Abstract
The article aims to document PhaPl that’s a teaching software to plot and research phase portraits of autonomous systems of 2 differential equations on a plane. Interactive computer teaching materials allow to demonstrate tasks describing large number of states of investigated systems clearly and involving students into solving. Plotting and research of phase portraits of autonomous system of 2 differential equations is an important task in “Differential equations” course and other courses that use dynamic systems. The software allows to visualize phase portraits and to perform analysis easily. Plotting of phase portraits needs a lot of routine computations. The software allows teacher to focus on analytical research of autonomous systems of 2 differential equations. The software supports linear and nonlinear autonomous systems of 2 differential equations. The software differs is very different compared with previously known programs: it has very easy graphical user interface and it gives clarity because it demonstrates all steps of solution. To get the full solution, it is enough to just enter a system to research. Initial conditions to plot phase trajectories are chosen automatically. Graphical representation of the phase plane is interactive and allows user to draw additional trajectories with specified initial conditions by mouse hovering over the phase plane. The software is based on popular Free Software (Maxima, Qt4, LaTeX) and it is Free Software itself, thus it is accessible to wider audience, including online students. The software is portable and works on Windows and Linux operating systems. The article describes advantages, disadvantages and peculiar properties of the software, and some aspects of teaching experience. The software is deployed in Moscow State University of Economics, Statistics, and Informatics (MESI), Lomonosov Moscow State University (MSU) since 2013, and in Plekhanov Russian University of Economics since 2016. Further development will be focused on reduction of binary size of the software, portability improvements, and number of analytical results showed.
Highlights
The article aims to document PhaPl that’s a teaching software to plot and research phase portraits of autonomous systems of 2 differential equations on a plane
The software allows teacher to focus on analytical research of autonomous systems of 2 differential equations
The software differs is very different compared with previously known programs: it has very easy graphical user interface and it gives clarity because it demonstrates all steps of solution
Summary
Основные этапы построения фазового портрета автономной системы дифференциальных уравнений на плоскости подробно описаны, например, в [13, §6.3]. У матрицы каждой из линейных систем необходимо найти собственные значения, собственные векторы, если они существуют, с их помощью построить фазовый портрет в окрестности особой точки 0. В других этапах есть небольшие отличия: фазовые портреты строятся приближённо при помощи метода Эйлера [14]. Как правило, в окрестности каждой точки строят фазовый портрет линейной системы, а затем сводят их вместе, Open education V. Программное обеспечение всегда строит сразу фазовый портрет исходной системы, но в выделяемой окрестности особой точки. Фазовые траектории строятся численно при помощи метода ломаных Эйлера. Построение фазовой траектории прекращается, когда она достигает границы области. Однако, границы области не достигают (например, циклы), поэтому построение любой траектории прекращается после определённого числа шагов, подобранного вручную таким образом, чтобы для всех задач учебного курса представлялась наглядная картинка
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