Summary

AutoPortrait is a free Mathematica package and user-friendly interface to automatically generate interactive phase portrait diagrams for multidimensional dynamical systems. The application only needs the user to type the differential equations defining the system and the program outputs a complete phase portrait for each pair of dependent variables. The portraits include a vector field, nullclines, and critical points with their stability. The user can interact with the portraits, zooming and panning the domain, changing parameters, and including trajectories that are visualized in the phase and time domains. The portraits can be saved as computable notebooks that retain all the interactive functionality. The method can aid in research reproducibility and improve learning in the classroom.

Mathematica Package

The method is also available as a Mathematica package that can be imported into a notebook or script. The main function in the package is GeneratePortrait, which takes two arguments. The first one is a list with the differential equations in the system. The second argument is a list with the dependent variables. The system of equations can contain parameters, which are automatically detected by the method. The function outputs the same interactive portraits that can be obtained with the graphical user interface. In addition, optional arguments can be defined to specify ranges, trajectories, and the displayed domain.

Generating phase portraits with the package is very simple. The example below shows the script necessary to produce the interactive phase portrait shown next. Notice that the equations do not need to include the time variable and the parameters (a and b in the example) are automatically detected by the method.