Gregory L. Baker

Inverting chaos: Extracting system parameters from experimental data

Given a set of experimental or numerical chaotic data and a set of model differential equations with several parameters, is it possible to determine the numerical values for these parameters using a least-squares approach, and thereby to test the model against the data? We explore this question (a) with simulated data from model equations for the Rossler, Lorenz, and pendulum attractors, and (b) with experimental data produced by a physical chaotic pendulum. For the systems considered in this paper, the least-squares approach provides values of model parameters that agree well with values obtained in other ways, even in the presence of modest amounts of added noise. For experimental data, the fitted and experimental attractors are found to have the same correlation dimension and the same positive Lyapunov exponent. (c) 1996 American Institute of Physics.

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.

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-

WHEN TWO COUPLED PENDULUMS EQUAL ONE: A SYNCHRONIZATION MACHINE

We show that two coupled pendulums that are coupled and can synchronize, are mathematically equivalent to one horizontal parametrically driven pendulum. We have fabricated a horizontal pendulum and present data from this horizontal pendulum which we believe to be the first physical realization of such a mechanical synchronization machine. A description of intermittent synchronization that can occur when two coupled pendulums are in a chaotic state is given in terms of the data from the horizontal pendulum. We discuss the relationship between the modes of the horizontal pendulum and the corresponding synchronization of the two coupled pendulums. Finally, we show that when a horizontal pendulum is driven by any random source, not necessarily chaotic, intermittent synchronization can occur.

Probability, pendulums, and pedagogy

Deterministic pendula exhibit a spectrum of behavior ranging from periodic to chaotic and provide an opportunity for an introductory discussion on the application of probability techniques to a deterministic system. Analytic and simulation techniques are used to determine probability distributions for a range of dynamical possibilities. In particular, we obtain probability distributions of the pendulum’s angular displacement and distributions of first return times for regular and chaotic motion. For chaotic motion, the latter distribution is modeled by a simple two-state Bernoulli process. Further considerations suggest that not all distributions are probability distributions.

EXPERIMENTAL OBSERVATION OF INTERMITTENCY IN COUPLED CHAOTIC PENDULUMS

We have experimentally realized on–off intermittency in a pair of mutually coupled damped driven pendula. The pendula interacted bidirectionally via a torque which was proportional to the difference in their angular velocities. Experimental data show that the intervals of synchronized chaotic motion are distributed in agreement with the theory for on–off intermittency: The conditional probability of short laminar times is found to follow a power law whilst the conditional probability of long laminar times follows the expected exponential law. The experimental data also reveal that the intermittent properties vary with the coupling strength, and that the average laminar time inferred from the results depends on the choice of threshold level.

Chaotic dynamics: Introduction

1. Introduction 2. Some helpful tools 3. Visualization of the pendulums dynamics 4. Toward an understanding of chaos 5. The characterization of chaotic attractors 6. Experimental characterization, prediction, and modification of chaotic states 7. Chaos broadly applied Further reading Appendix A. Numerical integration - Runge-Kutta method Appendix B. Computer program listings References Index.