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Chaos in a superconducting saucer
Physica B 165&166 (1990) 117-118 North-Holland
CHAOS IN A SUPERCONDUCTING
SAUCER
G DAVIES, C J LAMBERT, N S LAWSON, R A M LEE, R MANNELLA, P V E McCLINTOCK STOCKS.
and N G
Department of Physics, Lancaster University, Lancaster, LA1 4YB.
The dynamical behaviour of a very small permanent magnet levitating in a vacuum above a superconducting saucer at 4.2K has been investigated. The arrangement, which constitutes a near-Hamiltonian system, able to undergo both rotational and orbital motions, is found to exhibit a rich variety of behaviours, including chaos. Its relationship to chaotic motion in the solar system is discussed. 1. INTRODUCTION The motions of some members of the Solar System notably the tumbling of Hyperion (1) and, arguably, the orbital dynamics of the inner planets (2) - are believed to be chaotic. Such behaviour is difficult to investigate directly because of the extremely long time scales involved. In this paper we describe an experiment in which some aspects of the motion can, however, be modelled conveniently in the laboratory. 2. EXPERIMENTAL DETAILS Following a suggestion by A B Pippard (3), the investigation is based on measurements of the dynamical behaviour of a very small magnet levitating in a vacuum above a superconducting lead saucer, of 50 mm radius curvature, immersed in liquid helium at 4.2 K as sketched in Fig 1. Such an arrangement constitutes a near-Hamiltonian system, able to display both rotational and orbital motions, and with a finite spin-orbit coupling. In practice, the motion of the magnet is adjusted initially with the aid of a periodic drive applied to a set of four coils mounted slightly above the plane of the lead saucer. With appropriate choices of frequency, and of the phase shifts between the voltages applied to the coils, the magnet could be made to move in an elliptical orbit or (with some difficulty) to spin rapidly on its axis in the centre of the bowl. In the most general case, the magnet would be spinning as well as orbiting the inside of the bowl. Once any particular desired motion has been achieved, the drive wss switched off. An opposed pair of the same coils wss then used, instead, to pick up the induced emf from the moving magnet. The signal was taken to a high input impedance (to reduce damping of the motion) pre-ampliier and thence to a Nicolet 1180 data processor, able to record 512 K points of a discrete time
series at uniform (typically 1 ms) intervals over an interval of several minutes, storing the data continuously on disk. Results were recorded for several different neodymium magnets of different aspect ratio, all of which were of square cross-section and symmetric about their magnetic axes. The ratio of the lengths along their magnetic axes to either of the other two (equal) dimensions varied from 0.34 to 2.5: their shapes therefore varied between those approximating a “flake” and a “rod” respectively.
3. RESULTS Some six traces of typical data obtained from a N 1 mm cubical magnet are shown in Fig 2; the full It must be time series consists of 94 such traces. emphasized that the measurements are made in the absence of any externally applied driving force, with the magnet orbiting freely in the vacuum, almost without dissipation. Thus the various strikingly sudden changes in behaviour (intermittency) observable in Fig 2 are all occurring quite spontaneously and not as the result of outside intervention.
"Hyperion"
vacuum
superconductlng lead bowl FIGURE 1 Sketch of the experimental arrangement. The magnet labelled “Hyperion” orbits in an approximately horizontal plane; it also tumbles about its own axes.
0921-4526/90/$03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)
118
G. Davies et al.
Damping of the motion though small, was not zero: the rate of decay of amplitude was found to depend on the type of motion, but its time constant was typically N 15 - 30 minutes. In practice, unless the coils (of resistance _ 48) were deliberately short-circuited, so as to damp the motion and bring the system quickly to rest, the magnet was always found to be moving. It is believed that the observed damping arose mainly as the result of vertical movements, inducing dissipative eddy currents within the body of the magnet itself: this effect was naturally more pronounced in some types of orbit than in others. 4. DISCUSSION
To try to obtain some quantitative measure of the extent to which the motion was chaotic, the largest Lyapunov exponent was calculated for each time series by use of the algorithm proposed by Wolf, Swift, Swinney and Vastano (4). Essentially, an embedding space was constructed by extracting multidimensional “points” from the time series. This wss done by taking as a “point” an array of values separated by a constant delay; the multidimensional point wss then followed for a certain time and its rate of divergence was calculated. The dominant Lyapunov exponent, calculated in this way for each of the 94 traces of the (single, continuous) time series of which part is shown in Fig 2, is plotted against trace number (time) in Fig 3. It can be seen: (a) that the exponent is always positive, immediately indicating chaotic behaviour; and (b) that it tended to decrease with the passage of time. The latter effect probably arose because the more extravagant vertical tumbling motions suffer greater damping (see above) and consequently tend to die away relatively fast. The particular example illustrated (Fig 2) was selected because it looked especially chaotic to the eye.
FIGURE 3 The dominant Lyapunov exponent calculated for individual traces of the time-series shown in part in Fig 2, plotted against the trace number N. Each trace represents an interval of 2.8 s.
Other, more regular looking, motions were also analysed. They yielded smaller Lyapunov exponents that were still positive but which did not significantly decrease during the acquisition of a time series. The Hamiltonians of Hyperion and of the magnet, which are of a similar structure, are not of course, the same, even in the limits of zero damping and of dipole coupling between the magnet and its image in the saucer. One must be careful not to push the analogy too far. Nontheless, the experiments have revealed the characteristic chaotic behaviour of a body tumbling in its orbit, with a maximum of 12 degrees of freedom, under almost dissipationless conditions, in the presence of finite spin-orbit coupling. 5. CONCLUSION We have demonstrated dynamical chaos in the labo ratory in a near-Hamiltonian system. The observed phenomena may be of relevance to comparable behaviour observed in celestial mechanics. ACKNOWLEDGEMENTS It is a pleasure to acknowledge stimulating discussions or correspondence with N B Abraham, G P King and A B Pippard.
FIGURE 2 The emf induced in a pair of 6xed coils by the moving magnet, plotted against time. For convenience of plotting, the complete time series has been broken into 94 horizontal traces of which 6 are reproduced here.
REFERENCES 1. J. Wisdom, S. F. Peale and F. Mignard, Icarus 58 (1984) 137. 2. J. Lsskar, Nature 338 (1989) 237. 3. Suggestions made at the Royal Society Discussion Meeting on Dynamical Chaos, 5 February 1987. 4. A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Physica 16D (1985) 285.
CHAOS IN A SUPERCONDUCTING
SAUCER
G DAVIES, C J LAMBERT, N S LAWSON, R A M LEE, R MANNELLA, P V E McCLINTOCK STOCKS.
and N G
Department of Physics, Lancaster University, Lancaster, LA1 4YB.
The dynamical behaviour of a very small permanent magnet levitating in a vacuum above a superconducting saucer at 4.2K has been investigated. The arrangement, which constitutes a near-Hamiltonian system, able to undergo both rotational and orbital motions, is found to exhibit a rich variety of behaviours, including chaos. Its relationship to chaotic motion in the solar system is discussed. 1. INTRODUCTION The motions of some members of the Solar System notably the tumbling of Hyperion (1) and, arguably, the orbital dynamics of the inner planets (2) - are believed to be chaotic. Such behaviour is difficult to investigate directly because of the extremely long time scales involved. In this paper we describe an experiment in which some aspects of the motion can, however, be modelled conveniently in the laboratory. 2. EXPERIMENTAL DETAILS Following a suggestion by A B Pippard (3), the investigation is based on measurements of the dynamical behaviour of a very small magnet levitating in a vacuum above a superconducting lead saucer, of 50 mm radius curvature, immersed in liquid helium at 4.2 K as sketched in Fig 1. Such an arrangement constitutes a near-Hamiltonian system, able to display both rotational and orbital motions, and with a finite spin-orbit coupling. In practice, the motion of the magnet is adjusted initially with the aid of a periodic drive applied to a set of four coils mounted slightly above the plane of the lead saucer. With appropriate choices of frequency, and of the phase shifts between the voltages applied to the coils, the magnet could be made to move in an elliptical orbit or (with some difficulty) to spin rapidly on its axis in the centre of the bowl. In the most general case, the magnet would be spinning as well as orbiting the inside of the bowl. Once any particular desired motion has been achieved, the drive wss switched off. An opposed pair of the same coils wss then used, instead, to pick up the induced emf from the moving magnet. The signal was taken to a high input impedance (to reduce damping of the motion) pre-ampliier and thence to a Nicolet 1180 data processor, able to record 512 K points of a discrete time
series at uniform (typically 1 ms) intervals over an interval of several minutes, storing the data continuously on disk. Results were recorded for several different neodymium magnets of different aspect ratio, all of which were of square cross-section and symmetric about their magnetic axes. The ratio of the lengths along their magnetic axes to either of the other two (equal) dimensions varied from 0.34 to 2.5: their shapes therefore varied between those approximating a “flake” and a “rod” respectively.
3. RESULTS Some six traces of typical data obtained from a N 1 mm cubical magnet are shown in Fig 2; the full It must be time series consists of 94 such traces. emphasized that the measurements are made in the absence of any externally applied driving force, with the magnet orbiting freely in the vacuum, almost without dissipation. Thus the various strikingly sudden changes in behaviour (intermittency) observable in Fig 2 are all occurring quite spontaneously and not as the result of outside intervention.
"Hyperion"
vacuum
superconductlng lead bowl FIGURE 1 Sketch of the experimental arrangement. The magnet labelled “Hyperion” orbits in an approximately horizontal plane; it also tumbles about its own axes.
0921-4526/90/$03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)
118
G. Davies et al.
Damping of the motion though small, was not zero: the rate of decay of amplitude was found to depend on the type of motion, but its time constant was typically N 15 - 30 minutes. In practice, unless the coils (of resistance _ 48) were deliberately short-circuited, so as to damp the motion and bring the system quickly to rest, the magnet was always found to be moving. It is believed that the observed damping arose mainly as the result of vertical movements, inducing dissipative eddy currents within the body of the magnet itself: this effect was naturally more pronounced in some types of orbit than in others. 4. DISCUSSION
To try to obtain some quantitative measure of the extent to which the motion was chaotic, the largest Lyapunov exponent was calculated for each time series by use of the algorithm proposed by Wolf, Swift, Swinney and Vastano (4). Essentially, an embedding space was constructed by extracting multidimensional “points” from the time series. This wss done by taking as a “point” an array of values separated by a constant delay; the multidimensional point wss then followed for a certain time and its rate of divergence was calculated. The dominant Lyapunov exponent, calculated in this way for each of the 94 traces of the (single, continuous) time series of which part is shown in Fig 2, is plotted against trace number (time) in Fig 3. It can be seen: (a) that the exponent is always positive, immediately indicating chaotic behaviour; and (b) that it tended to decrease with the passage of time. The latter effect probably arose because the more extravagant vertical tumbling motions suffer greater damping (see above) and consequently tend to die away relatively fast. The particular example illustrated (Fig 2) was selected because it looked especially chaotic to the eye.
FIGURE 3 The dominant Lyapunov exponent calculated for individual traces of the time-series shown in part in Fig 2, plotted against the trace number N. Each trace represents an interval of 2.8 s.
Other, more regular looking, motions were also analysed. They yielded smaller Lyapunov exponents that were still positive but which did not significantly decrease during the acquisition of a time series. The Hamiltonians of Hyperion and of the magnet, which are of a similar structure, are not of course, the same, even in the limits of zero damping and of dipole coupling between the magnet and its image in the saucer. One must be careful not to push the analogy too far. Nontheless, the experiments have revealed the characteristic chaotic behaviour of a body tumbling in its orbit, with a maximum of 12 degrees of freedom, under almost dissipationless conditions, in the presence of finite spin-orbit coupling. 5. CONCLUSION We have demonstrated dynamical chaos in the labo ratory in a near-Hamiltonian system. The observed phenomena may be of relevance to comparable behaviour observed in celestial mechanics. ACKNOWLEDGEMENTS It is a pleasure to acknowledge stimulating discussions or correspondence with N B Abraham, G P King and A B Pippard.
FIGURE 2 The emf induced in a pair of 6xed coils by the moving magnet, plotted against time. For convenience of plotting, the complete time series has been broken into 94 horizontal traces of which 6 are reproduced here.
REFERENCES 1. J. Wisdom, S. F. Peale and F. Mignard, Icarus 58 (1984) 137. 2. J. Lsskar, Nature 338 (1989) 237. 3. Suggestions made at the Royal Society Discussion Meeting on Dynamical Chaos, 5 February 1987. 4. A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Physica 16D (1985) 285.