Phase Portrait Plotter

Systems evolving with time are undoubtedly ubiquitous in nature. From the evolution of the universe to the periodic flashing of a firefly, these topics have long been of academic interest. Remarkably, it is a recent realization (around the 1900s) that all such systems evolving with time have certain underlying commonalities. In technical terms, these systems are called dynamical systems. In essence, the study of dynamical systems is closely related to understanding the differential equations that govern the particular situation.

 

A fundamental idea to understand these dynamical systems is the use of geometrical concepts and qualitative reasoning based on theoretical knowledge. A central concept of this analysis is the phase portrait that is generated from the knowledge of the differential equations of the system. The phase portrait gives a qualitative description of all possible behavior that can be observed in the system. For a system that involves only two variables evolving with time, the phase portrait is simply a two dimensional image. Several real-life phenomena can be understood as the evolution of two variables. The corresponding phase portrait of these phenomena forms a powerful tool to fully understand them.

 

The current project aims to create a phase portrait for any two dimensional system given the governing differential equations. I developed a computer software named the ‘Phase Portrait Plotter’ on MATLAB to carry out the task and provide further information about the system. The software has an interactive graphical user interface (GUI) and has been developed with an intention to allow any interested user with a bare minimum knowledge of dynamical systems to generate phase portraits with ease. The software has a great scope of being used for pedagogical applications and self-study purposes.

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While the course and the project concluded successfully in November 2020, that has not marked the end of the software's lifespan. Since then, application has become quite popular in the MATLAB community through MATLAB File Exchange. As of 4th August 2021, the application has been downloaded over 1000 times. At its peak, it was downloaded 150+ times per month and was the 5th most popular file in the apps category on MATLAB File Exchange. This software was also used by Dr. Julien Lagarde (University of Montpellier, France) in an introductory course to dynamical systems.