A Tour of Second-Order Ordinary Differential Equations

This Demonstration is a tour of autonomous second-order ordinary differential equations (ODEs). The systems chosen represent most of the possible important qualitative behaviors. The general form of a second-order ODE is:

.

Some of the systems are most naturally described in polar coordinates:

.

The polar coordinates are then transformed to rectangular coordinates.

Phase portraits can be selected from a number of systems. Stable fixed points are indicated by solid disks, while unstable points are shown as open circles. Each system has a parameter that you can control using its slider bar. Drag the locator to highlight a single trajectory starting from any initial state. The dynamics of the selected trajectory can then be visualized using the slider bar for . To focus on a single trajectory only, set the density of the stream points to "none", select an initial state, and move the slider for .