Worked Problem Assignment 2#

Due 22 Sep 23

We have spent much of our time on a dynamical systems view of ODEs. In this assignment, you will choose an ODE and work through a dynamical analysis of the system.

Grading Rubric

Remember that a worked problem means the scale of a detailed (e.g., parts a-h) homework problem.

Places to find candidate problems:#

  • Classical mechanics textbooks; look at the problems in the Lagrangian chapters (you only need to the ODEs)

  • Differential equations textbooks; look at ODEs that represent real systems

  • Wikipedia has lists of named differential equations and problems

Starting your Analysis#

You will need to orient us to your problem as there are many ODEs available. You should limit yourself to systems with two degrees of freedom (either 3d with a constraint or 2d with no constraint).

  1. Write a paragraph or two describing the system you are investigating. Make sure to explain the parameters and variables. You do not need to solve for this ODE (e.g., with Euler-Lagrange); you can just start from one.

  2. Explain what you are going to do an why. If you can point us to where things are being done, that’s helpful.

Investigating the System#

After you set things up, please conduct an investigation using the tools we have developed in class. The order in which it proceeds is up to you, but make sure the flow is logical and clear. We want to understand how your investigations are going to help you be successful with your projects.

Assignment#

  1. Choose a mechanical (or dynamical) system with two degrees of freedom (e.g., \(x\) and \(\dot{x}\)). This could be a system you have already worked with, or something new.

  2. Explain the system including the parameters and variables.

  3. Plot a phase space diagram (phase portrait) of the system. Make sure to label the axes and any fixed points. What solution families exist?

  4. Find the fixed points for the system and describe their stability. What approach do you want to use?

  5. Numerical integrate the system to find trajectories and plot them on the phase portrait.

Looking ahead (earn extra credit)#

If you are looking into these systems and finding them interesting here’s two things you can do to earn extra credit that we will discuss later in the course.

  1. Work with a 3D system (note this can be quite challenging depending on the system)

  2. Explore an ODE with a tunable parameter and show how solutions change when that parameter is varied.

Submitting your work#

You will upload your work to D2L. It should be a single PDF of your work. If you also write code, please submit the associated files as well. If you do all of this in a Jupyter notebook, please export the notebook as a PDF and submit the notebook and PDF file. If you have any questions, please ask. We want you to enjoy working on these problems.