chaotic behaviour of a parametrically exciteddamped pendulum

7/28/2019 Chaotic Behaviour of a Parametrically Exciteddamped Pendulum

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Volume 86A, number 2 PHYSICS LETTERS 2 November 1981

CHAOTIC BEHAVIOUR OF A PARAMETRICALLY EXCITED DAMPED PENDULUM

R.W. LEVEN and B.P. KOCHSektionPhysik/Elektronik,E.-M.-Arndt-Universitt, DDR-2200 Greifswald, GermanDem. Rep.

Received 5 August 1981

The temporal behaviour ofa parametrically excited mathematical pendulum is examined by meansofthe numericalsolution of its equationofmotion. It is shown that for certain parameter intervalsthe pendulum behaves in an apparentlychaotic way. This kind of motion is studied by meansofthe stroboscopic phase representation, the Lyapunov character-

istic exponents and the power spectrum.

Recently it has been shown by several authors [1 it cycles and fixed points. But as the external modula-4] that the external periodic modulation of a nonlin- tion becomes larger, a description ofthe behaviour ofear oscillator may lead to a chaotic output behaviour, the system using perturbation theory becomes increasEspecially it was shown that there exists a range of ingly difficult. As will be shown below, the behaviour

parameter values for which the solutions ofthe corre- is subject togreat variations in the large-amplitude

sponding deterministic equation display a set ofcas- regime including both periodic and chaotic solutionscading bifurcations into a chaotic state, characterized and different transitions between turbulent and pen-

by a strange attractor in phase space, and associated odic states.broad-band noise in the spectral density. In all cases W e have numerically studied the time evolution of

the perturbation of the system was caused by an ad- eq. (1) by three methods: (1) stroboscopic phase rep-

ditive external force. resentation, (2) power spectrum, and (3) Lyapunov

In this letter we would like to demonstratethat characteristic exponents. For fixed damping and fre-

there is yet another class ofexternally driven systems quency of the external modulation the character ofwhich might display apparently chaotic behaviour, the motion changes as the amplitude of the driving

The systems we have in mind are parametrically ex- force is varied. As an appropriate set of parameterscited nonlinear oscillators. As an examplew~investi- we chose f 3 =0.15 and & 2 = 1.56.

gated a damped parametrically driven mathematical For small values ofA all solutions of eq. (1) are

pendulum which can be described by the equation attracted by the stable fixed point x =0, ~ =0. For

A 0.3 16 the application of the averaging theoremx+~1v+(1+Acos~t)sinx=O. (1) . .

yields two periodic solutions, a stable limit cycle and

Such an external perturbation which acts upon a pa- an unstable one, which is between the former and therameter the amplitude of the back-driving force stable fixed point (0,0). With increasingA the un-

is easy to realize by a periodic vertical displacement stable limit cycle approaches the origin giving rise to

of the suspension point of the pendulum. With ~2giv- an inverted bifurcation which makes the origin un-en,A depends upon the amplitude of this vertical os - stable. Indeed a stable periodic solution P

1 with thecillation and is, in principle, not limited, period 4in/&l is found numerically forA

~0.337,

For small driving forces and displacements the so- which is invariant under the symmetry S: (x,~)lutions ofeq. (1) can be obtained by means of tech- -~(x,~).The fixed point (0,0) is found to be sta-

niques such as averaging or harmonic balance [5]. The ble in the interval 0 ~A~ 0.9075,

phase portraits can be analyzed in terms of simple lim- ForA 0.65 the limit cycle P1 becomes unstable

0031-9163/81/00000000/s 02.75 1981 North-Holland 71


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and a symmetry-breaking bifurcation [6] takes placegiving rise to a pair ofstable periodic orbits with the 25

same period which are mutual images under S.

With increasing A each of these limit cycles under- xgoes period doubling bifurcations until at A

1 0.7028

an apparently non-periodic attracting motion appears.

The stroboscopic phase portrait for A-values in the 2 J 3

vicinity ofA1 strongly depends upon the initial con-dition (x0,~0).So almost any trajectory which starts

in the neighbourhood of the limit cycle described

above approaches one of two distinct sets ofcurvesin the (x,~)or (x,~)quadrants, respectively, each 45 -1 0 -0.5

of them representing a small strange attractorS1. Fig. 1. Stroboscopic phase representation ofthe small

Some trajectories tend asymptotically to the fixed strange attractor S2 forA =0.915.

point (0, 0). The former trajectories are observed to

approach an orbit which represents a rotating pendu- ton ~2 is shown in f ig . 1 . (Ifone changes the initiallum. With increasing A the attractor ~1 becomes un- values (x0,~0)to (x0,-~0)the corresponding at-

stable. So for A 0.705 we could not find any initial tractor is found in the fourth quadrant.) Fig. 2 shows

condition yielding a trajectory forming the attractor the largest Lyapunov characteristic exponent [8] as aS1. All phase points either approached the rest point function ofA in the range 0.91 ~A ~ 0.93. For 0.91

(0,0) or moved to the rotating pendulum regime. In ~A A~there is no stable fixed point or stable periodic

ble in the range 0.5 19 ~A ~ 0.793. For higher val- orbit. The phase point is wandering in the phase plane

ues ofA period doubling bifuncations are observed, forming a large strange attractor, which is shown

Table 1 gives the A-values at which bifurcations take in fig. 3 forA =0.94. (Remember that whenever the

place and the Feigenbaum coefficients 6,~=(A~~

obviously tend to a limit value which seems to be . .

close to the value 6 =4.669... as given by Feigenbaum.ForA >0.910 735 56... the system again exhibits 0.15 -

chaotic behaviour. The corresponding strange attrac-0.1

Table 1

n A~ 005-

0 0,5191 0.79338 2.7962 0.89150 6.291 0

3 0.907097 5.5644 0.909900 4.201

5 0.9105672 5.028 0.91 0.92 0.93 A

6 0.91069997 4.738

7 0 91072799 Fig. 2. Variationofthe largest Lyapunov characteristicexpo-____________________________ ________________ nent X with the value ofthe exciting parameterA.

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Volume 86A,number 2 PHYSICS LETTERS 2 November 1981

2 ~

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- 2 0 0.25 0.5 0.75 w 1 .0Fig. 4. Fourier transformP(w) ofthe autocorrelation func-

Fig. 3. Stroboscopic phase representation ofthe large strange tion ofthevelocity forA =0.94.attractor for A =0.94.

phase point leaves the region i r ~ x


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has much in common withthat of the Lorenz system One of the authors (R.W.L.) would like to thank[12] and nonlinear oscillatory systems with an addi- Prof. J.L. Klimontovich (Moscow) for bringing the

tional external force, Because of the very simple equa- problems connected with strange attractors tohisat-

tion ofmotion we believe that the pendulum system tention.

studied in this paper is a promising basis for the de-tailed examination of many phenomena connected References

with the chaotic response of nonlinear oscifiators tope-riodic perturbations. In addition to the aspects dis- [1] P. Holmes, Phios. Trans. R. Soc. A292 (1979) 1394.

cussed we would like to refer to further interesting in- [2] Y. Ueda, J. Stat. Phys. 20 (1979) 181.vestigations which are in progress: the examination of [3] K. Tomita an d T. Kai,J. Stat. Phys. 21(1979)65.

[4] B.A. Huberman, J.P. Crutchfield and N.H. Packard, Appthe stable and unstable manifolds ofsaddle points, the Phys. Left. 37 (1980) 750.construction ofan equivalent difference equation on [5] H. Kauderer, Nichtlineare Mechanik (Springer, Berlin,R

2, the detailed study of the transition from the 1958).

small to the large attractor and of the fine struc- [6] I. Shimada an d T. Nagashima, Prog. Theor. Phys. 61ture of the attractors. Very interesting is also the in- (1979) 1605.

[7] M.J. Feigenbaum,J. Stat. Phys. 19(1978)25.vestigation ofthe transition of the chaotic behaviour [8] V.1. Oseledec, Tr. Mosk. Ova. 19 (1968) 179 ;

from dissipative to conservative systems. G. Benettin, L. Galgani and J.M. Strelcyn, Phys. Rev,A14 (1976) 2338.

Note added in proof. In a recent paper J.B. Mc [9] M. Henon, Commun. Math. Phys. 50 (1976) 69.

Laughlin (J. Stat. Phys. 24(1981)375) presented [10] J.L. Kaplan and l.A. Yorke,Commun. Math. Phys. 67(1979) 93.

some numerical solutions of eq. (1~ [11] P. Manneville an d Y. Pomeau, Physica 1D (1980) 219.

[12] EN. Lorenz, J. Atmos. Sci. 20 (1963) 130.

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chaotic behaviour of a parametrically exciteddamped pendulum

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