## Interactive Software Demos for Learning Differential Equations

Since the time of Isaac Newton, differential equations have been useful for modeling of a wide variety of dynamical physical systems. For example, the motion of a mass acted upon by a force can be modeled by a second-order differential equation using Newton's second law.

The objective of this project is to visualize solutions to basic differential equations used in various undergraduate courses on applied mathematics, physics, and engineering. The visualization is provided by real-time web-enabled software technologies. It is expected that the interactive software demos smooth out the student's learning curve and help instructors in lecturing the undergraduate courses.

#### Nonlinear Damped Pendulum

Learning Goals: Classify critical points of the dynamical system and local stability of critical points. Match solutions of differential equations and trajectories on a phase plane of the system. Understand differences between finite and infinite trajectories on a phase plane. Identify the separatrix curves on the phase plane. Control behaviour of the system by changing initial values of the system. Understand the role of damping for motion of the pendulum.

#### Predator-Prey Population System

Learning Goals: Understand hypotheses and constraints of mathematical modeling. Classify critical points of the dynamical system. Understand differences between local and global stability of critical points. Match population cycles of the predator-prey system and periodic solutions of the dynamical system.

#### Van-der-Pol Nonlinear Oscillator

Learning Goals: Identify limit cycles on a phase plane of the system. Control the flow of trajectories that draws the global phase portrait of the system. Understand global stability of critical points and limit cycles. Utilize the Hopf bifurcation of the dynamical system.

Software Requirements: The software demos work with Microsoft Internet Explorer. They are read-only with other browsers such as Netscape. Before starting the demos, maximize the browser window and close other applications on your computer.

This project was prepared by the team of Prof. D. Pelinovsky:
• Guanrong Lou
• Le Bing
• X.-.F. Zhang
© McMaster University, Department of Mathematics, 2001-2002