Sensitivity of Parameters in Chaotic Systems

Namyong Lee, Minnesota State University, Mankato, Mathematics
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Summary

In this computer lab activity, students explore and discuss the nature of the sensitivity of parameters in a chaotic system. As a result, they understand the challenges of numerical simulation and/or parameters estimation in chaotic systems.

Learning Goals

Through this computer lab activity, students are expected to understand some nature of chaotic systems, especially the sensitivity of the parameters. Here, Matlab can be utilized as a virtual lab, without setting up a pendulum or biochemical system, one can explore what is happening from a mathematical model and its computer simulation. More critical thinking questions/ discussions can be added to this lab, such as how to define an error and how to quantify the uncertainty of such a system. If an instructor wants, one can demand a written group report or group oral presentation after the lab. One may also add extra exploring questions, such as how to identify the chaotic system and how to numerically simulate such a system.

Context for Use

This computer lab activity can be used in any college/ graduate level. The beginning undergraduate students may explore the nature of the sensitivity of parameters in chaotic systems through the computer simulation. More advanced level students may try to parameter estimations, parameter identifiability or even attempt to develop the concept of uncertainty quantification.

It is a computer lab activity. Depending on time availability, the instructor may provide pre-written Matlab code that students can explore and discuss. It can be extended by adding a series of activities to derive a parameter estimation or uncertainty quantification of the systems.

For beginning students, students only need to understand the pre-written Matlab code and how to modify a certain system parameter to explore the chaotic system through simulation. To more experienced students, they can modify the code to a nonlinear system that they might have an interest. For advanced students, they may perform an inverse problem (that is, parameter estimation).

It is best to use the subject specific examples. For instance, in physics, a pendulum is a popular choice.

Description and Teaching Materials

A sample {"Sensitivity of Parameters.pdf") of one hour computer lab that I used for both Differential Equations, and Mathematical Modeling class is attached.
Sensitivity of Parameters (Acrobat (PDF) 27kB Aug16 19)

Teaching Notes and Tips

This computer lab is intended to use as an active learning style class. (Each group will present at the end of the class their experiments.) One can easily modify the lab to a more lengthier written group report form.

Assessment

The proposed computer lab activity is intended for a part of active learning. Through the experiments, we will learn from each other. Active participation of the class will get the points. One can modify or add a few assessment questions.

References and Resources

[SZ] Sorensen, Jody & Zobitz, John. (2012). Chaos in Context: Behind the Scenes of the Lorenz Model. Math Horizons. 20. 25-28. 10.4169/mathhorizons.20.1.25.