James A. Blackburn

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.

Resonant Steps in the Characteristics of a Josephson Junction Coupled to a Transmission Line

A novel circuit is described which functions as an electronic analog of lumped element transmission line. The circuit requires only operational amplifiers, resistors, and capacitors. This module was coupled to a Josephson junction simulator and current voltage characteristics of the combined system were recorded. Steps were observed at voltages determined by the appropriate line resonances. When the transmission line was terminated with loads less than the characteristic impedance, chaos was seen in the lower steps. Similar results were obtained by numerical integration of the corresponding system of differential equations.

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.

Driven pendulum for studying chaos

An instrument suitable for experimental studies of chaotic motion is described. It consists of a mechanical pendulum driven by both steady and alternating torques, together with interface electronics for data logging. An optical shaft encoder serves as the transducer for measuring the instantaneous pendulum coordinate. The entire system is controlled from a personal computer. Sample phase‐plane and Poincare plots are presented.

Single and multiple superconducting weak-link systems

Using the one-dimensional Ginzburg-Landau equation and the boundary conditions of Zaitsev, we have evaluated the order parameters in single and multiple weak-link systems. From this information the current-phase characteristics were computed and it was found that Josephson-type behavior occurs for an intermediate range of link parameters. Other pertinent data, including associated thermodynamic free energies of the various solutions, have been obtained.

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.

A comparison of commercial chaotic pendulums

Chaos is an important and fundamental aspect of contemporary nonlinear physics. While numerical simulation of chaos is inexpensive and relatively straightforward, it is difficult and time consuming to construct physical apparatus that actually demonstrates chaos quantitatively. Most institutions must therefore resort to commercially available equipment. Yet these devices are fairly expensive for colleges to purchase and, furthermore, catalog descriptions, by nature, tend to emphasize the strongest features in a given design and to highlight visually pleasing aspects in the illustrations. As a consequence, there remain many questions. What physics does the device actually model? How well does it model the physics? Is it user friendly? Is it sturdy? Can students do a meaningful series of laboratory exercises with the pendulum? For what level is a particular device suited? These are some of the issues we raise in this paper. We have bench tested three commercially produced chaotic pendulums and report the results as a service to physics educators. While similar in their purposes as experimental chaos platforms, each is based on a slightly different paradigm and in consequence each presents a slightly different window on the chaotic world. Strengths and weaknesses of the design approaches are reviewed here. Since these units are moderately expensive, it is important to choose carefully the pendulum that best suits individual educational and possible research needs. We are aware of four commercially produced chaotic pendulums. These are manufactured by ~in alphabetical order! Daedalon Corp., Leybold, Pasco Scientific, and TELAtomic, Inc. We contacted all four manufacturers with invitations to participate in this project—all but Leybold ultimately did so. One author ~JAB! brings significant practical experience to the discussion as he is a codesigner of the pendulum manufactured by Daedalon. Testing of the apparatus was carried out in his laboratory at Wilfrid Laurier University. For the record, we acknowledge this special circumstance. As an assurance of evenhanded treatment for all participants, each company was given the opportunity to read a preliminary draft of this report and to suggest recommendations to that portion of the manuscript that dealt with their product. There exist various pendulums and other chaotic devices whose designs have appeared in the literature but which are not commercially available. The recent AJP resource letter on nonlinear dynamics gives references to a whole variety of chaotic devices, many of which have been in this journal. However, this article is directed at those who may have neither the time nor the facilities to construct the type of equipment described in the literature and are therefore interested in making an informed purchase of a pendulum. For the most part we treat each pendulum separately, fo-

A survey of classical and quantum interpretations of experiments on Josephson junctions at very low temperatures

For decades following its introduction in 1968, the resistively and capacitively shunted junction (RCSJ) model, sometimes referred to as the Stewart-McCumber model, was successfully applied to study the dynamics of Josephson junctions embedded in a variety of superconducting circuits. In 1980 a theoretical conjecture by A.J. Leggett suggested a possible new and quite different behavior for Josephson junctions at very low temperatures. A number of experiments seemed to confirm this prediction and soon it was taken as a given that junctions at tens of millikelvins should be regarded as macroscopic quantum entities. As such, they would possess discrete levels in their effective potential wells, and would escape from those wells (with the appearance of a finite junction voltage) via a macroscopic quantum tunneling process. A zeal to pursue this new physics led to a virtual abandonment of the RCSJ model in this low temperature regime. In this paper we consider a selection of essentially prototypical experiments that were carried out with the intention of confirming aspects of anticipated macroscopic quantum behavior in Josephson junctions. We address two questions: (1) How successful is the non-quantum theory (RCSJ model) in replicating those experiments? (2) How strong is the evidence that data from these same experiments does indeed reflect macroscopic quantum behavior?

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.