Barbara J. Breen

Background

My thesis research involved computational modeling of nonlinear systems. Most nonlinear systems cannot be analytically solved, and computational methods  are invaluable in understanding them. Working in the C/C++ environment, I brought the power of computational nonlinear dynamics to bear on two significant problems. The first was generalizing the phenomenon of monostable stochastic resonance to arrays (taking advantage of the Cluster Computing Initiative at Georgia Tech’s Interactive High Performance Computing Lab) and the second involved finding a better numerical algorithm for modeling a bursting neuron (applying the tools of nonlinear dynamical analysis to an existing model).

In 2011-2014 I was a research fellow at Auckland Bioengineering Institute (ABI) in the Lung and Respiratory Systems Group. With Merryn Tawhai I developed a 3D finite element model of an isolated human airway that incorporates passive (constitutive tissue properties) and active (contraction of airway smooth muscle) mechanics.  This work was done using CMISS, the finite element modeling software developed at ABI. A second project, with Graham Donovan, utilized MATLAB to simulate a multi-scale model of the human lung to investigate  aspects of asthmatic remodeling on whole-lung impedance. As part of that investigation I also sought to improve our algorithm for whole-lung impedance, which is based on an RIC circuit.

Computer modeling offers a scaffold between theoretical questions and experimental implementations. Collaborations are common, and I have found ways to involve students of almost any skill level in research. The opportunities for students range from acquiring programming skills to contributing to peer- reviewed articles. In collaboration with John Lindner of The College of Wooster, I have also been able to successfully develop research projects that incorporate experimental research in conjunction with computational simulations. At both The University of Portland (2006-2010) and Grinnell College (2014–present) I mentored undergraduate research assistants in collaborative endeavors where the computational aspect of the projects supported the development of experimental  implementations.

past work WITH UNDERGRADUATEs

At Grinnell College, my students and I modeled and built an elegant table-top experiment that demonstrated stochastic resonance (SR). Originally proposed as a mechanism for the ice ages, SR has been observed in electrical, optical, and biological systems. It is a seemingly counterintuitive nonlinear phenomenon where an increase in input noise improves a system’s ability to detect weak signals. Our experiment used a simple bistable torsional pendulum as a nonlinear device, a mechanical driver for a low amplitude signal, and harvested noise delivered by the ambient energy of a flapping flag driven by a high-velocity fan. The simulations were done in Mathematica, and they refined the design of the experiment at each step. Our experiment demonstrated, in essence, a cooperative effect in which a small periodic signal entrained external random noise – provided by the flapping flag. The final instantiation was made with 3D- printed parts designed in Mathematica and printed via shapeways.com. Our work was published in PRE in 2016 [1]. 

Most recently I have worked with two Grinnell students developing two unique computational projects. One project was inspired by analysis of light flux time series from the Kepler space telescope. As part of Kepler’s exoplanet search, the data archive contains over four years of light flux data from a formidable variety of stars. A University of Hawaii study demonstrated that light flux from some variable stars could be modeled by a nonlinear system driven by two incommensurate frequencies. This model allows construction of a fractal, non-chaotic attractor. However, the effect of data gaps raises the question: how well do nonlinear metrics remain robust in the presence of gaps in time series? The Kepler telescope shuts down irregularly. Events such as cosmic ray strikes, monthly science data downlinks, and safe modes cause shifts, skews and gaps in the data.

Our project exploited known metrics for the Lorenz and Rössler attractors. We simulated gaps in the time series, varying the mean gap size and mean frequency of gaps. We were able to document at what point nonlinear metrics such as correlation dimension failed. Subsequent stages of the research included testing algorithms for back-filling the gaps to find those that preserve as much original information as possible. This is an onging project to be pursued with more complex models that reflect characteristics of Kepler data, and ultimately analyze actual flux curves downloaded from the Kepler archive.

The second project involved simulations that “grow” point particle billiards to superballs: spheres with mass, dimension and angular momentum. Billiards mechanics have a long, illustrious history of providing meaningful insight into a variety of problems which have analogs in many fields of physics. A billiard dynamical system built of small rotating spheres rather than point particles is far less studied. Restricting motion to a plane, we enforce “no-slip” conditions that conserve energy, linear and angular momentum upon collision at the boundary.

Starting with simple canonical bounding spaces such as parallel plates, wedges, triangles, circles and stadia, we made excellent progress characterizing these orbits and generating phase space portraits. We learned a lot about the ways in which the behavior of point particles and superballs diverge under identical initial conditions. This in an ongoing project as I continue to seek nonlinear metrics that can elucidate the dynamical behavior. So far, we have found it useful to calculate fractal dimension of the phase space, find Lyapunov exponents of ergodic trajectories, and create bifurcation diagrams that elucidate transitions from periodic to ergodic trajectories.

While at the University of Portland (UP), my research focused on both computer simulations and experimental construction of arrays of nonlinear one-way coupled oscillators. One-way arrays of bistable elements might seem impossible at first as the equations describing them violate both energy and momentum conservation. However they are possible if the coupling is powered. The behavior of these arrays depends critically on their parity: in odd arrays solitary waves or solitons propagate endlessly; in even arrays, soliton pairs chase each other until they annihilate, leaving the array quiescent. They are mechanical analogs of electronic ring oscillators.

In 2007 we simulated computational models in the C/C++ environment that probed the parameter space in 1D and in the next two years, in collaboration with The College of Wooster, we built several versions of a water-powered mechanical array that confirmed the predictions of our simulations. My last summer at UP, 2010, my students improved the 1D hydro-mechanical array by using 3D-printed parts for uniformity, executed simulations of a 2D array, and built a functional prototype of a 2D electronic array. This work resulted in peer- reviewed publications that included undergraduate co-authors [1,2,3].

Future Work with Undergraduates

Stochastic Resonance & Noise Enhanced Propagation

One  of  my  research  proposals  for  ongoing  work  with  undergraduates  is  to continue the work begun at Grinnell this past summer. Based on what we have learned, we can use our Mathematica simulations to improve this novel bistable mechanical system that exhibits enhanced signal detection in the presence of noise generated by a flapping flag. I am interested in extending the simulation to investigate the propagation of a signal in an array of these oscillators. If we successfully improve the experimental device, we would attempt to build this array, an example of noise- enhanced propagation [5].

There is much interest in the concept of ambient energy harvesting as a way of powering small-scale mobile devices.  Vibration energy is being heavily investigated due to the almost universal presence of mechancial vibrations.

Standard approaches are overwhelmingly based on resonant linear oscillators that are acted on by ambient vibrations. If we succeed in exploiting the dynamical featues of stochastic nonlinear oscillators to harvest ambient vibrational energy it will be of great interest to this research community.

Aerodynamic One-Way Arrays

I am also interested in returning to the array of one-way coupled oscillators that we developed at UP. We powered our 1D array with falling water, and built a prototype of an electronic 2D array. My colleague at Wooster has implemented a short, 7-element wind- powered array prototype. The parity- specific behaviour of one-way coupling can’t be seen in an array this short, and the aerodynamic design of the individual elements (especially the wing and deflector) needs to be optimized either via computational fluid dynamics or wind tunnel testing, which can readily be accomplished with simple equipment.

With an improved design, we would attempt to build a longer array and observe soliton annihilation and creation dynamics in the presence of noise and disorder [2,3]. I anticipate that extensions of this work will include the simulation and design of tristable elements, with analysis leading to a generalization of parity- sensitive behaviour of such arrays.

References

1.   PRE 94: 062205 (2016)

2.   AJP 76(4/5), 347-352 (2008)

3.   PRE 78, 066604 (2008)

4.   PRE 83, 037601 (2011)

5.   PRL 81, 5048-5051 (1998)