H. J. T. Smith

Stability and Hopf bifurcations in an inverted pendulum

The inverted state of a simple pendulum is a configuration of unstable equilibrium. This instability may be removed if the pivot is harmonically displaced up and down with appropriate frequency and amplitude. Numerical simulations are employed to investigate the stable domains of the system. The associated basins of attraction, extracted by interpolated cell mapping, are seen to be fractal. Loss of stability at high excitation amplitudes is observed to follow a Hopf bifurcation.

Experimental study of chaos in a driven pendulum

Abstract Experimental results are reported for a driven, damped pendulum in which steady and / or alteranting torques were applied by means of a modified brushless, slotless linear motor. The pendulum coordinate was measured with an angular resolver which, in combination with an integrated circuit resolver-to-digital converter, produced readings with fourteen-bit precision. Hysteresis and ac induced steps are observed, as expected, in the torque-velocity characteristics for the pendulum. Poincare sections, power spectra and a state diagram confirm the existence of separate regions of periodic and chaotic behaviour. This system is of particular interest because it can also serve as a mechanical analog of a current biased Josephson junction.

A stochastic model of synchronization for chaotic pendulums

Abstract Uni-directionally coupled chaotic pendulums are studied with particular emphasis on the long term locking time distribution as a function of coupling strength. Numerical simulation data is analysed with a simple stochastic model that is also shown to satisfy a principle of maximum entropy.

The quantum pendulum: Small and large

The quantum pendulum finds application in surprising contexts. We use commercially available software to numerically solve the Schrodinger equation for a microscopic pendulum subject to molecular (electromagnetic) restoring forces, and a macroscopic pendulum subject to a gravitational restoring force. The dynamics of the microscopic quantum pendulum are closely related to molecular motions known as hindered rotations. We use standard probabilistic methods to predict whether this motion is weakly or strongly hindered at ambient temperature and test the prediction against experimental data for C2H6 and K2PtCl6. For the macroscopic gravitational pendulum, we examine the uncertainty in position and find, not surprisingly, that it is too small to measure physically, but is nevertheless relatively large compared to present-day limits in computation. The latter juxtaposition of computational precision with quantum uncertainty has consequences for the study of chaotic dynamics.

ELECTRON TUNNELING STUDY OF AMORPHOUS Pb--Bi SUPERCONDUCTING ALLOYS.

Abstract Electron tunneling studies carried out on amorphous Bi and Ga strong-coupling superconductors have been extended to a series of amorphous Pb-Bi superconducting alloys, ranging from 100% Pb to 100% Bi. The energy gap and transition temperature have been measured for each alloy composition, both in the amorphous state and after annealing to obtain the crystalline state. Phonon spectra have been derived for the amorphous state of the alloys by using the McMillan inversion program on our derivative tunneling data. Strong-coupling parameters derived from our phonon spectra have been used to test recent theories of the transition temperature of amorphous superconductors. The results are in reasonable agreement with the theories of Benda and of Garland et al.

Self‐Resonant Current Peaks in Josephson Junctions

A finite difference scheme is described which yields solutions of the Josephson phase equation. This simulation technique is used to calculate the self‐resonant current peaks of a Josephson junction for the two cases, R≫ZJ and R≪ZJ, where R is the current drive source resistance and ZJ is the junction impedance. The simulation current peaks are compared with ones derived by a perturbation technique for the same two regimes, and also with experiment. In terms of approximating experimental results, it is found that the perturbation solutions with constant damping, σ, are better than those with constant Q; although at small σ the magnitude and shape of the current peaks do not agree well with observations. However, simulation results at small σ are reasonably consistent with experiment. Simulation results for the case R≪ZJ indicate that self‐limiting will affect the magnetic field dependence of the resonances when L/λJ≥2, where L is the length of the junction and λJ is the Josephson penetration depth.

Dynamical properties of Josephson junctions coupled by a transmission line

A system composed of two Josephson junctions connected by a transmission line has been studied by means of electronic analog simulation. Under external current bias, the resistive component of the coupling induces frequency locking between the two junctions at commensurate ratios. The resonant modes of the transmission line give rise to steps in the I‐V characteristics of the system.

Josephson junctions with delayed feedback

Abstract We study a simple model of an overdamped Josephson junction coupled to a transmission line, which is regarded as a delayed feedback to the junction. It is demonstrated analytically how the nonlocal time dependence can give rise to hysteresis and steps in the current-voltage characteristics of the junction and the fundamental difference between positive and negative feedback is discussed. Excellent agreement between the analytical results and the results of numerical simulations is found.

Dynamics of double-Josephson-junction interferometers

The behavior of a superconducting loop containing two Josephson junctions has been studied by means of computer simulations. It is found that a current‐biased system can exhibit various dynamical modes in which flux quanta alternately enter and leave the loop through the junctions. The nature of these coupled oscillations and their dependence on device parameters are discussed.

STRONG-COUPLING SUPERCONDUCTIVITY IN AMORPHOUS BISMUTH.

Abstract Tunneling measurements indicate that amorphous bismuth is a strong coupling superconductor with 2 Δ / k B T c = 4.60 ± 0.05, and show evidence of phonon structure in the d V /d I versus V curves.