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Experiments on periodic and chaotic motions of a parametrically forced pendulum
Physica 16D (1985) 371-384 North-Holland, Amsterdam
EXPERIMENTS ON PERIODIC AND CHAOTIC MOTIONS OF A PARAMETRICALLY FORCED PENDULUM R.W. LEVEN, B. POMPE, C. WILKE and B.P. KOCH Sektion Physik / Elektronik, E. -M. -A rndt- Universiti~t, DDR, 2200 Greifswald, GDR Received 3 July 1984 Revised manuscript received 27 November 1984
An experimental study of periodic and chaotic type aperiodic motions of a parametrically harmonically excited pendulum is presented. It is shown that a characteristic rou~e to chaos is the period-doubling cascade, which for the parametrically excited pendulum occurs with increasing driving amplitude and decreasing damping force, respectively. The coexistence of different periodic solutions as well as periodic and chaotic solutions is demonstrated and various transitions between them are studied. The pendulum is found to exhibit a transient chaotic behaviour in a wide range of driving force amplitudes. The transition from metastable chaos to sustained chaotic behaviour is investigated.
1. Introduction If a nonlinear dissipative system is driven by an external force it leaves the equilibrium state and develops new orderings. The system either oscillates or even shows a more complicated temporal evolution, characterized by the notion of chaos. The chaos is connected with the existence of a chaotic or "strange" attractor: a surface in the phase space of the system to which orbits are attracted and on which they move pseudorandomly. An interesting question is how chaotic attractors arise as some parameter of the system is varied, and how and to what degree they are dependent on the driving parameter. Several routes to chaos have now been well documented. They include infinite period-doubling cascades [1], intermittency [2] and sharp transitions from a periodic or quasiperiodic regime to a chaotic one. Sudden qualitative changes, called crises, have been found numerically also in the chaotic regime [3, 4]. They have been investigated theoretically by Grebogi, Ott, and Yorke [5] and were observed experimen-
tally for a parametrically forced pendulum [6] and a driven nonlinear semiconductor oscillator [7]. If a system is characterized by only one parameter, it is relatively easy to study the routes to chaos as the parameter is varied. For real physical systems, however, the number of parameters is large. Even such apparently simple and physically transparent systems as externally driven nonlinear oscillators have, apart from the driving amplitude and the driving frequency, the natural frequency and the damping force as important parameters. Nevertheless, in many cases a "typical picture" can be obtained if all parameters are kept fixed and only the amplitude of the driving force is varied. This, for example, was done in the driven nonlinear semiconductor oscillator experiments of Testa, Perez and Jeffries [8], where the behaviour of the system was compared with the simple logistic model described by the one-parameter equation x , + x = ) ~ x , ( 1 - x , ) . A correspondence between the driving voltage V0 and the parameter h was assumed and a strong similarity between the predicted and the observed bifurcation diagram was found.
0167-2789/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
372
R. W. Leoen et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
In this connection, we would like to mention that, in general, for two- or more parameter systems the character of the bifurcation diagram, which is obtained if one continuously varies the "control parameter", depends upon the other parameters. So, for example, for the two-parameter H r n o n mapping the bifurcation diagram changes if the second parameter b is varied. This becomes obvious from fig. 3 of the paper of Arneodo et al. [9], where the threshold values of the parameter a for the onset of various subharmonics are traced as functions of b. There are m a n y intersections between the traces of different periodic attractors. This indicates that for b ~ 0 the sequence of periodic or chaotic solutions which one obtains with increasing a is not consistent with the universal U-sequence of Metropolis, Stein, and Stein [10]. Especially, the coexistence of two or more periodic (or chaotic) solutions with distinct basins of attraction is possible. As pointed out in [9], it is possible in such cases to observe rather exotic routes to chaos if one does not take much care of the adiabaticity in turning on the constraint. Besides numerical computations also many experimental investigations of the periodic and chaotic behaviour of externally driven nonlinear oscillators have been carried out, but so far only few experiments with mechanical systems have been made [11-13]. In a previous paper [6] we reported on experimental evidence for chaotic type nonperiodic motions of a parametrically forced mechanical pendulum. In this paper we present detailed measurements with an improved version of the equipment used in ref. [6] and {ry to answer the question of the typical behaviour of nonlinear oscillators driven by an external force. Our new system has the advantage of a higher resolution in the phase plane, and a well-defined damping force which, most importantly, could be varied continuously within an experimental run. This enabled us to vary adiabatically one of the parameters (the d a m p i n g coefficient) without interrupting the motion of the pendulum. Our experiments showed that the measured values vary adiabatically with a
sufficiently slow variation of the damping parameter. This adiabatic property is crucial for exactly following the various transitions from one type of motion to the other. As was mentioned already in [6], the Coulomb friction, which is the dominant damping force for sufficiently small values of k, makes the origin stable and attracting up to rather high values of the driving force. This gives the possibility to study the irregular motion of the pendulum in the transient regime over a large range of parameter values and to describe the transition from transient chaos to sustained chaotic behaviour, which was first studied in detail for the Lorenz model by Yorke and Yorke [14].
2. Experimental equipment
in
A sketch of the experimental apparatus is shown fig. 1. The parametrically excited damped
T 7
Fig. 1. Scheme of the mechanical part of the experimental set-up: 1-pendulum body; 2-pendulum rod; 3-disk with angular code; 4 - light barriers; 5 - eddy current brake; 6 - slide; 7 - frame; 8 - electromotor; 9 - transmission; 10- lever; 11 - balance weight.
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
pendulum consists of a body (mass = 200 g) which is fastened to a pendulum rod (length = 38.5 cm). The natural angular frequency too is adjustable from 5.08 to 6.73 s-1. A periodic vertical displacement of the pendulum occurs with an angular frequency to = 10.37 s-t. The adjustable amplitude a of this excitement ranges from 5.5 to 20.0 cm. A disk is rigidly connected to the pendulum. On the disk there is an angular code (GRAY-code) in the form of sectors which are pervious or impervious to light. Ten light barriers are radially arranged at some distance from the disk. The signals of these barriers are electrically analysed. They provide the electronic voltages Ux and Uy which are proportional to the angle x of the pendulum swing and to the angular velocity y = d x / d z (T = time) in the limits of measuring precision ( A x ~ +_0.013 rad, A y = ___0.63 tad/s). If the pendulum turns over at x = _ ~r rad, a transformation x -~ x ~- 2~r tad is carried out. Consequently, x varies from -~r to +~r rad. A stroboscopic phase representation of x and y by means of an oscillograph or rather a printer can be carried out in any position p of the point of suspension of the pendulum which is passed during its vertical displacement. In the following p = 0, 1 / 2 , or 1 / 4 , 3 / 4 means the lower, the upper dead point, or half the way from the lower to the upper and the upper to the lower dead point, respectively. For the value of to chosen in our experiments about 100 dots appear within one minute on the fluorescent screen of the oscillograph, which are photographed. The damping of the pendulum is first of all effected by means of an eddy current brake. By varying the current I giving rise to the magnetic field of the eddy current brake, the damping of the pendulum can be altered. The damped parametrically driven physical pendulum can be described by the equation j ( d E x / d ' r 2 ) + B o sgndx/d~" + Bx( d x / d , r )
+ m l ( g + ato2 cos to~") sinx = 0,
373
constants; m = mass; 1 = distance between centre of mass and rotation axis; g--- acceleration due to gravity. The damping term of eq. (1) Md=
B 0 sgn d x / d ~ + B 1 d x / d ' r
has its origin in experimental investigations. Fig. 2 shows the function Md(Y ) as established in experiments. Making use of the abbreviations
t = too~,
too = ( m l g / J ) t/2,
flo = B o / ( J°J~ ), ±=dx/dt,
[2 = to/too,
fix = B l l ( Jtoo),
£=d2x/dt
A = ato21g,
2,
M d [Nn~ I .221 .21
500 m A
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200 mA
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.15 100 mA
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50 mA
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.05
/
•
~ J
j
.03 .02 .01
y [2~'rad/s]
(1)
where J = moment of inertia; B0, B 1 -- damping
Fig. 2. Damping turning moment M d versus angular velocity y (experimental values). (Parameter: current I in the eddy current brake.)
374
R. IV. Leven et al./ Experiments on periodic and chaotic motions of a parametrically forced pendulum
eq. (1) can be written as
Yc+ floSgnJc + /31Jc+ ( l + Acosg2t)sinx=O.
(2)
If (x, y ) is a solution of eq. (2), then ( - x , - y ) is a solution too, because eq. (2) is invariant under the symmetry transformation S: (x, y ) (-x, -y). For fl0 = 0 this equation reduces to that studied numerically in [4] and [15]. As shown in [6], the main effect of the /3o sgn 2 part of the damping force is to hold the origin of the phase plane stable even for higher values of the external force. (Only if the eddy current brake was removed, the origin was found to be unstable in a certain parameter range.) As a consequence, almost every motion with the phase point visiting a small area around the origin has to be considered as a transient one. However, this is without importance for stable periodic motions or in a regime with a "small" bounded chaotic attractor lying far enough from the origin in the stroboscopic phase representation. It is true that our experimental investigations cannot be regarded as proof of the chaotic nature of the pendulum motion. Nevertheless, we call certain motions chaotic because in numerical examinations of an equivalent system [4] the existence of chaotic behaviour was made plausible by the computation of Lyapunov exponents and power spectra for some parameter sets. The observed period-doubling bifurcations between periodic and apparently aperiodic motions and the parabolic shape of a return map obtained for an aperiodic rotational motion are further hints at the chaotic nature of the aperiodic regime.
was kept as small as possible. For the largest 1"~= 2.04 we obtained fl0 = 0.033. With decreasing this value nearly linearly increases up to fl0 ~ 0.057 for 12 = 1.54. The damping coefficient fll could be varied from 0.008 to 0.106 for 12 = 2.04 and from 0.026 to 0.286 for 12 = 1.54. Our main intention has been to describe properly the various transitions between periodic motions with different periods, periodic and chaotic motions, and between chaotic motions of different character as one or more parameters are varied. In order to reduce the number of parameters which have to be varied we proceed from the assumption that a typical direction to chaos is that of increasing A or decreasing fl = fl0 + fix (the other parameters fixed). As shown in [16], the onset of the different subharmonics is characterized by a certain value of the relation A / f l for a given frequency I"L Fig. 3 shows the threshold values A~1) and A~21) for the first two period doublings of the rotational motion with period 1 (T = 2~r/~0) and the value A~a) at which the motion becomes chaotic (but
~-~=L57
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/~o=.057 /~o=.057 0) ,4
1.7
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u ~A2
(~)
Af
1.6 1.5 1.'~ 1.3 1.2 1.1 1.0
3. Experimental
results ,9
With the apparatus described above it was possible to carry out experiments within the following parameter intervals for A and f~: 0.6_
and
1.54<12_<2.04.
The value of the Coulomb friction coefficient fl0
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AN-'P Fig. 3. Experimentally obtained threshold values for the first two period doublings of the rotational motion (At 1) and A[1)) and the transition to a chaotic but continuously rotating motion (A(c1)) which becomes unstable at A~ ). A~s) marks the threshold value for a period-8 window. At A~ ) the chaotic attractor belonging to the window loses its stability.
R. 14/. Leven et al./ Experiments on periodic and chaotic motions of a parametrically forced pendulum
375
Y E27[rad /~
b)
2
c)
"i
•
>¢
0 - 7C
0
a)
o)
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Fig. 4. Stroboscopic phase representation ( p = O) of the transition from a regular rotational motion (a) via two period-doubling bifurcations (b and c) to a chaotic type rotational motion (d) including band mergings of the chaotic attractor (e and f). A = 1.42, ~2 = 1.57, fl0 = 0.057, a) fll = 0.243 (10 dots); b) fll = 0.215 (20 dots); c) fll = 0.187 (100 dots); d) fll = 0.182 (100 dots); e) fll = 0.178 {100 dots); f) fll = 0.176 (200 dots).
remains continuously rotating). A~ ~ marks the transition from the stable rotating chaotic regime to a transient chaotic behaviour, which is characterized by random changes in the rotational direction. In this regime the pendulum will reach the origin after a sufficiently long time and remains there. Due to saddle-node bifurcation at Ato8~, a stable period-8 window appears within this transient chaotic regime. A~ ~ marks the transition to the transient chaotic regime. All curves in the fl-A plane are monotonously increasing with ft. Evidently an increase of A is equivalent to a decrease of ft. Utilizing this property of the system together with the possibility of continuously changing the damping force without interrupting the motion of the pendulum, it was not difficult to follow the various transitions between the different types of motion in a well defined adiabatical manner. The evolution of a period-1 rotation via perioddoubling bifurcations to a chaotic rotational motion is shown on fig. 4. The control parameter fll was varied from 0.243 to 0.176, the other parame-
ters remaining fixed. From fig. 4a to fig. 4c two period doublings are observed, which are followed by a band merging process shown in figs. 4d-4f. For other parameter values (A = 2.08, 12 = 2.04, t0 = 0.033, 131 = 0.102 . . . . . 0.081) bifurcations up to period eight were observed. A real-time display of the periodic rotation with periods 1, 2, and 4 is shown in fig. 5, whereas fig. 6 presents a chaotic motion corresponding to the small 1-band attractor given in fig. 4f. Fig. 7 shows the return map of the linear part of a 2-band attractor similar to that of fig. 4e. In spite of the spread of the points caused by fluctuations of the external force the parabolic shape of the map is clearly visible. With further decrease of t : the rotation attractor loses its stability, the rotations of the pendulum become more and more irregular and the phase point in the stroboscopic representation leaves the bounded region of the rotation attractor. This effect was detected numerically in [4] and described there quantitatively by means of the Lyapunov exponents. In the light of the work of
376
R. IV. 1.even et aL/ Experiments on periodic and chaotic motions of a parametrically forced pendulum
,t
UUJ|IIIIIIf i l l ~ u ~ = ~ ~ ~ M ~ H ~ U ~ M ~ M / ~ U ~ t a ~ ` ~ ' ~ = ~ u ~ U ` j ~ M ~ U ~
L iiU=lJUii~lUllllillUtl~lliUtllUl~ UlUi Ullilll~l"aUUll/llilliUlillllllllt~U u ~F ~ ~ i ~ ~ i i ~ i ~ i ~ i ~ i i ~ i ~ u ~ i ~ i ~ i ~ i i i i i / ~ f
=. 795
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N =.17,~
|
_~
o
~b
,~o
60
~
1;o
I;o
.o
,~o
"c E2r / "] Fig. 5. Real time display of the angular velocity y of the pendulum. The upper display (fit = 0.224) corresponds to the rotational period-1 motion of fig. 4a, the middle one (ill = 0.195) to the period-2 motion of fig. 4b and the lower (ill = 0.174) to the period-4 motion of fig. 4c.
~ =1.57
J, [2r,~,~/,,]
A= I.~7
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R. W. Leoen et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
.65
y
A = 1.57
4*2
E27Crad/sJ • ' • ~ "~:" .-;,.':" •
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Fig. 7. R e t u r n m a p of the angular position x of the p e n d u l u m c o r r e s p o n d i n g to the rectilinear part of the chaotic 2 - b a n d a t t r a c t o r of fig. 4e.
Grebogi et al. [5] one would call this an external crisis. The attractor collides with one or more stable manifolds of other unstable periodic orbits. Some characteristic stages of the period-doubling route to chaos of an oscillating motion (i.e. without rotations) of the pendulum are shown in fig. 8. Fig. 8a shows the oscillation with period 2 for A = 0.83, /31 = 0.208. An increased A leads to a symmetry-breaking bifurcation, i.e. two different stable oscillations with period 2 appear which are transformed into each other by the transformation S: (x, y ) ~ ( - x , - y ) . Fig. 8b shows one of these different periods• With a further decrease of /31 each of them shows its own period-doubling bifurcation (fig. 8c) and a transition to a small 2-band chaotic attractor. Further decrease of/3x leads to a merging of the two small 2-band attractors. As a result we obtain a larger 2-band attractor which is again invariant under S. (In all cases I2 and fl0 are fixed at ~2 = 1.67, /30 = 0.051.) With decreasing/31 this attractor also becomes unstable and after some irregular motions the system passes into the rotating period-1 regime, which is stable in the whole parameter range where the described oscillating motion exists, or approaches the fixed point (0, 0). Another case of the coexistence of two different periodic solutions including an external crisis is
o
~/2
c)
d)
Illt ll BI ]III
Fig. 8. Stroboscopic phase representation ( p = 1 / 2 ) of the transition f r o m a regular symmetric oscillation (a) via a s y m m e try-breaking bifurcation (b) and period-doubting bifurcation (c) to a chaotic attractor (d). 12 = 1.67, flo = 0.051, a) A = 0.83, fll = 0.208 (20 dots); b) A = 1.10, fll = 0.208 (20 dots); c) A = 1.10, fl~ = 0.180 (20 dots); d) A = 1.10, fix = 0.170 (150 dots)•
demonstrated by figs. 9 and 10. Fig. 9a shows the stroboscopic phase portrait of a rotating period-3 solution. For the parameter values ~2 = 2, A = 1, /30 = 0 . 0 3 5 , /31 = 0.061 this solution coexists with the running period-1 solution. Depending upon the initial conditions it is possible to push the system into the period-3 regime as well as into the period-1 regime. If the system is in the period-3 regime and the damping is adiabatically decreased without interrupting the motion of the pendulum, a period-doubling bifurcation becomes visible (fig. 9b) and after some further decrease of/31 the system is characterized by a small 3-band chaotic attractor (fig. 9c). The crisis occurs after a further small decrease of fla. The 3-band attractor disappears and the system changes into the period-1 rotating motion. Fig. 9d shows this transient behaviour.
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
378
y
3
: •
•
-.~
-X/2
¢)
(~)
|
J I -
i ..-
0
9[/2
(fig. 10c).
b)
liinl|l Igilgm NmMm.. | Fig. 9. Stroboscopic phase representations ( p = 0) of the transition from the continuously rotating period-3 motion (a) via a period-doubling bifurcation (b) to a 3-band chaotic attractor (c) which becomes unstable and changes into the coexisting stable rotational period-1 attractor indicated by the arrow (d). A = 1.00, ~2= 2.00, fl0=0.035, a) fll =0.061 (30 dots); b) fll = 0.059 (60 dots); c) fll = 0.058 (100 dots); d)fl1 = 0.057 (130 dots).
y 3
°
-A"
If one starts with the same parameters as in fig. 9a and adiabatically increases ill, the three points in the stroboscopic phase portrait approach each other (fig. 10a), however, no coalescence takes place. If some critical value of fll is reached, the period-3 motion becomes unstable and the system passes (fig. 10b) into the stable period-1 solution
t
x
0
b) 41
c)
Fig. 10. Stroboscopic phase representations (p = 0) of the rotational period-3 motion (a) which becomes unstable and changes (b) into the coexisting stable rotational period-1 (c). A = 1.00, $2= 2.00, t0 =0.035. a) fll =0.077 (50 dots); b) fit = 0.0775 (60 dots); c) fll = 0.0775 (15 dots).
The period-3 solution appears with decreasing fll (or increasing A) due to saddle-node bifurcation, undergoes the period-doubling bifurcation route to a 3-band chaotic attractor which then is annihilated by an external crisis. For all parameter sets of figs. 9 and 10 the rotating period-1 motion, the oscillating period-2 motion and the fixed point (0, 0) can be observed if the motion is started under the proper initial conditions. We would like to emphasize that the period-3 solution, although appearing near the period-1 solution in the stroboscopic phase representation does not arise from a trifurcation of the latter but f r o m a saddle-node bifurcation. This becomes evident from the fact that the period-3 solution appears with a finite distance between the three points in the stroboscopic phase portrait and that both solutions (period-1 and period-3) coexist over a certain parameter range. We suppose that the "trifurcation" described by Ritala and Salomaa [17] is nothing else but the transition from a period-5 motion to a period-15 motion which is not related to the former (this could be made clear by determining the winding numbers of the two solutions). Figs. 11-13 show the stroboscopic phase portraits of the transient chaotic motion of the pendulum for different values of the damping coefficient ill. In fig. 11 the behaviour of the pendulum is characterized by a persistence of the sense of rotation. Nevertheless, after a sufficiently large, however unpredictable time the pendulum changes its sense of rotation. With decreasing fll the region in the phase space covered by the transient chaotic motion increases (fig. 12). Simultaneously, certain regions of the phase space are
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
379
Y
Y
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l-2 r,'ou/ 3 ~dh
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Fig. 12. Stroboscopic phase representation ( p = 0) of a transient chaotic motion. A = 1.42, 12 = 1.57, flo = 0.057, fin = 0.170 (1000 dots).
illa
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i,¢t
a4
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-2
visited more and more frequently as fll decreases (fig. 13). Fig. 14 shows a real time display of the motion illustrated in fig. 13 in the stroboscopic phase portrait. The "laminar" phases characterize the intermittent regime just before the onset of a periodic motion. It should again be emphasized that figs. 11-14 show transient chaotic behaviour, i.e. after a sufficiently large time the system approaches the stable origin (0,0). In such a case the pendulum
-3 -sE
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Fig. 11. Stroboscopic phase representation ( p = 0) of a transient chaotic motion. A = 1.42, 12 = 1.57, flo = 0.057, fll = 0.174 (1000 dots).
;"
llll
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-3 -:7/"
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. 0
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Fig. 13. Stroboscopic phase representation ( p = 0) of a transient chaotic m o t i o n in an intermittent regime. A = 1.42, 12 = 1.57, flo = 0.057, /31 = 0.090 (1000 dots).
was started again without caring much about the initial conditions (about 3-5 times per 2000 T). A period-8 motion is shown in figs. 15 and 16, where fll is a little bit smaller than in figs. 13 and 14, respectively. With decreasing fix a period-doubling bifurcation occurs, as shown in fig. 17. Then the periodic motion becomes unstable and the transient chaotic motion appears again (fig. 18). Fig. 19 shows two periodic oscillations with period 10 (fig. 19a) and 4 (fig. 19b). Both motions coexist with the period-1 rotation of the pendulum. Figs. 20a-d show the stroboscopic phase portraits of the transient chaotic motion for different Poincar6 sections in thephase space. Fig. 20a applies to p = 0 i.e. at the lower dead point of the periodic vertical displacement of the suspension point of the pendulum. In figs. 20b-d p takes the values 1/4, 1 / 2 and 3/4, respectively. Fig. 21 shows a chaotic attractor for very low damping. In order to obtain this attractor the eddy current brake was removed. In this case the origin proved to be unstable and the pendulum retained its irregular motion for arbitrarily long times. In order to study the transition from metastable chaos to sustained chaotic behaviour we measured
380
R. W. Lecen et al./ Experiments on periodic and chaotic motions of a parametrically forced pendulum
e [2r~a/4
~
,4 -_ 1.~7
= is7
/~..osz
F,, ..o82
01
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lilt
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,& ~o
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3Jo ~
~o
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96o ~o .Jo I~ 7"[2r/~]
Fig. 14. Real-time display of the angular velocityy of the pendulum in the intermittent regime shown in fig. 13. the decay time t d the pendulum needed to d a m p down to the stable fixed point (0,0) in the transient regime. This time could be determined directly because the pendulum came to rest after a finite time as a result of the Coulomb friction. We Y
3
I-2Z~ad/sJ u
E ~iii|****ia*a.i**.Ii
I*Uiaaili*iaiJllIm ..w,
~lwi
iwwv
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:
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0
x
3r
Fig. 15. Stroboscopic phase representation (p=0) of the period-8 motion. A = 1.42, 1"2= 1.57, flo = 0.057, fll = 0.089 (80 dots).
measured t d for/30 -- 0.15, fll ~ 0.02, ~2 = 1.56 and four values of A. For each A we pushed the pendulum about 80 times choosing the initial conditions at r a n d o m and determined the corresponding average decay time (td). Our results are given in table I ( T is the period of the driving force). It should be mentioned that the direct measurement of well-defined decay times in the transient regime is favoured by the fact that for the chosen parameter set the main periodic motions are all unstable so that the origin is the only attractor which could be detected under these conditions. Only for A = 0.616 the pendulum reached, in a few cases, a stable rotating regime with period T. These runs were disregarded in determining the average decay time. In analogy to the results of Yorke and Yorke [14] the four values of table I can be expressed by a function of the form (td)//T
= ~. ( A c - A)-l/13,
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
381
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2L,O
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]
r
1
~
~.
[
320
;i
::'
~iI
400
~
',i
180
'
~
~
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1
48
84
80
560
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6~0
i iI
]"
L~
][1'~[
98
m
'[2zM
,
1
,~
m
~80 "L-EJX/J--'----(2
Fig. 16. Real-time display of the angular velocity y of the pendulum for the period-8 motion shown in fig. 15.
w i t h a = 60 + 10, A¢ = 1.06 + 0.06. The e x p o n e n t in our case is 1 / f l = 0.8 + 0.2. This differs considerably f r o m the value 3 . 5 , . . . , 4.0 obtained in [14].
c h a o t i c m o t i o n o f a parametrically excited m e c h a n i c a l p e n d u l u m a n d the transitions b e t w e e n the different t y p e s o f m o t i o n . In all cases the periodd o u b l i n g r o u t e to c h a o s was f o u n d if the control p a r a m e t e r w a s varied s l o w l y w i t h o u t interrupting the m o t i o n o f the p e n d u l u m . Different sub-
4. Conclusions In this p a p e r w e h a v e presented e x p e r i m e n t a l results o n the periodic, c h a o t i c a n d transient Y
Y
3
",?.Xrad / sg
,:~ ¢.', L t B'
f
D.
i
,,
.
• I.
h ''''
mHH|H||lIHIH|l
!**,
V,,I
ills
**11
**1 I
'~t'
,~,
,,,w.
juuu|u,~wlu....
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Iiii
....
.
I.
,
J" p-
O r •m
X
-~
i
D.
I~.'.
~ ) •
X
0
Fig. 17. Stroboscopic phase representation ( p = 0 ) of the period-16 motion. A = 1.42, 1~ = 1.57, fl0 = 0.057, fl, = 0.088 (80 dots).
Fig. 18. Stroboscopic phase representation (p = 0) of a transient chaotic motion. A = 1.42, ~ = 1.57, fl0 = 0.057, fll = 0.086 (1000 dots).
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
382
Y
Y
a)
3 / 2 l iE2"lC'F°d/s]
b)
,'1
o :::: :i:t ::).: ..........
I .... , ....
E~o~3
o .
0
. . . . . . .I l.l
lilt
o
, I,I I,I . i ; ' ; t
i
X
"It
-3/2 -.IF/2
" 0
5V/2
Fig. 19. S t r o b o s c o p i c phase representations of two s u b h a r m o n i c oscillations, a) A = 0.80, ~2 = 2.04, ~o = 0.033, fl~ = 0.010 ( p = 0, 100 dots); b) A = 0.99, £ = ZOO, flo = 0.035, fli = 0.050 ( p = 1 / 2 , 40 dots).
Y
Y
a) [
i~
r
--'
c)
3
r
., I'~ ":
.~, ~ ~ ' , , t~
,#,
""
~--~- ~,s.:
01~I
;"" 'll~ ~ :
L,.--"
i . ,,',~
..t.
"2" "~" 4It
X
i
-2
?°*J /r
0
Y
Y
b)
d)
3
.d
~.~.: ........... ~. ~ , , .... ?2........
:----;.';.t ,-- ---
,d
.,#
-'-.7( ¸
0
D~J -2, -2"
Fig. 20. S t r o b o s c o p i c phase r e p r e s e n t a t i o n s of a transient chaotic m o t i o n w i t h different p h a s e s y n c h r o n i z a t i o n s p. = 1.57, fl0 = 0.057, fll = 0.117. a) p = 0; b) p = 1 / 4 ; c) p = 1 / 2 ; d) p = 3 / 4 (2000 d o t s for each case).
A = 1.42,
R. IV. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
Y • ~ "~/-;.;. ,! @.:r:;.~ -~:.;) i~:~
..,~;,,~..,~.~, .~;,'~...:. ~
""" "' '~*11"~,,"
'.2.~?,
~'"~': ~..:.,.
;~.,
'.,:',.¢','~,," :.~'~.... ~" 4:,,
0
~,......,
.
J
"~
"
::,,:. b.! ,@"
, ?
"e..i[. ~ ?.;'.:-.:* ,.., .:'. ~ ...
~.;~cj ..~r"
~.q~, r..,,,:, ..;',i~ X
-3
•,
4
g
.....
-Tr"
.,
0
Fig. 21. Stroboscopic phase representation ( p = 0) of a chaotic m o t i o n for relatively weak d a m p i n g (the origin (x, y ) = (0,0) was f o u n d to be unstable). A = 1.65, D = 1.52, fl0 + fll -< 0.026 (4500 dots).
harmonic periodic motions were detected which coexist with the periodic rotational motion of the pendulum as well as in the transient chaotic regime ("windows"). These subharmonics appear with increasing external force A (or decreasing damping coefficient fl) due to saddle-node bifurcation and evolve to chaos via a period-doubling cascade. This evolution can be terminated by external crises at different stages. In the transient chaotic regime the appearance of a periodic window is preceded by intermittency (in the sense described above).
383
The average decay time (ta) of chaotic motions was determined as a function of the amplitude A of the external periodic force. Within the accuracy of the measurements ~td) is proportional to (A c A ) - 1 / ~ where 1 / f l is between 0.6 and 1.0 for our choice of fl0, fix and 12. In all our measurements the motion of the pendulum was influenced by a certain amount of external noise (e.g. caused by random vibrations of the measuring apparatus). For this reason it was not possible to observe more than two or three period-doubling bifurcations, nor was it possible to evaluate characteristic numbers. In general the termination of bifurcation cascades by crises took place at lower A values (higher fl) than it should be the case for an undisturbed motion. On the other hand, the existence of periodic windows and the coexistence of different periodic motions in the presence of noise indicate that these subharmonics, having finite basins of attraction, are rather stable. Concerning the question whether the Usequence is characteristic of driven nonlinear oscillators we would assume that the behaviour of the parametrically forced pendulum is richer in form than shown by the logistic map (coexistence of different periodic motions, "intermittency" between small bounded and large attractors, combined rotational and oscillating motions), but typical properties of the U-sequence, e.g. perioddoubling route to chaos, existence of periodic windows, crises, intermittency, have been observed also in our experiments.
References
Table I
[1] F o r example, M.J. F e i g e n b a u m , J. Stat. Phys. 19 (1978) 25. [2] Y. P o m e a u a n d P. Manneville, C o m m u n . Math. Phys. 74
A
(ta)/T
0.616 0.770 0.847 0.924
120 166 216 292
_+ 10 + 13 + 18 + 33
(1980) 189. [3] Y. Ueda, J. Stat. Phys. 20 (1979) 181. [4] R.W. Leven and B.P. Koch, Phys. Lett 86A (1981) 71. [5] C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Lett. 48 (1982) 1507. [6] B.P. Koch, R.W. Leven, B. Pompe and C. Wilke, Phys. Lett. 96A (1983) 219.
384
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
[7] C. Jeffries and J. Perez, Phys. Rev. 27A (1983) 601. [8] J. Testa, J. Pgrez and C. Jeffries, Phys. Rev. Lett. 48 (1982) 714. [9] A. Ameodo, P. Coullet, C. Tresser, A. Libchaber, J. Maurer and D. d'Humi~res, Physica 6D (1983) 385. [10] M. Metropolis, M.L. Stein and P.R. Stein, J. Combinatorial Theory 15 (1973) 25. [11] F.C. Moon and P.J. Holmes, J. Sound Vibr. 65 (1979) 275.
[12] F.C. Moon, J. Appl. Mech. 47 (1980) 638. [13] V. Croquette and C. Poitou, C.R. Acad. Sci. B292 (1981) 1353. [14] J.A. Yorke and E.D. Yorke, J. Stat. Phys. 21 (1979) 263. [15] J.B. McLaughlin, J. Stat. Phys. 24 (1981) 375. [16] B.P. Koch and R.W. Leven, Physica 16D (1985) 1. [17] R.K. Ritala and M.M. Salomaa, J. Phys. 16C (1983) L 474.
EXPERIMENTS ON PERIODIC AND CHAOTIC MOTIONS OF A PARAMETRICALLY FORCED PENDULUM R.W. LEVEN, B. POMPE, C. WILKE and B.P. KOCH Sektion Physik / Elektronik, E. -M. -A rndt- Universiti~t, DDR, 2200 Greifswald, GDR Received 3 July 1984 Revised manuscript received 27 November 1984
An experimental study of periodic and chaotic type aperiodic motions of a parametrically harmonically excited pendulum is presented. It is shown that a characteristic rou~e to chaos is the period-doubling cascade, which for the parametrically excited pendulum occurs with increasing driving amplitude and decreasing damping force, respectively. The coexistence of different periodic solutions as well as periodic and chaotic solutions is demonstrated and various transitions between them are studied. The pendulum is found to exhibit a transient chaotic behaviour in a wide range of driving force amplitudes. The transition from metastable chaos to sustained chaotic behaviour is investigated.
1. Introduction If a nonlinear dissipative system is driven by an external force it leaves the equilibrium state and develops new orderings. The system either oscillates or even shows a more complicated temporal evolution, characterized by the notion of chaos. The chaos is connected with the existence of a chaotic or "strange" attractor: a surface in the phase space of the system to which orbits are attracted and on which they move pseudorandomly. An interesting question is how chaotic attractors arise as some parameter of the system is varied, and how and to what degree they are dependent on the driving parameter. Several routes to chaos have now been well documented. They include infinite period-doubling cascades [1], intermittency [2] and sharp transitions from a periodic or quasiperiodic regime to a chaotic one. Sudden qualitative changes, called crises, have been found numerically also in the chaotic regime [3, 4]. They have been investigated theoretically by Grebogi, Ott, and Yorke [5] and were observed experimen-
tally for a parametrically forced pendulum [6] and a driven nonlinear semiconductor oscillator [7]. If a system is characterized by only one parameter, it is relatively easy to study the routes to chaos as the parameter is varied. For real physical systems, however, the number of parameters is large. Even such apparently simple and physically transparent systems as externally driven nonlinear oscillators have, apart from the driving amplitude and the driving frequency, the natural frequency and the damping force as important parameters. Nevertheless, in many cases a "typical picture" can be obtained if all parameters are kept fixed and only the amplitude of the driving force is varied. This, for example, was done in the driven nonlinear semiconductor oscillator experiments of Testa, Perez and Jeffries [8], where the behaviour of the system was compared with the simple logistic model described by the one-parameter equation x , + x = ) ~ x , ( 1 - x , ) . A correspondence between the driving voltage V0 and the parameter h was assumed and a strong similarity between the predicted and the observed bifurcation diagram was found.
0167-2789/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
372
R. W. Leoen et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
In this connection, we would like to mention that, in general, for two- or more parameter systems the character of the bifurcation diagram, which is obtained if one continuously varies the "control parameter", depends upon the other parameters. So, for example, for the two-parameter H r n o n mapping the bifurcation diagram changes if the second parameter b is varied. This becomes obvious from fig. 3 of the paper of Arneodo et al. [9], where the threshold values of the parameter a for the onset of various subharmonics are traced as functions of b. There are m a n y intersections between the traces of different periodic attractors. This indicates that for b ~ 0 the sequence of periodic or chaotic solutions which one obtains with increasing a is not consistent with the universal U-sequence of Metropolis, Stein, and Stein [10]. Especially, the coexistence of two or more periodic (or chaotic) solutions with distinct basins of attraction is possible. As pointed out in [9], it is possible in such cases to observe rather exotic routes to chaos if one does not take much care of the adiabaticity in turning on the constraint. Besides numerical computations also many experimental investigations of the periodic and chaotic behaviour of externally driven nonlinear oscillators have been carried out, but so far only few experiments with mechanical systems have been made [11-13]. In a previous paper [6] we reported on experimental evidence for chaotic type nonperiodic motions of a parametrically forced mechanical pendulum. In this paper we present detailed measurements with an improved version of the equipment used in ref. [6] and {ry to answer the question of the typical behaviour of nonlinear oscillators driven by an external force. Our new system has the advantage of a higher resolution in the phase plane, and a well-defined damping force which, most importantly, could be varied continuously within an experimental run. This enabled us to vary adiabatically one of the parameters (the d a m p i n g coefficient) without interrupting the motion of the pendulum. Our experiments showed that the measured values vary adiabatically with a
sufficiently slow variation of the damping parameter. This adiabatic property is crucial for exactly following the various transitions from one type of motion to the other. As was mentioned already in [6], the Coulomb friction, which is the dominant damping force for sufficiently small values of k, makes the origin stable and attracting up to rather high values of the driving force. This gives the possibility to study the irregular motion of the pendulum in the transient regime over a large range of parameter values and to describe the transition from transient chaos to sustained chaotic behaviour, which was first studied in detail for the Lorenz model by Yorke and Yorke [14].
2. Experimental equipment
in
A sketch of the experimental apparatus is shown fig. 1. The parametrically excited damped
T 7
Fig. 1. Scheme of the mechanical part of the experimental set-up: 1-pendulum body; 2-pendulum rod; 3-disk with angular code; 4 - light barriers; 5 - eddy current brake; 6 - slide; 7 - frame; 8 - electromotor; 9 - transmission; 10- lever; 11 - balance weight.
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
pendulum consists of a body (mass = 200 g) which is fastened to a pendulum rod (length = 38.5 cm). The natural angular frequency too is adjustable from 5.08 to 6.73 s-1. A periodic vertical displacement of the pendulum occurs with an angular frequency to = 10.37 s-t. The adjustable amplitude a of this excitement ranges from 5.5 to 20.0 cm. A disk is rigidly connected to the pendulum. On the disk there is an angular code (GRAY-code) in the form of sectors which are pervious or impervious to light. Ten light barriers are radially arranged at some distance from the disk. The signals of these barriers are electrically analysed. They provide the electronic voltages Ux and Uy which are proportional to the angle x of the pendulum swing and to the angular velocity y = d x / d z (T = time) in the limits of measuring precision ( A x ~ +_0.013 rad, A y = ___0.63 tad/s). If the pendulum turns over at x = _ ~r rad, a transformation x -~ x ~- 2~r tad is carried out. Consequently, x varies from -~r to +~r rad. A stroboscopic phase representation of x and y by means of an oscillograph or rather a printer can be carried out in any position p of the point of suspension of the pendulum which is passed during its vertical displacement. In the following p = 0, 1 / 2 , or 1 / 4 , 3 / 4 means the lower, the upper dead point, or half the way from the lower to the upper and the upper to the lower dead point, respectively. For the value of to chosen in our experiments about 100 dots appear within one minute on the fluorescent screen of the oscillograph, which are photographed. The damping of the pendulum is first of all effected by means of an eddy current brake. By varying the current I giving rise to the magnetic field of the eddy current brake, the damping of the pendulum can be altered. The damped parametrically driven physical pendulum can be described by the equation j ( d E x / d ' r 2 ) + B o sgndx/d~" + Bx( d x / d , r )
+ m l ( g + ato2 cos to~") sinx = 0,
373
constants; m = mass; 1 = distance between centre of mass and rotation axis; g--- acceleration due to gravity. The damping term of eq. (1) Md=
B 0 sgn d x / d ~ + B 1 d x / d ' r
has its origin in experimental investigations. Fig. 2 shows the function Md(Y ) as established in experiments. Making use of the abbreviations
t = too~,
too = ( m l g / J ) t/2,
flo = B o / ( J°J~ ), ±=dx/dt,
[2 = to/too,
fix = B l l ( Jtoo),
£=d2x/dt
A = ato21g,
2,
M d [Nn~ I .221 .21
500 m A
.20"
PrnA .300 mA
.19 .18
200 mA
.I? .16
.15 100 mA
.13 .12 .11
.10" .09
50 mA
.08 .07 .06
.05
/
•
~ J
j
.03 .02 .01
y [2~'rad/s]
(1)
where J = moment of inertia; B0, B 1 -- damping
Fig. 2. Damping turning moment M d versus angular velocity y (experimental values). (Parameter: current I in the eddy current brake.)
374
R. IV. Leven et al./ Experiments on periodic and chaotic motions of a parametrically forced pendulum
eq. (1) can be written as
Yc+ floSgnJc + /31Jc+ ( l + Acosg2t)sinx=O.
(2)
If (x, y ) is a solution of eq. (2), then ( - x , - y ) is a solution too, because eq. (2) is invariant under the symmetry transformation S: (x, y ) (-x, -y). For fl0 = 0 this equation reduces to that studied numerically in [4] and [15]. As shown in [6], the main effect of the /3o sgn 2 part of the damping force is to hold the origin of the phase plane stable even for higher values of the external force. (Only if the eddy current brake was removed, the origin was found to be unstable in a certain parameter range.) As a consequence, almost every motion with the phase point visiting a small area around the origin has to be considered as a transient one. However, this is without importance for stable periodic motions or in a regime with a "small" bounded chaotic attractor lying far enough from the origin in the stroboscopic phase representation. It is true that our experimental investigations cannot be regarded as proof of the chaotic nature of the pendulum motion. Nevertheless, we call certain motions chaotic because in numerical examinations of an equivalent system [4] the existence of chaotic behaviour was made plausible by the computation of Lyapunov exponents and power spectra for some parameter sets. The observed period-doubling bifurcations between periodic and apparently aperiodic motions and the parabolic shape of a return map obtained for an aperiodic rotational motion are further hints at the chaotic nature of the aperiodic regime.
was kept as small as possible. For the largest 1"~= 2.04 we obtained fl0 = 0.033. With decreasing this value nearly linearly increases up to fl0 ~ 0.057 for 12 = 1.54. The damping coefficient fll could be varied from 0.008 to 0.106 for 12 = 2.04 and from 0.026 to 0.286 for 12 = 1.54. Our main intention has been to describe properly the various transitions between periodic motions with different periods, periodic and chaotic motions, and between chaotic motions of different character as one or more parameters are varied. In order to reduce the number of parameters which have to be varied we proceed from the assumption that a typical direction to chaos is that of increasing A or decreasing fl = fl0 + fix (the other parameters fixed). As shown in [16], the onset of the different subharmonics is characterized by a certain value of the relation A / f l for a given frequency I"L Fig. 3 shows the threshold values A~1) and A~21) for the first two period doublings of the rotational motion with period 1 (T = 2~r/~0) and the value A~a) at which the motion becomes chaotic (but
~-~=L57
A
/~o=.057 /~o=.057 0) ,4
1.7
I) r l )
u ~A2
(~)
Af
1.6 1.5 1.'~ 1.3 1.2 1.1 1.0
3. Experimental
results ,9
With the apparatus described above it was possible to carry out experiments within the following parameter intervals for A and f~: 0.6_
and
1.54<12_<2.04.
The value of the Coulomb friction coefficient fl0
.
.
.
.
.
.
.
.
.
.
.,o .12 .,, .I, .la
.
.
.
.
.
.2o .22
.
.
.
.
.
.2~ .2.
.
.
.
.28 .~o .32
--
AN-'P Fig. 3. Experimentally obtained threshold values for the first two period doublings of the rotational motion (At 1) and A[1)) and the transition to a chaotic but continuously rotating motion (A(c1)) which becomes unstable at A~ ). A~s) marks the threshold value for a period-8 window. At A~ ) the chaotic attractor belonging to the window loses its stability.
R. 14/. Leven et al./ Experiments on periodic and chaotic motions of a parametrically forced pendulum
375
Y E27[rad /~
b)
2
c)
"i
•
>¢
0 - 7C
0
a)
o)
r) ,f
I i
Fig. 4. Stroboscopic phase representation ( p = O) of the transition from a regular rotational motion (a) via two period-doubling bifurcations (b and c) to a chaotic type rotational motion (d) including band mergings of the chaotic attractor (e and f). A = 1.42, ~2 = 1.57, fl0 = 0.057, a) fll = 0.243 (10 dots); b) fll = 0.215 (20 dots); c) fll = 0.187 (100 dots); d) fll = 0.182 (100 dots); e) fll = 0.178 {100 dots); f) fll = 0.176 (200 dots).
remains continuously rotating). A~ ~ marks the transition from the stable rotating chaotic regime to a transient chaotic behaviour, which is characterized by random changes in the rotational direction. In this regime the pendulum will reach the origin after a sufficiently long time and remains there. Due to saddle-node bifurcation at Ato8~, a stable period-8 window appears within this transient chaotic regime. A~ ~ marks the transition to the transient chaotic regime. All curves in the fl-A plane are monotonously increasing with ft. Evidently an increase of A is equivalent to a decrease of ft. Utilizing this property of the system together with the possibility of continuously changing the damping force without interrupting the motion of the pendulum, it was not difficult to follow the various transitions between the different types of motion in a well defined adiabatical manner. The evolution of a period-1 rotation via perioddoubling bifurcations to a chaotic rotational motion is shown on fig. 4. The control parameter fll was varied from 0.243 to 0.176, the other parame-
ters remaining fixed. From fig. 4a to fig. 4c two period doublings are observed, which are followed by a band merging process shown in figs. 4d-4f. For other parameter values (A = 2.08, 12 = 2.04, t0 = 0.033, 131 = 0.102 . . . . . 0.081) bifurcations up to period eight were observed. A real-time display of the periodic rotation with periods 1, 2, and 4 is shown in fig. 5, whereas fig. 6 presents a chaotic motion corresponding to the small 1-band attractor given in fig. 4f. Fig. 7 shows the return map of the linear part of a 2-band attractor similar to that of fig. 4e. In spite of the spread of the points caused by fluctuations of the external force the parabolic shape of the map is clearly visible. With further decrease of t : the rotation attractor loses its stability, the rotations of the pendulum become more and more irregular and the phase point in the stroboscopic representation leaves the bounded region of the rotation attractor. This effect was detected numerically in [4] and described there quantitatively by means of the Lyapunov exponents. In the light of the work of
376
R. IV. 1.even et aL/ Experiments on periodic and chaotic motions of a parametrically forced pendulum
,t
UUJ|IIIIIIf i l l ~ u ~ = ~ ~ ~ M ~ H ~ U ~ M ~ M / ~ U ~ t a ~ ` ~ ' ~ = ~ u ~ U ` j ~ M ~ U ~
L iiU=lJUii~lUllllillUtl~lliUtllUl~ UlUi Ullilll~l"aUUll/llilliUlillllllllt~U u ~F ~ ~ i ~ ~ i i ~ i ~ i ~ i ~ i i ~ i ~ u ~ i ~ i ~ i ~ i i i i i / ~ f
=. 795
2
N =.17,~
|
_~
o
~b
,~o
60
~
1;o
I;o
.o
,~o
"c E2r / "] Fig. 5. Real time display of the angular velocity y of the pendulum. The upper display (fit = 0.224) corresponds to the rotational period-1 motion of fig. 4a, the middle one (ill = 0.195) to the period-2 motion of fig. 4b and the lower (ill = 0.174) to the period-4 motion of fig. 4c.
~ =1.57
J, [2r,~,~/,,]
A= I.~7
&=.05z
A=.162
&
,~ t|llmmlall
;
lr
.............................
..........................................................................................
3I
|
~llVl--ww-,,,. . . . . . i,=Wl|lllnlUlrllll~llllllll Wllnlm'qqlHi111"~= -,l",l'w t
l
.....
i a
•
•
01~ r ~ 1 r ~ | ~ i ~ 1 ~ n i r ~ 1 ~ n ~ t m | t m ~ 1 ~ m n n ~ v ~ w ~ n ~ ~ u
I
o
i
.k
a
J
=~ama,,,=J
,~p,w"~w~qnl'WKIlUli'nIWl~i~v,,v..... , . . . . . wnw wpWWlwllllllll
mmmmmwlmmwm~wf~
h
20
~o
60
8o
1~o
12o
~o
~6o
18o
2~o
220
2~o
26o
28o
'r r2r/,,] Fig. 6. Real-time display of the angular velocity y of the pendulum in the continuously rotating chaotic regime corresponding to the I - b a n d attractor of fig. 4f.
R. W. Leoen et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
.65
y
A = 1.57
4*2
E27Crad/sJ • ' • ~ "~:" .-;,.':" •
.n.~: ¢' ....
",.•.
"'d~ .. " ....
. :',. • .
/2o=
377
.01,3
3
a)
b)
fit = . 0 9 3
' - "..:....
:... ,,: ".,..:
• 60
•
55
".."7
×
:~.,.. ,,.o...
.50]
.,;5
0
• .,;'.l : • ..¢" :4"=.
.55
.55
•60
.65
-3 -7c/2
X. I
Fig. 7. R e t u r n m a p of the angular position x of the p e n d u l u m c o r r e s p o n d i n g to the rectilinear part of the chaotic 2 - b a n d a t t r a c t o r of fig. 4e.
Grebogi et al. [5] one would call this an external crisis. The attractor collides with one or more stable manifolds of other unstable periodic orbits. Some characteristic stages of the period-doubling route to chaos of an oscillating motion (i.e. without rotations) of the pendulum are shown in fig. 8. Fig. 8a shows the oscillation with period 2 for A = 0.83, /31 = 0.208. An increased A leads to a symmetry-breaking bifurcation, i.e. two different stable oscillations with period 2 appear which are transformed into each other by the transformation S: (x, y ) ~ ( - x , - y ) . Fig. 8b shows one of these different periods• With a further decrease of /31 each of them shows its own period-doubling bifurcation (fig. 8c) and a transition to a small 2-band chaotic attractor. Further decrease of/3x leads to a merging of the two small 2-band attractors. As a result we obtain a larger 2-band attractor which is again invariant under S. (In all cases I2 and fl0 are fixed at ~2 = 1.67, /30 = 0.051.) With decreasing/31 this attractor also becomes unstable and after some irregular motions the system passes into the rotating period-1 regime, which is stable in the whole parameter range where the described oscillating motion exists, or approaches the fixed point (0, 0). Another case of the coexistence of two different periodic solutions including an external crisis is
o
~/2
c)
d)
Illt ll BI ]III
Fig. 8. Stroboscopic phase representation ( p = 1 / 2 ) of the transition f r o m a regular symmetric oscillation (a) via a s y m m e try-breaking bifurcation (b) and period-doubting bifurcation (c) to a chaotic attractor (d). 12 = 1.67, flo = 0.051, a) A = 0.83, fll = 0.208 (20 dots); b) A = 1.10, fll = 0.208 (20 dots); c) A = 1.10, fl~ = 0.180 (20 dots); d) A = 1.10, fix = 0.170 (150 dots)•
demonstrated by figs. 9 and 10. Fig. 9a shows the stroboscopic phase portrait of a rotating period-3 solution. For the parameter values ~2 = 2, A = 1, /30 = 0 . 0 3 5 , /31 = 0.061 this solution coexists with the running period-1 solution. Depending upon the initial conditions it is possible to push the system into the period-3 regime as well as into the period-1 regime. If the system is in the period-3 regime and the damping is adiabatically decreased without interrupting the motion of the pendulum, a period-doubling bifurcation becomes visible (fig. 9b) and after some further decrease of/31 the system is characterized by a small 3-band chaotic attractor (fig. 9c). The crisis occurs after a further small decrease of fla. The 3-band attractor disappears and the system changes into the period-1 rotating motion. Fig. 9d shows this transient behaviour.
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
378
y
3
: •
•
-.~
-X/2
¢)
(~)
|
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i ..-
0
9[/2
(fig. 10c).
b)
liinl|l Igilgm NmMm.. | Fig. 9. Stroboscopic phase representations ( p = 0) of the transition from the continuously rotating period-3 motion (a) via a period-doubling bifurcation (b) to a 3-band chaotic attractor (c) which becomes unstable and changes into the coexisting stable rotational period-1 attractor indicated by the arrow (d). A = 1.00, ~2= 2.00, fl0=0.035, a) fll =0.061 (30 dots); b) fll = 0.059 (60 dots); c) fll = 0.058 (100 dots); d)fl1 = 0.057 (130 dots).
y 3
°
-A"
If one starts with the same parameters as in fig. 9a and adiabatically increases ill, the three points in the stroboscopic phase portrait approach each other (fig. 10a), however, no coalescence takes place. If some critical value of fll is reached, the period-3 motion becomes unstable and the system passes (fig. 10b) into the stable period-1 solution
t
x
0
b) 41
c)
Fig. 10. Stroboscopic phase representations (p = 0) of the rotational period-3 motion (a) which becomes unstable and changes (b) into the coexisting stable rotational period-1 (c). A = 1.00, $2= 2.00, t0 =0.035. a) fll =0.077 (50 dots); b) fit = 0.0775 (60 dots); c) fll = 0.0775 (15 dots).
The period-3 solution appears with decreasing fll (or increasing A) due to saddle-node bifurcation, undergoes the period-doubling bifurcation route to a 3-band chaotic attractor which then is annihilated by an external crisis. For all parameter sets of figs. 9 and 10 the rotating period-1 motion, the oscillating period-2 motion and the fixed point (0, 0) can be observed if the motion is started under the proper initial conditions. We would like to emphasize that the period-3 solution, although appearing near the period-1 solution in the stroboscopic phase representation does not arise from a trifurcation of the latter but f r o m a saddle-node bifurcation. This becomes evident from the fact that the period-3 solution appears with a finite distance between the three points in the stroboscopic phase portrait and that both solutions (period-1 and period-3) coexist over a certain parameter range. We suppose that the "trifurcation" described by Ritala and Salomaa [17] is nothing else but the transition from a period-5 motion to a period-15 motion which is not related to the former (this could be made clear by determining the winding numbers of the two solutions). Figs. 11-13 show the stroboscopic phase portraits of the transient chaotic motion of the pendulum for different values of the damping coefficient ill. In fig. 11 the behaviour of the pendulum is characterized by a persistence of the sense of rotation. Nevertheless, after a sufficiently large, however unpredictable time the pendulum changes its sense of rotation. With decreasing fll the region in the phase space covered by the transient chaotic motion increases (fig. 12). Simultaneously, certain regions of the phase space are
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
379
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visited more and more frequently as fll decreases (fig. 13). Fig. 14 shows a real time display of the motion illustrated in fig. 13 in the stroboscopic phase portrait. The "laminar" phases characterize the intermittent regime just before the onset of a periodic motion. It should again be emphasized that figs. 11-14 show transient chaotic behaviour, i.e. after a sufficiently large time the system approaches the stable origin (0,0). In such a case the pendulum
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Fig. 11. Stroboscopic phase representation ( p = 0) of a transient chaotic motion. A = 1.42, 12 = 1.57, flo = 0.057, fll = 0.174 (1000 dots).
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was started again without caring much about the initial conditions (about 3-5 times per 2000 T). A period-8 motion is shown in figs. 15 and 16, where fll is a little bit smaller than in figs. 13 and 14, respectively. With decreasing fix a period-doubling bifurcation occurs, as shown in fig. 17. Then the periodic motion becomes unstable and the transient chaotic motion appears again (fig. 18). Fig. 19 shows two periodic oscillations with period 10 (fig. 19a) and 4 (fig. 19b). Both motions coexist with the period-1 rotation of the pendulum. Figs. 20a-d show the stroboscopic phase portraits of the transient chaotic motion for different Poincar6 sections in thephase space. Fig. 20a applies to p = 0 i.e. at the lower dead point of the periodic vertical displacement of the suspension point of the pendulum. In figs. 20b-d p takes the values 1/4, 1 / 2 and 3/4, respectively. Fig. 21 shows a chaotic attractor for very low damping. In order to obtain this attractor the eddy current brake was removed. In this case the origin proved to be unstable and the pendulum retained its irregular motion for arbitrarily long times. In order to study the transition from metastable chaos to sustained chaotic behaviour we measured
380
R. W. Lecen et al./ Experiments on periodic and chaotic motions of a parametrically forced pendulum
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3
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Fig. 15. Stroboscopic phase representation (p=0) of the period-8 motion. A = 1.42, 1"2= 1.57, flo = 0.057, fll = 0.089 (80 dots).
measured t d for/30 -- 0.15, fll ~ 0.02, ~2 = 1.56 and four values of A. For each A we pushed the pendulum about 80 times choosing the initial conditions at r a n d o m and determined the corresponding average decay time (td). Our results are given in table I ( T is the period of the driving force). It should be mentioned that the direct measurement of well-defined decay times in the transient regime is favoured by the fact that for the chosen parameter set the main periodic motions are all unstable so that the origin is the only attractor which could be detected under these conditions. Only for A = 0.616 the pendulum reached, in a few cases, a stable rotating regime with period T. These runs were disregarded in determining the average decay time. In analogy to the results of Yorke and Yorke [14] the four values of table I can be expressed by a function of the form (td)//T
= ~. ( A c - A)-l/13,
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
381
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w i t h a = 60 + 10, A¢ = 1.06 + 0.06. The e x p o n e n t in our case is 1 / f l = 0.8 + 0.2. This differs considerably f r o m the value 3 . 5 , . . . , 4.0 obtained in [14].
c h a o t i c m o t i o n o f a parametrically excited m e c h a n i c a l p e n d u l u m a n d the transitions b e t w e e n the different t y p e s o f m o t i o n . In all cases the periodd o u b l i n g r o u t e to c h a o s was f o u n d if the control p a r a m e t e r w a s varied s l o w l y w i t h o u t interrupting the m o t i o n o f the p e n d u l u m . Different sub-
4. Conclusions In this p a p e r w e h a v e presented e x p e r i m e n t a l results o n the periodic, c h a o t i c a n d transient Y
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Fig. 18. Stroboscopic phase representation (p = 0) of a transient chaotic motion. A = 1.42, ~ = 1.57, fl0 = 0.057, fll = 0.086 (1000 dots).
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
382
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A = 1.42,
R. IV. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
Y • ~ "~/-;.;. ,! @.:r:;.~ -~:.;) i~:~
..,~;,,~..,~.~, .~;,'~...:. ~
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harmonic periodic motions were detected which coexist with the periodic rotational motion of the pendulum as well as in the transient chaotic regime ("windows"). These subharmonics appear with increasing external force A (or decreasing damping coefficient fl) due to saddle-node bifurcation and evolve to chaos via a period-doubling cascade. This evolution can be terminated by external crises at different stages. In the transient chaotic regime the appearance of a periodic window is preceded by intermittency (in the sense described above).
383
The average decay time (ta) of chaotic motions was determined as a function of the amplitude A of the external periodic force. Within the accuracy of the measurements ~td) is proportional to (A c A ) - 1 / ~ where 1 / f l is between 0.6 and 1.0 for our choice of fl0, fix and 12. In all our measurements the motion of the pendulum was influenced by a certain amount of external noise (e.g. caused by random vibrations of the measuring apparatus). For this reason it was not possible to observe more than two or three period-doubling bifurcations, nor was it possible to evaluate characteristic numbers. In general the termination of bifurcation cascades by crises took place at lower A values (higher fl) than it should be the case for an undisturbed motion. On the other hand, the existence of periodic windows and the coexistence of different periodic motions in the presence of noise indicate that these subharmonics, having finite basins of attraction, are rather stable. Concerning the question whether the Usequence is characteristic of driven nonlinear oscillators we would assume that the behaviour of the parametrically forced pendulum is richer in form than shown by the logistic map (coexistence of different periodic motions, "intermittency" between small bounded and large attractors, combined rotational and oscillating motions), but typical properties of the U-sequence, e.g. perioddoubling route to chaos, existence of periodic windows, crises, intermittency, have been observed also in our experiments.
References
Table I
[1] F o r example, M.J. F e i g e n b a u m , J. Stat. Phys. 19 (1978) 25. [2] Y. P o m e a u a n d P. Manneville, C o m m u n . Math. Phys. 74
A
(ta)/T
0.616 0.770 0.847 0.924
120 166 216 292
_+ 10 + 13 + 18 + 33
(1980) 189. [3] Y. Ueda, J. Stat. Phys. 20 (1979) 181. [4] R.W. Leven and B.P. Koch, Phys. Lett 86A (1981) 71. [5] C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Lett. 48 (1982) 1507. [6] B.P. Koch, R.W. Leven, B. Pompe and C. Wilke, Phys. Lett. 96A (1983) 219.
384
R. W. Leven et al. / Experiments on periodic and chaotic motions of a parametrically forced pendulum
[7] C. Jeffries and J. Perez, Phys. Rev. 27A (1983) 601. [8] J. Testa, J. Pgrez and C. Jeffries, Phys. Rev. Lett. 48 (1982) 714. [9] A. Ameodo, P. Coullet, C. Tresser, A. Libchaber, J. Maurer and D. d'Humi~res, Physica 6D (1983) 385. [10] M. Metropolis, M.L. Stein and P.R. Stein, J. Combinatorial Theory 15 (1973) 25. [11] F.C. Moon and P.J. Holmes, J. Sound Vibr. 65 (1979) 275.
[12] F.C. Moon, J. Appl. Mech. 47 (1980) 638. [13] V. Croquette and C. Poitou, C.R. Acad. Sci. B292 (1981) 1353. [14] J.A. Yorke and E.D. Yorke, J. Stat. Phys. 21 (1979) 263. [15] J.B. McLaughlin, J. Stat. Phys. 24 (1981) 375. [16] B.P. Koch and R.W. Leven, Physica 16D (1985) 1. [17] R.K. Ritala and M.M. Salomaa, J. Phys. 16C (1983) L 474.