Phase Plane and Slope Field

or

You can use the Phase Plane and Slope Field apps to qualitatively analyze ordinary differential equations (ODEs).


Phase Plane app: Analyze two-dimensional autonomous ODE systems.	Slope Field app: Analyze single variable ODEs.

These apps capture the functionality of the traditional PPlane and DField apps created by John C. Polking in MATLAB between 1995 and 2003 [1]. While similar in function to the original apps, the Slope Field and Phase Plane apps have been written entirely from scratch in MATLAB App Designer using modern MATLAB coding practices. This makes the new apps easier to maintain, edit, and use.

Latest update: 1.2.0

Updated to offer both dark mode and light mode for plotting region.
Removed unused function in Phase Plane app.
Improved user message when changing the solver field density slider.
Fixed case choices on slider labels to be consistent with each other.
Removed ode15i from offered solvers because it is an implicit solver that requires a different equation structure than these apps are designed to study.

Previous update: 1.1.0

Updated field arrows with solid arrowheads. These fix the distortion that was visible when the axes were scaled differently.
Default field color is darker. Also, an option has been added to set a custom field color (in the Appearance menu).
The solver now allows complex solutions but only plots the real part.
Added options for the numerical estimation of separatrices.
Added options to export the field to a figure.

Acknowledgements

Thank you to Roy Goodman at NJIT for his support of this project and many insightful suggestions.

Setup

MATLAB®

Ensure that you have MATLAB R2021a or newer installed.
Download and unzip the entire repository.
Double-click each app installer (SlopeField.mlappinstall and PhasePlane.mlappinstall) and follow the installation instructions.
Access the apps from the APPS tab in the MATLAB toolstrip. Use the dropdown to expand the menu.

MATLAB Online™

Double-click each app installer (SlopeField.mlappinstall and PhasePlane.mlappinstall) and follow the installation instructions.
Access the apps from the APPS tab in the MATLAB toolstrip. Use the dropdown to expand the menu.

Getting Started

Tutorial Videos

Learn the basics of the Phase Plane and Slope Field apps in these 3-minute tutorial videos.

Phase.Plane.Tutorial.mp4

Slope.Field.Tutorial.mp4

Quick Start Guides

As an alternative to the videos, you can use these PDF quick start guides to get up and running quickly.

MathWorks Products

Requires MATLAB release R2021a or newer

License

The license for the Phase Plane and Slope Field apps is available in the LICENSE.md file in this GitHub repository.

Educator Resources

Find an issue or need help? Email the MathWorks teaching resources team: [email protected]

If you want to contribute directly to this project, you can find information about how to do so in the CONTRIBUTING.md page on GitHub.

Full Capabilities: Phase Plane App

The details of the Phase Plane app are documented here for reference. The Phase Plane app has four main areas you can interact with:

Each of these areas is described below.

Differential Equation System Panel

Functionality	Action
Define an ODE system	Type the two dependent variable names in the first two fields and the ODE expressions in terms of the dependent variables and any parameters you defined.
Define a parameter	Type the name of the parameter in the first field. Type the value in the second field. You can use a valid MATLAB expression, such as log(2), but you cannot use other parameters or variables.
Update the phase plane field with the edited ODE system	Click Update
Clear the differential equation system and parameter fields	Click Clear
Use the default ODE system	Click Default

Phase Plane Panel

Functionality	Action
Generate a solution	Click the phase plane
Delete a solution	Right-click a solution curve
Highlight a solution	Left-click a solution curve
Remove highlighting	Left- or right-click a highlighted curve
Solve from a numerically defined initial condition	Set the initial values in the x0 and y0 edit fields. Then press Solve from (x0,y0).
Clear solution curves	Click Clear solutions
Show the analysis for an equilibrium point	Click an equilibrium point (equilibria are generated from the Analysis menu)

Appearance Tab

Functionality	Action
Change the phase plane horizontal axis limits	Type values of xmin and xmax
Change the phase plane vertical axis limits	Type values of ymin and ymax
Change the time series horizontal axis limits	Type values of tmin and tmax
Speed up or slow down the animation	Move the Animation speed slider
Toggle the time series plots	Click the Time series checkbox
Increase or decrease the widths of solution curves	Use the spinner or type a new value for Line width
Increase or decrease the number of field arrows	Use the spinner or type a new value for Field density
Increase or decrease the size of the field arrows	Use the spinner or type a new value for Field scale

Solution Tab

Functionality	Action
Change the solver time span. The ODE solver will start at t=0 and solve both forward and backward in time based on the defined values.	Edit the Forward solution tmax and Backward solution tmin fields.
Allow the solver to continue beyond the axis limits	Uncheck Terminate solutions at axis limits
Change how many solutions are generated when the Solve from region functionality is used	Adjust the Solve from region density slider

Solver Tab

Functionality	Action
Change the variable step ODE solver	Select a solver from the Solver dropdown
Change the ODE solver relative tolerance	Enter a new value in the Relative tolerance field
Change the ODE solver absolute tolerance	Enter a new value in the Absolute tolerance field
The ODE solver automatically terminates if it runs for too long (in real time). You can adjust how long the solver will run.	Type a new value for Max solver wall clock (s)
Use a fixed step solver	Press the Fixed step button
Use a different fixed step solver. Note that implicit methods use a Newton iteration at each step and, as a result, solve slowly.	Select a solver from Solver dropdown
Use a different step in the numerical integration	Type a new value for Step size

Analysis Menu

Functionality	Action
Numerically solve for an equilibrium of the system. Equilibria occur where x'(t)=0 and y'(t)=0. This function uses the Newton-Raphson method with a finite-difference Jacobian to solve for zeros of the differential equation system.	Select Find nearby equilibrium. Then, click the phase plane near the suspected equilibrium point.
Find equilibria in the phase plane. This method scans the plane using a grid of initial guesses and records all equilibria found. The Newton-Raphson method is used to solve for zeros of the differential equation system.	Select Scan for equilibria
Clear the equilibria	Select Clear equilibria
Solve for saddle separatrices. Saddle separatrices are numerically estimated by generating a solution an increment away from each saddle equilibrium in the directions of the eigenvectors. Before solving for saddle separatrices, you should scan for equilibria.	Select Solve for saddle separatrices
Clear the separatrices	Select Clear separatrices
Show nullclines or hide nullclines. Nullclines are curves along which x'(t) = 0 or y'(t) = 0. Nullclines with x'(t) = 0 are blue and those with y'(t) = 0 red. Intersections of the nullclines are equilibria since x'(t) and y'(t) are both zero.	Select Show nullclines
Automatically generate isoclines. Isoclines are curves along which the phase plane field directions are constant: y'(t)/x'(t) = m.	Select Auto-generate isoclines. Then, enter an integer for how many isoclines you want to generate.
Draw an isocline curve through a point.	Select Draw isocline through a point. Then, click the phase plane.
Draw isoclines with specified values.	Select Draw several isoclines. Then, type a list of slope values. For example: -1 3 5
Clear the isoclines	Select Clear isoclines

Solve and Draw Menus

Functionality	Action
Generate several solutions starting within a region	Select Solve > Solve from region. Then, click once on the phase plane to start drawing, draw your region, and click again to stop drawing.
Draw a solution on the phase plane	Select Draw > Draw solution. Then, click once on the phase plane to start drawing, draw your solution, and click again to stop drawing.
Draw a solution on the phase plane and compare it to the numerical solution	Select Draw > Draw and compare solution. Then, click once on the phase plane to start drawing, draw your solution, and click again to stop drawing.

Appearance Menu

Functionality	Action
Toggle solution animations	Select Animate solution
Toggle initial value labels	Select Point labels
Toggle the location of the axes	Select Axis through origin
Toggle dark mode	Select Dark mode
Toggle light mode	Select Light mode
Toggle between showing the field arrows with magnitude and orientation and orientation only	Select Field orientation only

Note: At most, one of the Dark and Light modes can be selected simultaneously. Selecting one will clear the other. The original theme colors are used if both Dark mode and Light mode are unselected.

Library and Custom Library Menus

Functionality	Action
Set the differential equation to a standard system	Select one of the systems from the Library menu
Add the current system to the Custom library tab	Select Custom library > Add current system
Save the current custom library to a MAT file	Select Custom library > Save
Load a custom library MAT file (note: the custom library should be one created by the Phase Plane app)	Select Custom library > Load
Clear the current custom library	Select Custom library > Clear

Export Graphics Menu

Functionality	Action
Export the phase plane portrait and time series to an image file. This method includes the equations in the exported image.	Select Export to PNG
Export the phase plane portrait and time series to a scalable vector graphics file (this format is useful for editing or high resolution website display). This method includes the equations in the exported image.	Select Export to SVG
Export the phase plane portrait and time series to a PDF. This method includes the equations in the exported image.	Select Export to PDF
Export the phase plane portrait to an image file	Select Export portrait only to PNG
Export the phase plane portrait to a scalable vector graphics file	Select Export portrait only to SVG
Export the phase plane portrait to a PDF	Select Export portrait only to PDF

Full Capabilities: Slope Field App

The details of the Slope Field app are documented here for reference. The Slope Field app has four main areas you can interact with:

Each of these areas is described below.

Differential Equation Panel

Functionality	Action
Define an ODE	Type the dependent variable name and the ODE expression in terms of the dependent variable and the independent variable t.
Define a parameter	Type the name of the parameter in the first field. Type the value in the second field. You can use a valid MATLAB expression, such as log(2), but you cannot use other parameters or variables.
Update the slope field with a new equation	Click Update
Clear the differential equation and parameters	Click Clear
Use the default ODE	Click Default