Instructor Interview
Below, Prof. Daniel Rothman describes various aspects of how he taught 12.006J Nonlinear Dynamics: Chaos in the fall of 2022.
OCW: For the uninitiated, what is (are?) nonlinear dynamics about? In what disciplines are nonlinear dynamics and chaos especially relevant?
Daniel Rothman: In this course, “dynamics” refer to systems composed of just a few variables that vary with time. “Nonlinear” dynamics means that the variables may depend on each other in nonlinear way, so that, for example, the motion of an object is not simply proportional to the force exerted on it. For example, if you push lightly on a heavy block at rest on a table, it does not move. But it does move if you push hard enough. Nonlinearity exists in virtually all systems—not just physical and biological systems, but also the social sciences. It is especially relevant in any system where “tipping points” appear to exist.
OCW: 12.006J is taught every year, but not usually by the same professor/instructor. To what extent are the content and structure of the course fixed from year to year? To what extent did you custom-design it when you taught the subject?
Daniel Rothman: I created the course myself, long ago (in 1989!). Eventually it became co-listed in Math and MechE (Mechanical Engineering), and faculty in Math and MechE have also taught it.
Fall 2022 was the first time I taught the course since 2006. In returning to it, I significantly revised the content, not only to reflect problems of significant current interest (such as tipping points in climate science and ecology) but also because my own view of the subject has evolved. Although the basic content of the subject does not vary from instructor to instructor, each instructor ends up emphasizing the sorts of problems and viewpoints they consider to be most important.
OCW: Did all students, by the time they took this course, already have experience using Python and/or MATLAB to model real-world problems? Did you have to provide any supports to students who came to the course without this prior experience?
Daniel Rothman: Many but not all of the students came to the course with this experience. As needed, we worked with students individually to help them acquire computing skills.
OCW: The final project is intended to encourage students to think imaginatively about nonlinear dynamics. Do many students have difficulty with this?
Daniel Rothman: I thought some would, but nearly all our final projects were independently motivated and very well done. Indeed, it was inspiring to see how good they were. So I plan to give a bit more attention to this in the future, because it is the one part of the course where students genuinely have a chance to express their own interests.
OCW: How can an instructor nurture imaginative thinking, as opposed to competent mastery of a given body of material?
Daniel Rothman: That’s a good question! We try to do that by asking questions that require exploration of how a system works. This typically involves a combination of mathematical theory and numerical simulation.
Nonlinear systems exhibit much surprising, non-intuitive behavior, so learning how they work generates many questions for students to explore. Indeed, many of the most important discoveries in the subject followed a similar path. So one of our goals is to recreate that sense of discovery in the problem sets.
OCW: What would you like to share about teaching 12.006J that we haven’t yet addressed?
Daniel Rothman: It is tremendously fun to teach!
Curriculum Information
Prerequisites
- 8.02 Physics II: Electricity And Magnetism, 8.021 Physics II: Electricity And Magnetism, or 8.022 Physics II: Electricity And Magnetism
- 18.03 Differential Equations or 18.032 Differential Equations
Requirements Satisfied
- 12.006J can be applied toward a Bachelor of Science in Earth, Atmospheric, and Planetary Sciences with a concentration in Planetary Science and Astronomy, but is not required.
- 12.006J can be applied toward a Bachelor of Science in Mechanical Engineering, but is not required.
- 12.006J can be applied toward a Bachelor of Science in Mathematics with a concentration in Applied Mathematics, but is not required.
Offered
Once a year, in the fall semester
Assessment and Grading
Students’ grades were based on the following activities:
- 80% Problem sets
- 20% Final project
Student Information
Enrollment
Fewer than 10 students
Student Background
About one-third of the students were Math majors, another third were either Physics or Earth and Planetary Sciences majors, and the remainder came from other departments. All students were familiar with ordinary differential equations; i.e., they had successfully completed 18.03.
How Student Time Was Spent
During an average week, students were expected to spend 12 hours on the course, roughly divided as follows:
Lectures
- Met twice per week for 1.5 hours per session; 26 sessions total; mandatory attendance.
Out of Class
- Outside of class, students completed the assigned problem sets and worked on their final project and presentation.
