 orbit when 1.56 < B < 2.09 (cf 1.346 < B < 2.09 given by (4)) and the m - - 2 orbit is not found. In both cases careful choice of the initial point will locate the orbit for 'the whole range given by (4). Violation of the righthand inequality occurs as an eigenvalue of Df(x*) leaves the unit circle at - 1 in a flip bifurcation; the sink becomes a saddle point and a stable period-2 orbit is born. If F(x) is a mapping on a two-dimensional manifold with Idet DF(x)I < 1 everywhere (such as f or ftP)) then the eigenvalues of the Jacobian can only cross the unit circle on the real axis. Hence the newly created period two orbit can only lose its stability by a reverse bifurcation back to the original period-1 orbit, a further flip bifurcation or a tangent bifurcation. Experience with one-dimensional maps of the line [5] and the per-

R.M. Everson / Chaotic dynamics of a bouncing ball

fectly quadratic Htnon map [6] leads one to expect the period doubling sequence to be continued indefinitely as B is increased. Indeed, this is often the case for many fixed points of this map. However, the sequence is frequently interrupted or modified. At B ~- 2.09 the m = 1 periodic orbit loses stability and a period-2 orbit is born, which in turn loses stability as a period-4 orbit is created at B -- 2.82. However, the sequence is not continued; there is a tangent bifurcation at B ~ 2.9107 as an eigenvalue of Df ~4) leaves the unit circle at + 1. Motion becomes noisily periodic with intermittent bursts during which the trajectory moves on a (strange) attractor. Transient trajectories followed onto the stable periodic orbits (at B < 2.9107) suggest that this attractor is the one depicted in fig. 4, but the periodic orbits dominate until B = 2.9107. We discuss its structure and evolution in section 6. The period-four orbit rapidly loses its influence as B is increased above 2.9107; "laminar" sequences shorten until becoming indistinguishable. Again, in section 6, we discuss the mechanisms at work here. It is apparent from the plots of Lyapunov exponent versus reduced acceleration that a negative exponent signals a periodic orbit, whilst a positive exponent indicates a chaotic orbit and the divergence of initially close trajectories. In the periodic region bifurcations in period coincide with the exponent being just zero, as one orbit loses stability and another gains it. A particularly striking feature is the baselevel (-- -0.458) to which the exponent falls for large parameter intervals, regardless of B and independent of period (e.g.periods 1, 2 and 4 for the m = 1 orbit and its subharmonics in fig. lb). Equally conspicuous are the smooth humps in Lyapunov exponent which do not appear to correspond to qualitative changes in the orbits of fig. la. Fig. 8 shows the trajectories and Lyapunov exponent for the m - - 1 orbit and its subharmonics, but at e = 0.385. The tangent bifurcation has disappeared to be replaced by the expected sequence of period doubling bifurcations culminating in

361

chaos. A new period four orbit has appeared, coincident with the swelling and splitting of the hump in Lyapunov exponent at B -- 2.55. At a still lower coefficient of restitution (0.365) the new orbit splits repeatedly to leave four localised chaotic bundles as shown in fig. 9. The period doubling sequence does not extend to arbitrarily high period at intermediate e. Rather the sequence is truncated, at some finite period and negative Lyapunov exponent, before the series of reverse bifurcations back to period four. Priifer [7] has observed similar chaotic bundles in discrete twodimensional maps while Knobloch and Weiss [8] have noted similar features in 5-dimensional flows. Denoting by Bn the reduced acceleration for which the period 2 n orbit bifurcates to period 2 "+1, we have confirmed numerically that as n becomes large the ratio (B~-B~_I)/(B~+ 1 - B , ) approaches Feigenbaum'suniversal ratio ~ for maps of an interval [9]. Note also that the chaotic bundles contain windows of periodicity analogous to those observed for one-dimensional maps. Before showing how the constant baselevel and smooth humps in Lyapunov exponent arise we need to collect a few facts about Lyapunov exponents to be used here and in later sections. For a trajectory {xi}, x~=f(O(x)on a m-dimensional manifold M and a vector I ~ Tx, the tangent space to M at x, the following limit may be defined: l i n a n1

lnlDf¢=)(/) 1= o(x,/).

(5)

The tangent vector ! may be thought to join ( x i } to an initially close trajectory ( y i }, y - x = / , so that o(x, l) measures on average logarithmic rate of divergence of trajectories. Osledec [10] and Benettin et al. [11] have shown that the limit (5) exists and is finite under very general conditions. The number o(x, i) is called the Lyapunov exponent of the vector ! and there are m, not necessarily distinct exponents. They have also shown that, for almost all vectors l, o(x, i)---, ol(x ), the largest of the m exponents. It is generally assumed (and Grebogi et al. [3] provide good numerical

R.M. Everson / Chaotic dynamics of a bouncing ball

362

a

d

| 1.7

1.9

2.1

2.3

2.5

2.7

2.9

b 0.t.

1/

t~2

0.0

"

,

'

~

'

I

-0.2

-0.t.

J

1.7

1.9

2.1

2.3

2.5

2.7

2.9

Fig. 8. (a) Phases visited by m = 1 orbit and its subharmomcs as B varies at r = 0.385 and (b) corresponding Lyapunov exponents

R.M. Everson/ Chaotic @namics of a bouncing ball

363

a

...J"~'JJ i : ~ . ~ 1 1 m U ~ ~:~

1~7

1.9

2'.1

2.3

2.5

217

2.9

b

0.6 0./.

0.2 0.0

-0.2

J 1.7

1.9

2.1

2.3

2.5

2.7

2.9

Fig. 9. (a) Phases visited by m = 1 orbit and its subharmonics as B varies at ~ = 0.365, showing bifurcation to chaotic bubbles and back to a period-2 orbit and (b) corresponding Lyapunov exponents.

364

R.M. Everson / Chaotic dynamics of a bouncing ball

evidence) that all trajectories that eventually move on the same attractor have the same spectrum of exponents, though distinct attractors generally have different exponents. Note that for a map f of a two-dimensional manifold with det D f = const, the second, smaller exponent is easily found from the first since o , ( x ) = lnldet Dfl.

(6)

i=1

Numerical calculations of the exponents seldom use (5) directly; instead, by noting that D f (") = D f ("- 1)Dr, the limit may be written o ( x , !) = lim 1 In

n

IDf(")(l)l

(7)

IOf(.-1)(t)l"

The exponential growth of the tangent vector may be followed in this formulation, renormalising ! at each iteration to reduce numerical error. A slightly different aspect is illuminated by writing A = [Df(n)(x)]X/n,

the Lyapunov exponents are given in terms of the eigenvalues h1,E(n) of A.: o i = lim lnl~i(n)[. A, is some sort of average Jacobian along the trajectory. Now if the trajectory is periodic with period rn, A, may be written as [( m

~n/m']l/n

m

l/m

where the J,. = Df(x~), i = 1 . . . . . m are the Jacobian matrices evaluated at the m points on the orbit. If Idet (Dr(x)) I = E is constant, then Idet A~'~)[ = e". The eigenvalues v~ of A(,'~) lie either on the real axis (in the intervals ( - 1 , - e '~) or (era, l) when the orbit is stable) or they form a conjugate pair: Pi ~ ~m/2e-t: i~,

for some q0 depending upon the exact form of f.

Thus when the eigenvalues A~'~) are not purely real the Lyapunov exponents are equal and given by oi=lnlel/2e+-i~/ml=½1n[e[,

i = 1,2.

(8)

Hence the regions of constant Lyapunov exponent correspond to D f ( ' ) ( x ) having imaginary eigenvalues, the exponent rising from the baselevel when both eigenvalues are real. Numerically calculated baselevels agree well with eq. (8) and numerical calculation of the Pi confirms that the humps in Lyapunov exponent begin as they coalesce at + e m/2 and move along the real axis. The above is not restricted to the particular map (1), but is applicable to any map on a two-dimensional manifold with I d e t ( D f ( x ) ) l = e < 1. If an eigenvalue reaches the unit circle a bifurcation occurs, a new orbit is born and the observable motion moves to the new orbit. However, f may be sufficiently complicated that the eigenvalues reach a maximum separation, corresponding to the peak in the hump, before moving back towards gin~2. For example, the peak at B ~ 2.6 in fig. lb corresponds to an eigenvalue of D f (2) moving along the real axis towards + 1 and then back. The period-4 orbit has two humps, one corresponding to negative eigenvalues and another for which the eigenvalues are positive. When e = 0.4 the excursion of one eigenvalue towards + 1 is so large as to actually reach the unit circle and there is a tangent bifurcation. Changing the coefficient of restitution to 0.385 (fig. 8) reduces the size of the excursion and an eigenvalue leaves the unit circle at - 1 in a flip bifurcation. As an orbit of period 2" is born via a flip bifurcation in a sequence of period doubling bifurcations the eigenvalues of Df(2")(x) lie at 1 and e2". By the next doubling they have moved continuously (though not necessarily "undirectionally") to - e2" and - 1. Since neither eigenvalue may pass through zero they form a conjugate pair for some interval of the bifurcation parameter. We thus infer that the ditches in the Lyapunov exponent actually reach the baselevel for some

R.M. Everson / Chaotic dynamics of a bouncing ball

parameter interval no matter how high the orbit period, though the numerical resolution is usually too low to show it.

4. Horseshoes We now turn attention to the map's behaviour at moderate coefficients of restitution (e --- 0.4) and moderate reduced accelerations, B - - 5 . A typical trajectory is shown in fig. 6. At e = 0.4, B = 5.6 there are no stable simple periodic sinks and numerical calculations suggest that any more complicated periodic orbits are also unstable. Consequently all motion follows a trajectory similar to fig. 6. The largest Lyapunov exponent calculated for this trajectory is 1.57, implying that initially close trajectories diverge and information about initial conditions is quickly lost. Although no stable periodic orbit has been located numerically, the existence of such orbits must not be discounted. We may just be observing a very long "chaotic transient" before the motion actually settles down to regular behaviour. Gambaudo and Tresser [12] and Sparrow [13] give examples of this "preturbulence", "metachaos" or chaotic transients exhibited by both discrete maps and flows. In Gambaudo and Tresser's example a two-dimensional map adopts a period-11 orbit only after 1.5 × 106 iterations! We shall examine quite short chaotic transients for this system. Nonetheless, the trajectory in fig. 6 has not settled down, even after 2 × 106 iterations, so for a long time, exceeding the duration of many physical experiments, the observed behaviour is chaotic. When the e = 0.4 and the reduced acceleration is greater than 5.6 it can be shown that the map possesses a Smale horseshoe [14]. The presence of a Smale horseshoe allows a direct hnk with symbohc dynamics to be made and assures us of a Cantor set, A, invariant under f, such that the action of f restricted to A is homeomorphic to a shift on an alphabet of two symbols. Proofs and a detailed discussion of this link may be found in Guckenheimer and Holmes [15], Chillingworth [16]

365

also gives an excellent exposition. We note some important properties of A: (i) A contains a countable set of periodic orbits with arbitrarily long periods; (ii) A contains an uncountable set of aperiodic motions; (iii) A contains a dense orbit. A, however, is not an attractor, rather it exerts its influence on the motion by virtue of its stable and unstable manifolds, which have the local structure of the product of an interval and a Cantor set. It is an approximation to the unstable manifold that is generated by the trajectory in fig. 6 and is often known as a strange attractor. Many leaves of the manifold are already apparent and further magnifications reveal a self similar structure on all scales (cf. H~non [6]). A behaves like an uncountable collection of saddle points which, like a sieve, split the neighbouring orbits as they pass. It is this divergence of neighbouring trajectories that remain close to the unstable manifold that is characteristic of chaotic motions and is measured by Lyapunov exponents. To demonstrate the existence of a horseshoe it is easiest to work with the map in a slightly different form. If the velocity of the ball as it leaves the platform at the nth bounce is B u n, then an equivalent formulation of (1) is ~.+1 = O. + u.

mod2~r,

(9)

u.+ 1 = e u . + B(1 + e)(1 + sin (#~ + un) ). The form of (9) resembles a map also describing a bouncing ball (but with negative velocities) employed by Holmes [4] and a proof of a horseshoe's existence uses constructions based on his. Rather than detail a proof here we merely state that when e = 0.4 and B _> 5.6 the map has an invariant set A and consequently possesses chaotic trajectories. However, numerical calculations show that there is chaos (a positive Lyapunov exponent) at much lower accelerations. The invariant set located in the proof is associated with the m = 2 simple periodic points in that it is located close to them and they form two of the countable set of periodic

366

R.M. Everson / Chaotic dynamics of a bouncing ball

orbits mentioned above as property (i). There is chaos associated with the m = 2 orbits at reduced accelerations as low as 3.21, however, a horseshoe in f cannot be found because the map stretches domains too slowly. This chaos arises from a homoclinic intersection of the stable and unstable manifolds of the orbits which implies there is a horseshoe in f(") for some n. A proof of a horseshoe in f thus assures us of the existence of chaotic trajectories at sufficiently high B, but it does not predict when chaos first appears, nor does it yield any information about the shape of the attractor. We give partial answers to these problems in section 6 as we consider the evolution of the attractor.

5. Quasiperiodic trajectories When the reduced acceleration is small, corresponding to slow, weak platform oscillations, the phase advance between bounces is much less than 2~- and the ball receives many impulses from the platform during each platform cycle. The trajectories are dominated by the fixed point (v*, 19")= (0, 3~r/2). This is the m = 0 member of the series of simple fixed points, but it has anomalous character because D f ( x * ) has one eigenvalue lying on the unit circle for all e and B, the other is within the unit disc at e. A typical trajectory is shown in fig. 2. Approaching the fixed point a ball receives smaller and smaller kicks, whereas after ~9 exceeds 3*r/2 larger and larger kicks are imparted as it recedes. The great majority of the trajectory is spent close to the fixed point with brief excursions at other phases during which the ball makes a few relatively large bounces before returning to the vicinity of the fixed point. Though the ball is frequently in the neighbourhood of (0,3~r/2) this type of trajectory never approaches the fixed point arbitrarily closely, but repeatedly cycles throughout all phases with non-zero velocity. The insert to fig. 2, in rectangular coordinates, shows the trajectory's path in the neighbourhood of the fixed point.

In contrast, trajectories on the centre manifold of the fixed point may approach the fixed point arbitrarily closely. Writing % = ~9, - 3rr/2 and approximating for small % allows a power series expansion for the centre manifold in the neighbourhood of the fixed point to be made: v = h(q0) = ~cp 2

By2

cp3 + ¢ ( ¢ ) '

where ~/= (1 + e ) / ( 1 - e). Thus to tP(q02) motion on the centre manifold is described by Bey % +1 = % + T

2 %"

A ball perturbed to q0> 0 thus accelerates away from the fixed point as it receives increasingly large kicks from the platform, but small perturbations to q0 < 0 return, asymptotically, to the fixed point. On the other hand, if the ball is perturbed by more than -2~BET, it will overshoot the fixed point on the first bounce and accelerate away. Thus, on the centre manifold the basin of attraction of the fixed point grows like 1~Be. At small B or e all trajectories are captured by the fixed point, as can be seen from fig. la, where at B below about 0.25 trajectories have failed to pass ~-3*r/2. At larger B or e the attracting region shrinks rapidly and most trajectories miss it. When e and B are large enough for most trajectories to avoid the centre manifold, they appear to lie upon a one-dimensional manifold, invariant under f and topologically equivalent to a circle. Magnifications of the trajectory do not reveal the multi-leaved structures associated with strange attractors, rather iterates appear to fill out a onedimensional curve. If such a manifold exists, the trajectory is driven by f restricted to it which is the map of a circle. A good deal is known about circle maps and we will be able to infer much about the behaviour of f. When B = 0 there is an invariant manifold given by the cardioid U(v~) = y(1 + sin v~).

367

R.M. Everson/ Chaotic dynamics of a bouncingball

B = 0 is a special case as the curve includes the fixed point and the invariant circle coincides with the centre manifold. For sufficiently small, nonzero e or B there exists an invariant, attracting manifold; the result is summarised in the following theorem:

with u ~ U, a new manifold f ( M ) is obtained by acting on M with f. Under the hypotheses of the theorem we show that f ( M ) again has the form { r = ~(t~)} for some ~ U. Thus a nonlinea! mapping F: U ~ U is defined by

Fu= ~. Theorem. Consider the map r-,,r f:

+ (1 +

Ber+ B(l +e)g(#)

mod2~r,

where g ( # ) is 2¢r periodic, g(0) = 0 and g(O) > 0 for all ~. Under either of the following conditions f possess an attracting, invariant curve encircling the origin. (i) e ~ [0,1) and for all sufficiently small positive B. (ii) B < C and for all sufficiently small positive e. C is a positive number, usually less than 1, but not arbitrarily small. Furthermore, under these conditions f restricted to the invariant curve is an orientation preserving diffeomorphism of St. The map in the theorem is just (1) with a change of angle coordinates to bring the fixed point to # = 0, in which case g(#) = 1 - cos ~. The proof of the theorem is based on Ruelle and Takens proof of the Hopf bifurcation for diffeomorphisms [17]. The strategy of the proof will be described here; details are given in the appendix. We will seek an invariant manifold of the form { r = u(0)}, where u has the following properties: (i) u(~) = u(O + 2~r); (ii) 0 _~
Iu ( o t ) - u(O:)I_< vc210t - 021. Denoting the space of functions satisfying (i), (ii) and (iii) by U, we seek a contraction mapping on U. Starting with a manifold

M= {r-- u(0)),

We then show that, with respect to the supremum norm, F is a contraction on U and hence has a unique fixed point u*, which is the sought invariant manifold. The mapping F is constructed by first showing there is a unique ~ such that the 0-component ot f(u(O), ~) is 0; that is, such that the equation

=

= b+ a

u(b) + B(1 +

(10)

has a unique solution for # given #. Then set Fu(~) equal to the r-component of f ( u ( ~ ) , ~); namely

Fu(#) = eu(~) + (1 + e)g(~). The remainder of the proof consists of showing Fu(a?) is also in U and then that F is a contraction on U. That the manifold is attracting follows naturally from the contractivity of F. We remark here that when e is strictly positive we can make the space of functions in which we work smaller by requiring that 0 < u(#) < "/C1 in place of condition (ii). This shows that the invariant manifold does not touch the fixed point when e is positive (cf. insert of fig. 2). Having established the existence of an invariant curve at small B or e we can proceed to discuss the dynamics on it. The theorem gives no precise information on the point at which the manifold loses its smoothness or conjugacy to a circle. The breakup of the manifold is discussed below, but when the numerics suggest it we will assume the manifold is topologically equivalent to a circle. Since the 0-component of the full map is just the s ( # ) given in (10), f restricted to the invariant curve is s(t~): $ 1 ~ Sk In the course of showing that (10) has a unique solution we find that

R.M. Eoerson / Chaotic dynamics of a bouncing ball

368

s(#) is Lipschitz continuous, strictly increasing in O and that s(# + 2~r) = s(#) + 2~r, (see appendix). Hence, under the hypotheses of the theorem, f restricted to the invariant manifold is an orientation preserving map of the circle. We define the rotation number O(s) by

O(s) = lim

1

o),

n ---* o O

where s denotes the lift of s from St to R. Numerous authors [18] show that for orientation preserving homeomorphisms O(s) exists and is independent of ~. The rotation number distinguishes periodic and ergodic orbits; O(s) is rational if and only if s(O) has a periodic orbit. In this case one says that the system is in resonance and the bali's bounces are phase locked or entrained to the platform oscillations. More precisely, if O(s)=p/q (where p and q are relatively prime) then s has a periodic orbit of period q and during one cycle of the orbit the platform oscillates p times. A simple periodic orbit of order m would have a rotation number m/1. Numerical evidence suggests that (except for simple periodic orbits) orbits with rotation numbers greater than 1 are rare, or at least have small basins of attraction. Before the m = 1 orbit appears the platform is not supplying sufficient energy for rotation numbers to exceed 1 and when the invariant manifold exists there is a profusion of periodic orbits. Diffeomorphisms of a circle that depend upon a parameter have been extensively studied by Herman [19], who greatly extends Arnold's work [20]. Herman's results imply that if s(O) depends continuously on a parameter (such as B or e) and if varying the parameter changes O(s) from a t to a2, it must en route pass through all rotation numbers in between. Fig. 10 shows a magnification of fig. la between the 1/7 and 1/5 resonances. Each of these and the 1/6 resonance are very dear. The lowest order resonance between a pair of resonances, a/b and b/c is their mediant: (a + b)/(c + d). Thus the lowest order resonance between the 1/7 and 1/6

resonances is the 2/13 resonance, visible at B 0.542. Since s(~) depends continuously on B, Herman'swork implies that there is a resonance with rotation number a for every rational number between, say, 1/7 and 1/6. Herman also shows that there is an open interval in B for which each resonance occurs. However, the length of these intervals decreases rapidly as the denominator ot the resonance increases (cf. fig. 10a and the plot o! Lyapunov exponents in fig. 10b). Also many of the resonances between 1/6 and 1/5 are corrupted by period doubling; we discuss this below. The argument to establish the existence of resonant intervals is easy if Os/OB~O for all 0 and B (cf. Guckenheimer and Holmes [15]), but Herman requires a careful discussion of function spaces if not. So far we have not discussed trajectories with irrational rotation number and there are an uncountable number of irrationals between any pair of rationals. When the rotation number is irrational, say a, Denjoy's theorem [21] states that s(O) is topologically conjugate to a rotation by 2~ra. That is, there exists a homeomorphism, h (which Herman has shown to be analytic for almost all irrational a), such that

s ( O ) = h o R a o h -1

withRa: ~ + 2 ~ r a .

This implies that the orbit is ergodic and eventually comes arbitrarily close to any given point on the invariant manifold. Because the homeomorphism h is not linear the ball spends more time at some phases than others; in fig. 3 there is bunching close to the fixed point, while in fig. 4 the bunching is close to the kink in the manifold. Denjoy's theorem also implies that one Lyapunov exponent is exactly zero and the other is In e. This follows from the definition (5) and the fact that two C t conjugate maps have the same spectrum of Lyapunov exponents. Thus s(0) has the same exponent as R , and DRy= 1. The smaller exponent for the full map f follows from eq. (6). At resonance the Lyapunov exponents are, of course, both negative.

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370

R.M. Everson / Chaotic dynamics of a bouncing ball

In contrast to resonant orbits the parameter sets for which the rotation number is irrational consists only of disconnected points. Hence any orbit with p(s) irrational may be brought to a periodic orbit by an arbitrarily small, but carefully chosen, perturbation of B or e. On a computer with finite wordlength it is impossible to adjust either e or B to obtain a truly ergodic trajectory. However, it is not difficult to find trajectories for which one Lyapunov exponent has modulus less than 10 -6. The trajectory shown in fig. 3 is one. Herman has also shown that the Lebesgue measure of the set of parameter values which yield ergodic trajectories is positive. When e is small the periodic orbits seem to occupy a smaller fraction of the B parameter space, while as e approaches 1 periodic orbits seem to be more common. However, neither a detailed or quantitative study has been made. At e = 0.4 a significant portion of B parameter space is occupied by both periodic orbits with low period and by ergodic trajectories and periodic orbits with high period. In macroscopic physical systems there is always a small amount externally imposed noise, even if it is only due to thermal fluctuations. Noise may be added to this map to produce a perturbed trajectory { x i } by modifying the iteration scheme as follows:

xi+ 1 = f ( x / ) + ~i,

(11)

where ~i is a random variable symmetrically distributed about 0 with small standard deviation. In numerical simulations the ~ are generated by a pseudo-random number generator. Except when close to bifurcation points, the noise level (variance) has a much greater influence on the trajectory than the nature of the noise distribution. Quantities analogous to Lyapunov exponents may be calculated for a noisy trajectory by using (7) in which Dr(x) is calculated at each point on the noisy trajectory, but no extra noise is added in the tangent space. It is not clear how one would usefully add noise in the tangent space. These exponents measure an average rate of divergence along the perturbed trajectory.

Adding noise to an ergodic trajectory (that is one with a Lyapunov exponent very close to zero when noise free) has the surprising effect of driv. ing the Lyapunov exponent more negative. Fol example adding noise of standard deviation 0.005 to the trajectory shown in fig. 3 yields an exponenl of -0.0047 and an exponent of -0.017 is pro. duced by noise with standard deviation 0.02. Th~ time series is little changed from the noiseless cas~ and the only change in a plot like fig. 3 is a slighl broadening of the path. However, adding nois~ produces a distinct sharpening in the peaks of the power spectrum! Both the sharpening of the peak., and the lowering of Lyapunov exponent are con. trary to intuition as they appear to indicate a more ordered system, while adding noise is generall 3 thought to induce disorder. The change in Lyapunov exponents may b~ understood in terms of the averaging of parameters technique described by Crutchfield et al [22] This theory depends upon an equivalence of external noise and parametric noise. Suppose that j depends upon a vector of parameters p, whicl~ may be emphasised by writing f(x; p). One assumes that the trajectory produced by equatior (11) could instead have been produced by parametric noise. In other words, there is a sequence { vl, } of noises so that the iteration scheme

x.÷l = f ( x . ; p + n . ) produces exactly the same sequence { x n } as (11) The statistical properties of {~1.} may be quit~ different to the (~n }, indeed the sequence may no! be ergodic. Nonetheless, it is clear that the same trajectory could be produced by varying the parameters at each stage. The statistical effect ot noise is as if f were replaced by f averaged ovez nearby parameter values. Now, if the parameter`` are set so that O(s) is irrational, there are nearby intervals in parameter space that produce periodic orbits and hence negative exponents. The parameter set yielding irrational rotation numbers consists of points and at these points the Lyapuno~ exponent can only be zero, not positive. Therefor~

R.M. Everson/ Chaotic dynamics of a bouncing ball

as the Lyapunov exponent is averaged over nearby parameter values it can only be decreased. The amount by which it decreases depends upon the fraction of parameter space yielding periodic orbits and the statistics of (~ln}. Rounding due to finite wordlength may be viewed as noise superimposed on an exact computer. Thus even when noise is not deliberately added some averaging of nearby parameter values occurs. This may account for the observation that Lyapunov exponents calculated for trajectories which are good computer approximations to ergodic trajectories being very small but negative. In contrast, the sharpening of the power spectrum peaks is explained by examining the direct effect of fluctuations rather than the statistical effect. When noise is added to a system close to bifurcation the result is intermittency. Typically a trajectory consists of alternate bursts of the two types of behaviour found on each side of the bifurcation point. Each burst is noisy, but they are distinct and recognisable; transitions between the two being fairly infrequent. The long term statistical properties are still an average over both sides of the bifurcation point, but the actual trajectory separates into distinct portions. The circle map with irrational rotation number is just on the point of bifurcation to neighbouring periodic orbits. We suggest that the trajectory adopted is a series of the neighbouring periodic orbits, each one slightly noisy. Fluctuations will knock the trajectory into neighbouring periodic orbits which are stable for relatively wide parameter intervals and so are maintained for a fairly long time. These bursts of periodicity show up in the Fourier transform as fairly pure periodic sequences. Increasing the noise past some optimum level means that the trajectory is knocked more frequently into different periodic orbits and so is not in any single one long enough to appear periodic, consequently the power spectrum peaks begin to broaden again. Until now we have assumed the invariant manifold to be topologically conjugate to a circle whenever we needed. However, the manifold loses and regains its integrity many times as B is in-

371

creased. Aronson et al. [18] give a detailed discussion of the break up of an invariant circle spawned from a Hopf bifurcation. Our numerical studies suggest that the mechanisms they describe are at work here and we follow their description of the manifold's break up. Suppose that as B is increased the rotation number becomes rational ( p ( f ) = p / q , say) and a periodic sink appears. At birth the eigenvalues associated with the sink are + 1 and eq. Further increasing B causes the eigenvaiues to move towards each other along the real axis until some minimum separation is reached, after which they diverge again. As an eigenvalue reaches the unit circle at + 1 the sink disappears. It may happen that as B is increased the eigenvalues meet at e q/2, form a conjugate pair and move around a circle of radius e q/2 in the complex plane. Aronson et al have shown that the manifold is no longer conjugate to a circle when the eigenvalues are not purely real. Hence, using the results of section 3, we can be sure that if the Lyapunov exponents reach the baselevel, ½In e, the manifold is not conjugate to a circle. The map may be sufficiently nonlinear that the eigenvalues meet again at - e q/2 and subsequently separate along the negative real axis, possibly reaching - 1 and - eq, at which point a flip bifurcation occurs and a sink of period 2q is born. Notice that the platform oscillates 2p times during each 2q cycle, so the rotation number is unchanged. By exactly the same route as described in section 2, there may be further period doubling bifurcations; chaotic bundles may be formed or the sequence may terminate at finite period to be followed by reverse bifurcations back to the original period-q sink. When the eigenvalues eventually arrive back on the positive real axis the manifold is again conjugate to a circle and s(#) is once more an orientation preserving homeomorphism on it. Chaotic bundles emanating from the 1/3 and 1 / 2 resonances can be seen in fig. 1. Period doubling within higher order resonances, both to chaos and stopping at finite period are shown in fig. 10 at B --- 0.62.

372

R.M. Everson/ Chaotic dynamics of a bouncing ball

The other principal mechanism by which ordered dynamics on a circle are lost is via a homoclinic intersection of the stable and unstable manifolds of a periodic orbit. As depicted in fig. 11, an invariant manifold in resonance actually possess two periodic orbits. If the orbit followed in a computer simulation is a sink of period q, there also exists an unstable orbit which is a saddle point of f~q). At resonance the unstable manifolds of the saddle points form the strong stable manifolds of the sinks. Each of the two branches of the saddle point's stable manifold lies entirely inside or outside the invariant circle and one thinks of iterates being drawn along the stable manifold, into the neighbourhood of the saddles, after which they are repelled along the saddle's unstable manifold to approach the sink. However, a branch of the stable manifold may become tangent to and then transversely intersect the stable manifold, forming a homoclinic orbit. As we noted in section 4, a homoclinic intersection indicates a Smale horseshoes in some f~n)

Fig. 11. An invariant manifold in resonance (p(s) = 1/4)just before a homoclinic intersection occurs. The manifold is composed of alternating saddles (&) and sinks (Q), the unstable manifolds of the saddles forming the strong stable manifolds of the sinks. A small increase in B causes the stable manifolds to intersect the unstable manifolds.

with consequent complex dynamics. If a homoclinic intersection is present Aronson et al. are able to show that, for fixed parameters and some non-trivial interval I, it is possible to locate a periodic orbit with rotation number a for every ot ~ I. Any invariant manifold certainly cannot be equivalent to a circle if orbits on it with distinct rotation numbers are present. As a parameter changes the intersecting manifolds usually separate or pass right through each other leaving the original pair of periodic orbits. This type of breakdown occurs at higher B than period doubling: an example is the band at B = 0.83. Pictures of the full trajectory are very similar to a quasiperiodic trajectory; a narrower region close to the position of the recently destroyed resonant manifold is filled. But closer examination reveals a multi-leaved structure and the attracting set has a dimension greater than one. We remark that a homoclinic intersection does not necessarily destroy periodic motion. Usually the approach to the sink will be chaotic, but long term motion lies on the periodic orbit. For example, the homoclinic intersection responsible for the chaotic dynamics at B = 0 . 8 3 occurred at B=0.7967, but was masked until the sink lost stability. The final, irrevocable breakdown is at B ~- 1.35. This transition is very complicated and we have been unable to pin down the exact mechanism; all we can attempt here is to report the phenomena observed. The 1/3 resonance undergoes period doubling bifurcations to chaos and then reverse bifurcations back to period 3 (fig. 1). However, the period 3 orbit coexists with the 1/2 resonance and in an intermediate region the orbit adopted depends upon the initial phase and velocity, as B is increased the period-2 orbit eventually predominates. When two distinct rotation numbers are present the manifold cannot be conjugate to a circle, but the manifold may recover its integrity when only the period-2 orbit is present. In the familiar manner, the period-2 orbit period doubles to chaos. The attractor lies on part of the invariant manifold which is becoming increasingly deformed

R.M. Everson/ Chaotic dynamics of a bouncingball

(fig. 12). Although the components of the attractor appear as arcs on this scale, closer examination reveals a multi-leaved structure. The action of the map here is similar to the stretching and folding encountered in one humped maps of an interval. We demonstrate this in fig. 12 by labelling the ends and middle of an "arc" and then following their progress under 4 iterations of the map. It is important to notice that all mixing is a consequence of the entire arc being folded. Exact details of further developments are very complicated and sensitive to the coefficient of restitution's exact value: we describe only the main features. A series of subductions and interior crises (see Grebogi et al. [3]) produce orbits with several different periods interspersed with chaotic regions. Eventually a period-3 orbit predominates, to undergo period doubling to chaos on an attractor consisting of 12 spatially distinct pieces. As B is increased further the separate pieces expand along the vestiges of the invariant manifold until they meet and intertwine. At this stage the attractor occupies a similar region to the one resulting from the 1 / 2 resonance (fig. 12), but the dynamics on it are far more complicated, because each "arc" is

¢

.,/]

Ji

Fig. 12. The fate of an "arc" ABC under four applicationsof the map. The arc is successivelyfolded,elongatedand squashed to bring A4 and C4 together.

373

composed o f interlocking portions of the 12 piece attractor. Viewed on a large scale there is mixing within each of the arcs. Components of the attractor expand until an interior crisis [3] results in the attracting set becoming the whole region originally occupied by the invariant manifold. The characteristic shape of the manifold can be seen underlying the attractor in fig. 4 at B = 1.55, though the manifold is certainly no longer a single invariant curve surrounding the origin! The attractor is contained in the closure of the unstable manifold of p 1, and further large changes in the dynamics are organised by the simple fixed points Pl and ql.

6. Evolution of an attractor

In the preceding sections periodic, quasiperiodic, and chaotic trajectories have been examined in isolation. There is a marked difference in the shape of the chaotic attractors when B -- 1.55 (fig. 4), immediately following the destruction of the invariant manifold, at B = 3.0 and at B = 5.6, (figs. 5 and 6), where a horseshoes in f can first be found. In this section we attempt to link these by following the evolution of the attractor from B = 1.55, showing how the shape at B = 3.23 is built up and what trajectories are to be expected as the attractor evolves. The attractor resulting from the breakup of the invariant manifold, which we will call A1, suddenly gives way to the m = 1 periodic sink at B -- 1.56. This sink has actually existed since B = ~r/~, = 1.346... and by choosing an initial phase and velocity carefully a trajectory that converges to the sink can be found. However, at B -- 1.56 the sink's basin of attraction suddenly swells to inelude the whole trapping region. This type of transition has been described and named as a boundary crisis by Grebogi et al. [3]. We review it here from a different aspect as it shows how the attractor evolves. Fig. 13 shows the unstable manifolds of the m = 1 saddle point (Pl) before the crisis. One branch of the unstable manifold spirals

374

R.M. Everson / Chaotic dynamics of a bouncing ball

Fig. 13. Stable ( m ) and unstable ( - - ) manifolds of Pl when B = 1.55. (cf. fig. 4).

around and into qx, the m = 1 sink, which has imaginary eigenvalues. The chaotic attractor, A1, lies in the closure of the other branch. The stable manifold forms the boundary between the basins of attraction of the' sink and AI: initial conditions lying on one side of the manifold spiral into the sink, whilst those on the other side move chaotically on A1. The crisis occurs as the stable and unstable manifolds of Pt collide. As the manifolds become tangent and then intersect infinitely many times a homoclinic orbit is formed. A homoclinic orbit implies that there is a horseshoe in f(") for some n and hence a hyperbolic invariant set with all the chaotic properties listed in section 3. To see how the basin of attraction has changed with the crisis examine fig. 14, which shows the

stable and unstable manifolds at B -- 1.62. Focus first on the "three banded arc" of the unstable manifold labelled AB. Its image under f is the arc A'B', the image of A'B' lies even closer to the saddle point and is labelled A"B". Notice how the finger projecting through the stable manifold, labelled C, is carried along with the arc. Further images are packed in closer to the saddle point and since A and B both lie on the stable manifold their images under f approach the saddle point arbitrarily closely. The arc of the unstable manifold, on which A1 lies, is squashed along the stable manifold and stretched in the direction of the unstable manifold. (The local expansion and contraction rates are just the eigenvalues of the Jacobian, Df(pl). ) In fact, the images of the arc extend all the way along one branch of the unsta-

R.M. Everson/ Chaotic dynamics of a bouncing ball

375

pl

B'

/; /

,I '/// / /

Fig. 14. Stable ( - - ) and unstable ( - - ) manifolds of Pl when B = 1.62, following a boundary crisis.

ble manifold to wind around the sink, but the base of each tongue, at the images of A and B, remains on the stable manifold. Some of the more prominent tongues are visible close to the "corners" in the unstable manifold's spiral. The way in which the sink gathers points is now evident. Suppose that the initial point of a trajectory lies on the arc AB or on the finger C. Under successive applications of f it is carried to the arcs A'B', A " B " and so on until it is arbitrarily close to the sink. If the initial point does not lie in the arc or one of its images, the trajectory moves chaotically on the major part of A1 until it falls into AB. From there it follows the same route as a trajectory initialised in AB. The invariant set associated with the homoclinic orbit means that motion on A1,

before AB is reached, is chaotic. Initially close points may take very different routes to AB. This view is somewhat arbitrary, because one could consider a preimage of AB instead of AB itself to be the starting point of the route to the sink. It is difficult to locate preimages of AB because generating the stable manifold is a numerically unstable procedure. Preimages become convoluted and elongated as they approach the saddle point and the segment forms part of the homoclinic tangle. If two initially close points are separated by part of the stable manifold they lie in different preimages of AB and the one lying in the earlier preimage has to take a longer route to the sink as it must first reach the later preimage. In this manner the trajectory starting in the earliel preimage returns to the neighbourhood of its ini.

376

R.M. Everson/ Chaotic dynamics of a bouncing ball

tial point but then follows a completely different route onto the sink. In spite of this, it is convenient to think of a trajectory consisting of two parts: a chaotic transient followed by motion (still on the unstable manifold of Pl) very close to the sink. With increasing reduced acceleration the unstable manifold's simple spiral about the sink grows increasingly convoluted, but remains an infinite spiral while ql has imaginary eigenvalues. The route followed onto the sink is thus increasingly complicated, but eventually all motion is periodic. When the sink loses stability the unstable manifold of the resultant saddle runs into the two points lying on the period-2 orbit. The unstable manifold of the original saddle, pl, now winds closely about the period-2 orbit so that a trajectory is transferred from the original attractor A1 to the period-2 orbit. By exactly the same mechanism motion is driven onto the period-4 orbit when the period-2 orbit loses stability. However, as we described in section 2, the period-4 orbit does not period double, but becomes a saddle point with two positive eigenvalues at B = 2.9107... Before the sink loses stability the branches of the unstable manifold of Pl spiral around the sink, intersecting its strong stable manifold. When the sink becomes a saddle it develops stable and unstable manifolds which, because like manifolds of different fixed points cannot intersect, are constrained to spiral away from the sink. The location of the manifolds is sketched in fig. 15. Now, a trajectory initialised on A1 is drawn by the usual route close to the unstable orbit. As the manifolds are tightly wound about the orbit the trajectory resides in its vicinity for some time and so is almost periodic. Eventually, however, the iterates follow the saddle's unstable manifold away from the fixed point back onto A1. There are thus "laminar" sequences as iterates move close to the period-4 orbit interrupted by chaotic transients on A1. Increases in B make the period-4 orbit more strongly repelling; consequently iterates leave more rapidly, the "laminar" sequences shorten and the trajectory visits all of A1, which has evolved to the form shown in fig. 5.

Fig. 15. Sketch of invariant manifoldsclose to a periodicorbit, q, just after it has lost stability to become a saddle point. A trajectory that was previouslydrawn onto the sink along the unstable manifold (--) of a saddle now approaches q, close to the saddle's unstable manifold and q's stable manifold (---), before being repelled along its unstable manifold(--). By now the m = 2 simple periodic orbit has appeared. In exactly the same way as the m = 1 orbits, the two attractors with distinct basins of attraction coexist until a boundary crisis between the unstable manifold of P2 and A1 occurs at B = 3.06327. Until B = 3.064155, when the sink q2 loses stability by period doubling, all trajectories consist of a chaotic transient followed by periodic motion very close to the sink. A series of period doublings to period 64 follows the first and all persistent motion is periodic. However, the period doubling does not continue smoothly to chaos; instead it is interrupted as the unstable manifold of q2 curves around to intersect the stable manifold forming a homoclinic orbit. Again there is chaotic motion, but now on a small attractor lying in the closure of the unstable manifold of q2, very close to the fixed point. This attractor, which we call A2, is entirely distinct from the original attractor A1. Fig. 16 shows the invariant manifolds of both the m = 2 fixed points at B = 3.219, after the homoclinic intersection has occurred. The original attractor, A1, lies in the closure of the unstable

377

R.M. Everson/ Chaotic dynamics of a bouncing ball

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Fig. 16. B ~ 3.219. Stable and unstable manifolds of q2, showing their homoclinic intersection, and the stable manifold of P2 (~). The unstable manifold of P2 is not shown, but winds closely around the unstable manifold of q2. manifold of P2 and the stable and unstable manifolds intersect so there is a homoclinic orbit associated with P2. Also, the stable manifold of P2 intersects A1, so by the mechanism described for the m = 1 fixed points iterates are drawn into the vicinity of q2. A typical trajectory is thus a chaotic transient on A1 followed by localised chaos on the new attractor A2. Unfortunately, we cannot be sure that A2 is really a chaotic attractor. It is certain that the invariant manifolds of q2 intersect and there is a hyperbolic invariant set on which f is chaotic, but we do not know that the period-64 orbit is de-

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stroyed. For example, the invariant manifolds of ql intersect at B ~ 2.83, while the period-2 orbit is stable. Both it and the succeeding period-4 orbit survive the intersection, and form the eventual motion. It m a y be that the period-64 orbit survives the homoclinic intersection but the proximity of the chaotic invariant set means that a chaotic transient in the neighbourhood of the orbit is extremely long. Again we draw attention to G a m b a u d o and Tresser's work on very long chaotic transients [12]. We have been unable to find a trajectory converging to a sink, even after 6 × 10 6 iterations, but its existence cannot be discounted. Nonetheless, all trajectories are chaotic in character for very long times and we shall call A2 a chaotic attractor. Another crisis is imminent. This involves a collision between the unstable manifold of q~ and the stable manifold of P2. These two manifolds are shown just before the crisis in fig. 16. Recall that the boundary crisis which led to trajectories being drawn into the vicinity of A2 involved the unstable manifold of P2 and the stable manifold of q2. F r o m this we m a y expect that the effect of the new crisis will be to draw a trajectory on A2 onto A1. Indeed, this is the observed behaviour. As illustrated in fig. 17, the trajectory consists of a succession of chaotic transients. Iterates initialised on A1 bounce around on A1 until they are drawn onto A2, but once on A2 the new intersection means that, after another chaotic transient, they are drawn back onto A1. The observed motion is similar to the intermittent behaviour except that almost periodic, "laminar" sequences are replaced

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by localised chaos on A2. The overlap between A1 and A2 grows as B is increased and the trajectory is ejected more rapidly from A2, thus spending a larger fraction of its time on A1. Eventually sequences on A2 become indistinguishable and the trajectory visits all of A1 and A2 equally• It is useful to compare fig. 5 and 18 which show the path visited by a trajectory before and after the two crises involving the manifolds of the m = 2 fixed points• The region occupied by the localised attractor A2 is visible in fig. 18 as a darker band close to the q2 saddle point. Notice how the two crises have been responsible for extending the attractor around and through the two m = 2 fixed points• A series of similar crises and localised chaotic motion draws the attractor around and

through the each new pair of fixed points until it has the spiral structure shown in fig. 6 and a horseshoe can be located in f rather than some f(n)•

7. Stochastic approximation When the reduced acceleration is sufficiently large, corresponding to rapid, violent oscillations of the platform, the ball makes large, long bounces. The velocity on arriving at the platform at each bounce depends mostly on the impulse given at the previous bounce and not on its arrival velocity at the previous bounce. Phase correlation with earlier bounces is small because the ball has a long

R.M. Everson/ Chaotic dynamics of a bouncingball

time of flight between bounces; the ball quickly "loses memory" of its history. If phase correlations are completely lost, the next bounce depends only on the present bounce and the dynamics may be described by first order Markov statistics. We have shown how the attractor is built up around the simple periodic orbits into the tightly wound spiral of fig. 7. The convoluted stable manifolds of invariant sets resulting from horseshoes and homoclinic intersections serve to separate initially close trajectories so that phase correlations decay rapidly. Numerical computation of autocorrelations confirms that phase correlations are very small, though appreciable velocity autocorrelation remains. In this region we can hope to use an approximation to an "invariant probability density to describe the motion. An implicit assumption of stationarity has been made; that is, we assume the "density" generated by collecting successive iterates of f(x) is that density towards which an ensemble of balls, starting with an arbitrary density, would evolve. The Frobenius-Perron operator, Pf, describes the evolution of an arbitrary density 0n(x)= On(O, v) under f:

379

invariant density, fo~ p(v,

O)dv=y~. 1

(13)

Recall that the Lyapunov exponents for a trajectory { x~ } and tangent vector ! are given by (7)

o(x,l)= lim

n ---} OQ

llnlDf(')(01 1 i

= n-.oolim n i = l

In IDf(n)(t)l

(14)

Writing the tangent vector l, = Df(n)(l), we assume the sequence ( x i ) to be stationary and replace the average in (14) along the sequence with an ensemble average:

o(x,i)= f ln ~ p ( x ' ) d x ' , the integration being over all phase space. Specialising to the mapping (1), we follow a route pioneered by Chirikov [23] to obtain a scheme for the growth of the tangent vector i n = (x,, y,): xn+ 1 = ex o + (1 + e) cos 0,y,,

= ezpn(x) =

Y0+1=Yn + BXn+I.

z y~f-l(x)

(12) The sought invariant density is obviously a fixed point of Pf. When the coefficient of restitution is closer to 1 than 0.4 it is possible to use Markov statistics to show that the velocity density at any particular phase is approximately Gaussian. When e = 0.4 the density is not Gaussian. An iterative scheme based on (12) may be used to approximate the invariant density to any required scale, but expressions for the density are extremely unwieldy. However, because phase correlations are rapidly lost, all phases are visited equally and the attractor has the fairly uniform appearance of fig. 7. In order to calculate Lyapunov exponents we can embody the observation that all phases are visited equally in the following assumption about the

Since the exponents measure rates of divergence, they are independent of the precise form in which the map is written and we may rescale by writing p = Bx, ~1=Y: ;o+1 = evo + B(1 + e) cos 0 j l , ,

(15)

*/o+1 = ~/. + v.+l.

Writing B(1 + e) as K, we compare (15) with the simpler tangent mapping Pn+I = evo + k*/., 1 1 o + 1 = 17o q" P o + I ,

(16)

in which the coefficients do not depend upon the original trajectory. Eisenvalues for (16) are h_+ =½(l+e+k+~(l+k)2+2e(k-1)+e 2) and

R.M. Everson / Chaotic dynamics of a bouncing ball

380

writing 1, as a projection onto the eigenvectors e _+, (! o = Uoe + + roe_ ) yields a solution to (16): !,, = Uoh"+e + + VohLe_.

In cases where Ih~l< 1 there is local stability. But when h+> 1, (h_-- e / h + < 1) the tangent vector grows rapidly with n and initially close trajectories diverge. For n large the direction of the tangent vector swings to approach that of the eigenvector e+ and its magnitude varies as 10lh~ I. Thus the Lyapunov exponent corresponding to all directions except exactly e_ may be written as

Table I Comparison of Lyapunov exponents calculated using equation 18 (%) and numerical estimates of the limit 7 using 50000 iterates ( on)

0.4 0.4 0.4 0.4 0.1 0.9 0.4 0.4 0.4 0.4

B

o.

oo

,,./o.

100.0 50.0 30.0 20.0 20.0 10.0 10.0 6.0 4.0 3.0

4.2480 3.5590 3.0536 2.6431 2.4009 2.2545 1.9472 1.4715 1.0269 0.5843

4.2485 3.5553 3.0445 2.6391 2.3979 2.5513 1.9459 1.4351 1.0296 0.7419

1.000 1.000 1.002 1.002. 1.001 1.001 1.001 1.025 0.997 0.787

o ( x ) : fo~ fo2'~lnlh+ It)( v, O) dO dr. Now we use the assumption that all phases are sampled equally (eq. (13)) and replace k with K cos 0. The eigenvalue is independent of velocity which allows direct integration over the velocity. o=~l

1 r2~r "0

lnlh+ldO.

(17)

correlation becomes significant and phases are visited only approximately equally. In (17) and (18) the dependence of Lyapunov exponents on a particular orbit { x~ } is no longer sho~a because this scheme, assuming that all phases are sampled equally, cannot account for it. Indeed the true orbit may be periodic with low period, the density being a set of delta functions, however, such orbits are rare at high B and e.

Also, if K >> 1, h+ = K cos O and Acknowledgements

!

(2,

°=2rrJo

K B(1 + e) lnlKc°s01 d0 = In ~ = l n 2 (18)

Table I compares exponents calculated using (18) and found numerically by following directly the growth of a tangent vector. Good agreement is found even to quite low B and e, where the phase

Advice and encouragement from Dr. J. Brindley, Dr. H. Morris and Dr. P. Scheuer is gratefully acknowledged. Valuable comments and suggestions by a referee are greatly appreciated. This work was supported financially by grants from the Inner London Education Authority and the Science and Engineering Research Council.

Appendix A

We supply here some more details of the proof that there exists an invariant curve surrounding the origin when either e or B is small enough. The strategy of the proof was discussed in section 5; we detail here the construction of F, a proof that Fu(O) ~ U if u(O) ~ U, and that F is a contraction on U. The following quantity will be used in several estimates:

where the supremum is taken over the range 0 < # < 2~r.

R.M. Everson / Chaotic dynamics of a bouncing ball

381

As we described in section 5, the construction of F depends on showing that

# -- s ( # ) = 8 + s e ~ ( ~ ) + s(1 + e)g(~) has a unique solution 8 given ~. If we can show that s(#) traverses an interval of exactly 2¢r as # runs from 0 to 2¢r and s(~) is strictly increasing, then s(#) has a unique inverse. Since g ( ~ ) is 2~r-periodic s(~ + 2~r)= s(~) + 2¢r. Let ~1 < ~5, then

s(#2)- s(81)=

#5 - 81 + Be( u(#5) - u(81)) + B(1 + e)( g(~2) - g(bl)).

Ig(bs) - g(b,)I_< Xlb2 - ~1 l, and, from Lipschitz continuity of u(#),

lu(85) - u(81)I- ./c51 b 5 - 61 I. Whence

S(82) -- S(~1) ~ {1

-- B X - B e . / C 2 - B e X

}(85 - 81).

So provided that 1 - B ~ - B e . / C 2 - B e ~ > 0, s(~) is strictly increasing and ~ = s - l ( # ) is a well-defined function. It is evident that we can either fix B < h-1 and choose e sufficiently small or fix e and choose B sufficiently small. From the above estimates it also follows that, under the hypotheses of the theorem, s - l ( 0 ) is Lipschitz continuous:

1S-1(02) -- S-1(1~1) ]_~<{1 -

BX - Be./C 5 - BeX

} -11~ 2 - 011.

Hence our definition of F u ( # ) makes sense, but we must check that from periodicity of g(#) and u(#): (ii)

F u ~ U.

The first condition follows

0 < Fu(#) < ./C1,

F~(~) = eu(b) + (1 + e)g(8) >_0, since g ( O ) > 0 and We require F u ( # e./C1 +

Note that if e > 0 this condition may be replaced by 0 < so

u ( O ) > O. ) < ./C1,

(1 + e)?~ < ./C1

which is satisfied for all e and B by choosing ~ = ./. (iii)

Lipschitz continuity of

Fu(O),

IF u ( 0 , ) - F u ( 0 2 ) I = le(u(82) - u(81)) + (1 + e)(g(85) - g(~x))l -< { evC2 + (1 + e ) X } l ~ 2 - ~11 e./C 2 + (1 + < 1 - Be'/C 2-B(1

e)h + e)A 1#2- bll,

Fu(O).

R.M. Everson / Chaotic dynamics of a bouncing ball

382

where we have used the Lipschitz continuity of s-1(#). So for Lipschitz continuity of F u we need e y C 2 + (1 + e)~, < yC2{1 - B e y C 2 - B(1 + e)h }.

Gathering terms in B onto the left-hand side yields By T z - - ~ + h

}


If we choose C2 > h and fix e, B may be made sufficiently small for the inequality to hold. Alternatively, expanding the inequality in powers of e, to ¢(e2), shows that if C 2 > h/(1 - hB) the inequality is satisfied for all e sufficiently small. F~= F u is thus an element of U for all e or B sufficiently small; it remains to show that F is a contraction on U. Let Ul, u 2 e U, choose a # and define #1 and ~2 to be the solutions of

o = #1 + s~Ul(#l) + s O + ~)g(#l) and

o - #2 + B~u2(#:) + a ( l + ~)g(#~). Subtracting and taking absolute values yields

1#2 - #1 1= S(1 + e)h[# 2 - #11 + B e l u , ( # l ) - u2(#2)I. Also

I~,(#1) - u2(#2)I~ I~1(#1) - u~(#l)I+ lug(#1) - us(#2)l < IlUl - u211+ YG[ #1 - #2[. Using this in the previous equation gives 1#2 - #1 ]< B(1 + e)h I#2 - #1 ]+ BeJ]ul - u2[I + y C 2 B e ] #2 - #11 or

1~2- #1 I~ B e { 1

- B e ~ C 2 - B(1 + e)h}

-lllu2 - ulll

Bllu2 - UlllNow, using the definition of Fu: I f u l ( O ) - fu2(O)l_< e[ul(#1) - u2(&2)I+ -<

(x + ~)lg(bl) -- g( 02)l

e { Ilua - u2l[ + YC2] #1 - ~2 [} + (1 + e) Xl #1 - ~2 [

_< ( ~ + eyC2B + (1 + e)hB } Ilul - u211 -< alia1 - u211.

ILM. Everson/ Chaotic dynamics of a bouncing ball

383

So F is a contraction if a < 1. Fixing B and expanding ~8 in e shows that Be

and so a = B h + ~(e). Thus if B h < 1, a may be made less than 1 for all e sufficiently small. Similarly, fixing e and expanding/8 and a in terms of B, we find

a=e+O(B). Again, we have a contraction if B is chosen small enough.

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[13] C. Sparrow, The Lorenz Equations (Springer, Berlin, 1982). [14] S. Smale, Bull. Amer. Math. SOc. 73 (1967) 747. [15] J. Guckenheimer and P.J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, Berlin, 1983). [16] D.R.J. Chillingworth, Differential Topology with a View to Applications (Pitman, London, 1976). [17] D. Ruelle and F. Takens, Comm. Math. Phys. 20 (1971) 167; 23 (1971) 343. J.E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications (Springer, Berlin, 1976). [18] D.G. Aronson, M.A. Chory, G.R. Hall and R.P. McGeehee,Comm. Math. Phys. 83 (1982) 303. [19] M.R. Herman, Publ. Math. IHES 49 (1976). [20] V.I. Arnold, AMS Transl. Set. 2, 46 (1965) 213. [21] A. Denjoy, J. Math. 17 (1932) 333. [22] J.P. Crutchfield, J.D. Farmer and B.A. Huberman, Phys. Rep. 92 (1982) 45. [23] B.V. Chirikov, Phys. Pep. 52 (1979) 265.