Lecture 11
Phase plane analysis of 2d ODE systems
Public resources and restricted materials are identified below. Access information.
Core Readings
- Phase Plane Analysis. Chapter 15 of Cellular Biophysics and Modeling: A primer on the systems biology of excitable cells. [DOI]
Slides & Notes
Recordings & Tutorials
- The following three videos will walk you through phase plane analysis of a 2d nonlinear system. Here is a link to the phase plane graphing software used in this video: pplane. [Software]
Tutorial: Phase plane analysis tutorial
- Lecture recording: phase-plane analysis of two-dimensional nonlinear systems
- Van der Pol phase-plane example — part 1
- Van der Pol phase-plane example — part 2
Models & Demonstrations
Practice
Further Reading & 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. [Software]
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).
- MIT Mathlet on Vector Fields. [Mathlet]
Many phase-plane exercises 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. Use this procedure to work through phase-plane exercises.
Helpful videos for phase-plane flow
These videos show how to determine direction of flow in the phase plane when one variable is fast, such as an activation variable, and the other is slow, such as a recovery variable.
This is a useful skill for interpreting two-dimensional models of excitable membranes, including Morris-Lecar-style phase-plane diagrams.
R(t) = Romeo, J(t) = Juliet - dynamic models of love affairs
Two-dimensional nonlinear dynamics of the Morris-Lecar and Fitzhugh-Nagumo models provide a deep understanding of membrane potential excitability and oscillations, much as one-dimensional models give insight into membrane potential bistability.
Two-dimensional linear systems are useful because they allow classification of steady states, or fixed points, in nonlinear systems: nodes, saddles, spirals, and related cases. This is the significance of the 2x2 matrix of partial derivatives called the Jacobian. The classification of steady states by linear stability analysis is summarized by the trace-determinant plane.
A lighthearted review of two-dimensional linear systems is provided by the following paper and video on dynamical models of love affairs. Romeo’s love for Juliet is given by R(t), and Juliet’s love for Romeo is given by J(t). The pair (R, J) is the state space for a two-dimensional linear system.
Restricted reading copies in the PDF companion — private repository access required.