Nonlinear and Complex Systems

As a Ph.D. student, I perform research under the guidance of Professor Daniel Gauthier. My research is focused on nonlinear systems, in particular chaotic systems: systems that are predictable in theory, but not in practice (like the weather, where we can only trust forecasts for a few days not years). The question I am interested in is: if you are given several nonlinear systems that all have the potential to interact with and influence each other’s dynamics (a network of nonlinear systems) how and how much information can you glean about the network properties just by measuring the dynamics of a single unit? In general, this is called the “network inverse problem” [1] and has applications in many different fields. In neuroscience, for example, people are interested in how the different neurons in the brain are connected, and developing methods to determine that topology are important.

To answer this question, I have built and experimented with a particular type of nonlinear element: an optoelectronic oscillator (OEO). I couple several of these units together and measure how their dynamics change with different coupling strengths and configurations. I am particularly interested in what happens when all of the nodes are operating in what I call the "broadband chaotic regime." In this regime the dynamics are extremely complicated, with a broad, flat power spectrum, similar to that of white noise. The discovery and analysis of this type of behavior led to a first-author publication in Physical Review Letters [2] and a highlight article in Nature News and Views [3].  Additionally, I was asked to write a book chapter about the topic [4] and co-authored another paper about a different type of behavior I discovered with this device [5].

I am now using what I have discovered about the behavior of isolated OEOs to determine the topological properties of a network of coupled OEOs. Specifically, I have developed a method to simultaneously determine (1) which link is affected and (2) by how much when one of the coupling strengths in a small network of unsynchronized dynamical nodes is altered. After proper calibration, realizing this method involves only measurements of the dynamical features of a single node. I have also investigated ways to decrease the uncertainty in detecting both the position and strength of the attenuating element. I find that a node has enhanced sensing performance when it has: self-feedback, chaotic (rather than stochastic) dynamics, and parameter mismatch with respect to the other nodes in the network.

Physics Education Research (PER)

In my future research endeavors, I plan to take what I learned about analyzing data from complex systems and apply it to progressing our understanding of best teaching practices in physics.  The need for improvement over traditional teaching methods has been well documented [6], and recent progress has been made in the development and assessment of novel teaching methods [7]. At the same time, the gender gap and high attrition rates of females that have persisted in the physics discipline are also being investigated, which has lead to surprising findings. For example, a recent study found that a simple values affirmation writing exercise eliminated the gap in an introductory physics course [8].

I am interested in studying the intersection of these two ideas: the influence that new teaching methods have on the gender gap and attrition rates of female students, along with other minority populations in the field. In particular, I would like to study “flipped classrooms” (where the content delivery is done outside of class, and class time is instead used to apply the material using various activities) and massive online open courses (MOOCs, which bring educational opportunities to the public free of charge). One of the elements these two disparate educational approaches have in common is that they both deemphasize the role of the teacher as the sole authority on the subject at hand, while emphasizing the role of student-to-student interactions in the learning process.

While there are many different methods for flipping a classroom, I am particularly intrigued by the Team-Based Learning (TBL) method developed by Larry Michaelsen [10]. I helped a faculty member in my department try the TBL method in her introductory course for freshman physics majors. Anecdotally, we have both been extremely pleased with the kind of learning that takes place in this highly collaborative environment. In addition, before this method was implemented the course had a reputation for weeding out female physics majors. After implementing this method, however, it seems that there will be a much greater percentage of females sticking with the physics major than in recent years. I would like to investigate: (1) if this is true at other institutions that use flipped classrooms, (2) what factors of flipped classrooms contribute to the retention of females, and (3) what impact flipped classrooms have on the gender gap.

Additionally, if the opportunity arose, I would like to be involved in the development and assessment of MOOCs. Significant progress has been made in this field in the past few years, and there are many success stories. Elite institutions are rushing to jump on board so as not to be left behind, but so far controlled studies of the impact of MOOCs on students, educators, educational institutions and the public seem to be lacking. For example, as with flipped classrooms, I am curious what effects this type of educational experience has on females taking physics. Are the effects of stereotype threat altered? If so, why? The advantage of studying MOOCs is that data will not be hard to come by. Currently, the Coursera: Introduction to Astronomy course offered by a professor at Duke University has an active enrollment of over 15,000 people. The results gleaned from analyzing this wealth of data will certainly help make MOOCs better, but could also help inform best practices in other educational settings.
 
While my future research plans do not include traditional physics research, I still plan to provide opportunities for undergraduate physics majors to be involved. These students would benefit by learning skills common to any researcher, like how to extract meaningful conclusions from complicated data sets, in addition to having a chance to reflect on their own approach to learning. Not only will this help prepare them for future research in any physics subfield, but it will also extend the conversation about teaching and learning to the students and bring about a new level of mutual investment in their physics education.

References

  1. D. Yu, M. Righero, and L. Kocarev, ‘Estimating Topology of Networks,’ Phys. Rev. Lett. 97, 288701 (2006).
  2. K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schoell, ‘Broadband Chaos Generated by an Optoelectronic Oscillator,’ Phys. Rev. Lett. 104, 113901 (2010).
  3. L. Larger and J. M. Dudley, ‘Optoelectronic chaos,’ Nature 465, 41-42 (2010).
  4. K. E. Callan, L. Illing, and D. J. Gauthier, ‘Broadband Chaos,’ in Nonlinear Laser Dynamics: From Quantum Dots to Cryptography, K. Ludge (John Wiley-VCH Verlag, Weinheim, 2012), pp. 317-332.
  5. D. P. Rosin, K. E. Callan, D. J. Gauthier, E. Schoell, ‘Pulse-train solutions and excitability in an optoelectronic oscillator,’ Europhys. Lett. 96, 34001 (2011).
  6. Rising Above the Gathering Storm: Energizing and Employing America for a Brighter Economic Future. 2007, Washington, D.C.: National Academies Press; R. R. Hake, ‘Interactive-engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses,’ Am. J. Phys. 66, 64-74 (1998).
  7. J. Handelsman, et al., ‘Scientific Teaching,’ Science 304, 521-522 (2004).
  8. Akira Miyake, et al., ‘Reducing the Gender Achievement Gap in College Science: A Classroom Study of Values Affirmation,’ Science 330, 1234 (2010).
  9. D. X. Parmelee and L. K. Michaelsen, ‘Twelve tips for doing effective Team-Based Learning (TBL),’ Med. Teach. 32, 118-122 (2010).