Phase plane analysis of 2D ODE systems
Required Reading
- Phase Plane Analysis. Chapter 15 of Cellular Biophysics and Modeling: A primer on the systems biology of excitable cells.
Required Watching
The following three videos will walk you through phase plane analysis of a 2d nonlinear system. Here is a link to phase plane graphing software used in this video: pplane (Phase Plane) Java Software Download http://math.rice.edu/~dfield/dfpp.html
Fall 2025 Slides
Optional Reading and Resources
- Another example of determining vector fields with the help of nullclines
- Two dimensional linear systems by Steve Strogatz
- Two dimensional nonlinear systems and fixed points by Steve Strogatz
- Here is a link to phase plane graphing software: pplane (Phase Plane) Java Software Download http://math.rice.edu/~dfield/dfpp.html
Alternatively, if you have any trouble running Java on your computer, you may use any other web based phase plane tool you can find.
- Linearization (Example of finding and classifying fixed points in an ODE system).
Many of the problems discussed in class and on homework can be investigated by selecting the ODE system dx/dt = (e+a*x+b*y)*x and dy/dt = (f+c*x+d*y)*y and choosing appropriate values for the six constants a, b, … , e.
For more information on phase plane methods and qualitative solutions see Chapter 5 of Edelstein-Keshet & Segel.
For more information on phase plane analysis see Sections 7.1-7.5 by Edelstein-Keshet & Segel.
Phase plane analysis in five easy steps is a very brief overview of how to approach phase plane analysis of 2D autonomous nonlinear ODEs. Follow this procedure when doing Problem Set 5.